PDF (printable) Manual

PDF (printable) Manual

Fractal Zplot Application Help

© 2003 - 2010 ... Mystic Fractal

I Fractal Zplot Application Help

Table of Contents

Foreword

1 Main Index

3 Size ................................................................................................................................... 2

4 Move ................................................................................................................................... 2

1

0

9 Close ................................................................................................................................... 3

2 Fractal ViZion Remote

5

© 2003 - 2010 ... Mystic Fractal

Contents II

3 File menu

© 2003 - 2010 ... Mystic Fractal

15

II

III Fractal Zplot Application Help

4 Edit menu

19

Neg Button

Map Button

H/R Button

.......................................................................................................................................................... 31

.......................................................................................................................................................... 31

.......................................................................................................................................................... 31

Copy Button .......................................................................................................................................................... 32

SRG Button

SRB Button

.......................................................................................................................................................... 32

.......................................................................................................................................................... 32

Okay Button .......................................................................................................................................................... 32

Reset Button .......................................................................................................................................................... 32

Red Slider .......................................................................................................................................................... 32

Green Slider .......................................................................................................................................................... 33

Blue Slider .......................................................................................................................................................... 33

Scram ble .......................................................................................................................................................... 33

12 Size ................................................................................................................................... 33

5 Fine-Tuning menu

35

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Contents IV

Escape

Level

......................................................................................................................................................... 38

......................................................................................................................................................... 39

Set Only ......................................................................................................................................................... 40

Use Palette ......................................................................................................................................................... 40

Log High

Log Low

......................................................................................................................................................... 41

......................................................................................................................................................... 41

Ray Trace ......................................................................................................................................................... 41

Palette m enu .......................................................................................................................................................... 41

Anti-Alias

Biom orph

.......................................................................................................................................................... 43

.......................................................................................................................................................... 43

Biom orph Off .......................................................................................................................................................... 44

New ton Set

New ton Off

.......................................................................................................................................................... 44

.......................................................................................................................................................... 44

Renorm alize .......................................................................................................................................................... 44

Convergence .......................................................................................................................................................... 45

Level Curve .......................................................................................................................................................... 46

Spin

Torus

Torus Off

Map m enu

Z-Real

Z-Imag

.......................................................................................................................................................... 48

.......................................................................................................................................................... 48

.......................................................................................................................................................... 49

.......................................................................................................................................................... 49

......................................................................................................................................................... 49

......................................................................................................................................................... 49

© 2003 - 2010 ... Mystic Fractal

IV

V Fractal Zplot Application Help

Abs(Z) ......................................................................................................................................................... 52

Surface Filter .......................................................................................................................................................... 53

Add Noise

Factors

.......................................................................................................................................................... 53

.......................................................................................................................................................... 53

15 Julia ................................................................................................................................... 58

16 Invert ................................................................................................................................... 59

19 Filter ................................................................................................................................... 60

21 Pilot ................................................................................................................................... 61

6 View menu

61

toolbar .......................................................................................................................................................... 62

status bar .......................................................................................................................................................... 63

6 Zoom ................................................................................................................................... 65

8 Reset ................................................................................................................................... 66

7 Window menu

67

2 Tile ................................................................................................................................... 67

© 2003 - 2010 ... Mystic Fractal

Contents VI

8 Help menu

68

2 Index ................................................................................................................................... 69

Chronology .......................................................................................................................................................... 97

Index 100

© 2003 - 2010 ... Mystic Fractal

VI

1

1 Fractal Zplot Application Help

Main Index

Fractal ViZion User Manual

Getting Started

Fractal V iZion Remote

Channel Guide

Command menus

File menu

Edit menu

Fine Tuning

View menu

Window menu

Help menu

1.1

Title Bar

Title Bar

The title bar is located along the top of a window. It contains the name of the application and drawing.

To move the window, drag the title bar. Note: You can also move dialog boxes by dragging their title bars.

A title bar may contain the following elements:

Application Control-menu button

Drawing Control-menu button

Maximize button

Minimize button

Name of the application

Name of the drawing

Restore button

© 2003 - 2010 ... Mystic Fractal

Main Index 2

1.2

Scroll bars

Scroll bars

Displayed at the right and bottom edges of the drawing window. The scroll boxes inside the scroll bars indicate your vertical and horizontal location in the drawing. You can use the mouse to scroll to other parts of the drawing.

1.3

Size

Size command (System menu)

Use this command to display a four-headed arrow so you can size the active window with the arrow keys.

After the pointer changes to the four-headed arrow:

1. Press one of the DIRECTION keys (left, right, up, or down arrow key) to move the pointer to the border you want to move.

2. Press a DIRECTION key to move the border.

3. Press ENTER when the window is the size you want.

Note: This command is unavailable if you maximize the window.

Shortcut

Mouse: Drag the size bars at the corners or edges of the window.

1.4

Move

Move command (Control menu)

Use this command to display a four-headed arrow so you can move the active window or dialog box with the arrow keys.

Note: This command is unavailable if you maximize the window.

Shortcut

Keys: CTRL+F7

1.5

Minimize Command

Minimize command (application Control menu)

Use this command to reduce the Fractal ViZion window to an icon.

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3 Fractal Zplot Application Help

Shortcut

Mouse: Click the minimize icon on the title bar.

Keys: ALT+F9

1.6

Maximize Command

Maximize command (System menu)

Use this command to enlarge the active window to fill the available space.

Shortcut

Mouse: Click the maximize icon on the title bar; or double-click the title bar.

Keys: CTRL+F10 enlarges a drawing window.

1.7

Next Window

Next Window command (drawing Control menu)

Use this command to switch to the next open drawing window. Fractal ViZion determines which window is next according to the order in which you opened the windows.

Shortcut

Keys: CTRL+F6

1.8

Previous Window

Previous Window command (drawing Control menu)

Use this command to switch to the previous open drawing window. Fractal ViZion determines which window is previous according to the order in which you opened the windows.

Shortcut

Keys: SHIFT+CTRL+F6

1.9

Close

Close command (Control menus)

Use this command to close the active window or dialog box.

Double-clicking a Control-menu box is the same as choosing the Close command.

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Main Index 4

Shortcuts

Keys: CTRL+F4 closes a drawing window

ALT+F4 closes the application

1.10

Restore

Restore command (Control menu)

Use this command to return the active window to its size and position before you chose the

Maximize or Minimize command.

1.11

Switch to

Switch to command (application Control menu)

Use this command to display a list of all open applications. Use this "Task List" to switch to or close an application on the list.

Shortcut

Keys: CTRL+ESC

Dialog Box Options

When you choose the Switch To command, you will be presented with a dialog box with the following options:

Task List

Select the application you want to switch to or close.

Switch To

Makes the selected application active.

End Task

Closes the selected application.

Cancel

Closes the Task List box.

Cascade

Arranges open applications so they overlap and you can see each title bar. This option does not

Tile

affect applications reduced to icons.

Arranges open applications into windows that do not overlap. This option does not affect applications reduced to icons.

Arrange Icons

Arranges the icons of all minimized applications across the bottom of the screen.

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5 Fractal Zplot Application Help

2 Fractal ViZion Remote

Fractal ViZion Remote

The remote provides access to many of the most-used commands in Fractal ViZion. Info about each button can be obtained by using the '?' located near the close box in the top right-hand corner.

2.1

Channel Guide

Channel Guide

The nineteen channels accessed via the FV remote:

J1: Random Julia -- basic Julia sets, using one formula and many different rendering options

J2: Random Julia 2 -- includes more advanced composite-type fractals, using two formulas

ES: Random Escher-like fractals -- Julia fractal set using Escher mapping

S/B: Random Bubbles and Stalks -- based on Paul Carlson's bubble and stalk methods

Q1: Random Quaternion/Hypernion -- traditional 3D quaternion and hypercomplex quaternion fractals

Q2: Random Quaternion/Hypernion 2 -- extended search through non-traditional formulas

C1: Random Cubic Mandelbrot fractals

C2: Random Cubic Mandelbrot2 -- relaxed formulas/parameters

CJ: Random Cubic Julia fractals

O1: Random Octonion fractals

O2: Random Octonion2 fractals -- extended dimensional search

QT: Random Quat-trap fractals

HF: Random Julia height fields -- 3D rendering of Julia sets using ray-tracing and continuous potential smoothing

LD: Random Landscapes -- 3D rendering using ray-traced mid-point displacement algorithm

LS: Random Lsystems -- 3D turtle graphics with ray-traced highlights

OR: Random Orbital fractals -- 3D rendering of strange attractors

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Fractal ViZion Remote 6

NT: Random Newton/Halley -- Julia sets using Newton's method and Halley's methods, with various convergent techniques and other rendering options

HY: Random Hypercomplex -- Julia sets using hypercomplex extensions

IF: Random IFS -- IFS slide show using techniques developed by Michael Barnsley

2.2

New button

New button

Use this button to create a new drawing window in Fractal ViZion.

2.3

Undo button

Undo button

Use this command to undo the last action. An image can be continued after an undo, if continue was enabled before the last action, but not after an image is resized. Color-cycling is disabled after using

Undo.

2.4

Size button

Size button

This allows you to set the drawing area for a picture, independent of the Windows screen size. It also shows which size is currently in use. The aspect for the drawing is based on the ratio of X

(horizontal width) to Y (vertical height.) The custom setting allows for any size/aspect that system memory will permit. Videos are limited to the standard 4/3 vga aspect or 1/1. The minimum size for an image is 40X30. Note: if the image is less than 100 width, the aspect must be 4/3 for solidguessing to work properly.

2.5

Color button

Color button

Use the palette editor to modify the palette(s) in use.

2.6

Batch button

Batch button

Here you set parameters for batching and saving random-generated images to disk.

2.7

Fvr button

FVR button

The window opened varies with the fractal type selected, and contains all the major variables that

Fractal ViZion now scales between key frames of an avi stream.

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7 Fractal Zplot Application Help

2.8

Draw button

Draw button

Use this button to draw or redraw the image for the current fractal variables. Clicking inside the draw window with the left-mouse button stops all plotting. Use the Continue button on the toolbar to restart plotting from the current column. When the lsystem option is selected, a custom dialog is opened to enter lsystem draw options. Note: the draw window that a plot is initialized in (by clicking

on an Okay or Apply button or Draw) must have the focus to begin drawing. With a lot of parameter or formula windows open for different draw windows, it is easy to mistake one window for another. So if the plot doesn't start immediately in the window that has the focus, check to see that that window is actually the one you wanted to start a new plot in. If it isn't, click on the title bar for the window that the plot is activated in. The plot should then commence.

2.9

Abort button

Abort button

Use this command to stop drawing. Clicking inside a window's drawing area or close box (or the program close box) will also stop the drawing. Note: once a plot has started Fractal ViZion continues to draw the image for that window regardless of which drawing window has the input focus, until done or aborted. You can open and close other drawing windows without affecting the current drawing, but only one drawing is active at any time.

2.10

Scan button

Scan button

This generates a Julia set or quaternion Julia set from a formula's Mandelbrot 'P' space. Random points in a formula's current Mandelbrot space are scanned for an interesting Julia set. Rendering options are maintained in the current fractal. Equivalent to the 'F' hot keys. Not used with lsystem, landscape or orbital fractals.

2.11

Rend button

Rend button

For quaternion, lsystems, landscape, orbital and 3D height fields, the current coloring filter and lighting variables are applied. This allows you to see what the surface texture looks like before the fractal is finished drawing. Note: to randomize the coloring filter, select the Random Render button.

2.12

View button

View button

Displays the entire plot, expanding or shrinking the image to fit in a maximized window without title bar, scroll bars or menu bar. At all other times, part of the picture is hidden by the inclusion of the title bar, toolbar, scroll bars and menu bar.

To exit full-screen mode, press any key or click the left-

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Fractal ViZion Remote 8

mouse button.

2.13

Help button

Help button

Use this button to open the help index for Fractal ViZion.

2.14

Pilot button

Opens the Pilot window to adjust key parameters, rotate, zoom and redraw the figure interactively.

The current image is reduced to one quarter normal for faster redraw. Each click on a Pilot button increments or decrements a parameter. The Speed slider controls the rate at which the buttons operate (default is 10.)

Press the space bar or Click on Ok to open a new window and draw the altered image full-size.

Press Esc or click on Cancel to exit this mode without opening a new window. Note: when using this option while an AVI stream is open, a new window isn't opened, but the altered figure is drawn in the current draw window, the changed parameters replacing the previous ones.

2.15

Formula button

Formula button

Use this button to change

formulas or type

.

2.16

Light button

Light button

Edit

lightpoint and viewpoint variables

2.17

Channel J1

Random Julia (Channel J1 button)

A random Julia fractal is generated. Many of the rendering options of Fractal ViZion are selected on a random basis, and the Mandelbrot space for one of the 180 built-in formulas is scanned for an interesting Julia set. The palette used is also randomized. Note: In most case the Julia search is a short one, but sometimes the "seek" mode can seem to get stuck when the criteria for an interesting

Julia set fails to match the formula used. In the latter case, either click the left mouse button and restart the search process, or pressing a palette key will sometimes override the current search and restart it (if HSV filtering has been selected.) Tip: some things remain to be done after the Julia set is drawn. The algorithm doesn't check the type of split palette that is used, so a mismatch in the

"Divide by" color option may need correcting, e.g. Divide-by-four may be selected for a divide-byeight palette. Feel free to experiment with all the parameters, reframe the image, change palettes etc.

This routine provides a fast intro to many options in Fractal ViZion that the user may be unfamiliar with: no knowledge of fractal science/math required!

© 2003 - 2010 ... Mystic Fractal

9 Fractal Zplot Application Help

2.18

Channel J2

Random Julia (Channel J2 button)

A random Julia fractal is generated. Many of the rendering options of Fractal ViZion are selected on a random basis, and the Mandelbrot space for one of the 180 built-in formulas is scanned for an interesting Julia set. The palette used is also randomized. Note: In most case the Julia search is a short one, but sometimes the "seek" mode can seem to get stuck when the criteria for an interesting

Julia set fails to match the formula used. In the latter case, either click the left mouse button and restart the search process, or pressing a palette key will sometimes override the current search and restart it (if HSV filtering has been selected.)

This is like the Random Julia command, except that more options are randomized, including spin and switch, and the Formula Type can be composite or Escher, so the search/draw time may be somewhat longer, and the results not as certain. But the images can be quite weird!

Tip: some things remain to be done after the Julia set is drawn. The algorithm doesn't check the type of split palette that is used, so a mismatch in the "Divide by" color option may need correcting, e.g.

Divide-by-four may be selected for a divide-by-eight palette. Feel free to experiment with all the parameters, reframe the image, change palettes etc. This routine provides a fast intro to many options in Fractal ViZion that the user may be unfamiliar with: no knowledge of fractal science/math required!

2.19

Channel ES

Random Escher (Channel QT button)

A random Julia fractal is generated using Escher-like tilings (Type 10 in the Formula window.) The

Rise variable in the Parameters window sets the degree of tiling (1-235). Note: Sometimes it helps to reduce iterations if the tilings do not appear to be distributed throughout the circular plane. 20-25 iterations is sufficient for many Escher plots.

2.20

Channel SB

Random Stalks and Bubbles (Channel S/B button)

A random Julia fractal is generated using one of Paul Carlson's orbit traps or bubble method.

2.21

Channel Q1

Random Quaternion (Channel Q1 button)

A random quaternion/ hypernion fractal is generated. Like Random Julia, a set of formulas appropriate for quaternions is scanned to find an interesting Julia set, and then the parameters are adjusted to produce a quaternion or hypernion image. The ranges are reset, Hj is set to 2.0, and the lighting is set for optimum viewing.

Note: for some images an hj value of 2.0 may result in a partially clipped image. Sometimes it helps to increase this value to 2.5 or 3.0, but too high a value may interfere with Solid guessing.

© 2003 - 2010 ... Mystic Fractal

Fractal ViZion Remote 10

2.22

Channel Q2

Random Quaternion2 (Channel Q2 button)

A random quaternion/hypernion fractal is generated. Like Random Julia, a set of formulas appropriate for quaternions is scanned to find an interesting Julia set, and then the parameters are adjusted to produce a quaternion or hypernion image. The ranges are reset, Hj is set to 2.0, and the lighting is set for optimum viewing.

This option uses an extended set of formulas, not all of which may produce useable images all the time. The images can be quite different from the traditional quaternion.

Note: for some images an hj value of 2.0 may result in a partially clipped image. Sometimes it helps to increase this value to 2.5 or 3.0, but too high a value may interfere with Solid guessing.

2.23

Channel C1

Random Cubic Mandelbrot (Channel C1 button)

A random cubic Mandelbrot fractal is generated. Like the inverse of Random Julia, the essential cubic parameters are randomly adjusted to point into the four-dimensional formula G0, and then the

Mandelbrot set is scanned to find an interesting area to zoom into. The ranges are reset, and the lighting is set for optimum viewing.

2.24

Channel C2

Random Cubic Mandelbrot2 (Channel C2 button)

A random cubic Mandelbrot fractal is generated. Like the inverse of Random Julia, the essential cubic parameters are randomly adjusted to point into the four-dimensional formula G0, and then the

Mandelbrot set is scanned to find an interesting area to zoom into. The ranges are reset, and the lighting is set for optimum viewing.

This option uses the cubic formulas G0 and G1, with relaxed parameters to create cubic-

Mandelbrot like fractals that may extend to six or more dimensions.

2.25

Channel CJ

Random Cubic Julia (Channel CJ button)

A random cubic Julia fractal (the Julia analog of a cubic Mandelbrot fractal) is generated. The essential cubic parameters are randomly adjusted to point into a four-dimensional formula. Like

Random Julia, a set of formulas (G0 and G1) appropriate for cubic Julias is scanned to find an interesting Julia set, and then the parameters are adjusted to produce the cubic image. The ranges are reset, Hj is set to 2.0, and the lighting is set for optimum viewing. Note: This is a quasi-Julia

© 2003 - 2010 ... Mystic Fractal

11 Fractal Zplot Application Help

approximation that doesn't follow traditional cubic Mandelbrot theory. The "quaternions" produced by this method do exhibit characteristics of cubic Mandelbrots, but here I am more interested in esthetics than mathematical conformity.

2.26

Channel O1

Random Octonion (Channel O1 button)

A random octonion Julia fractal is generated. The essential octonion parameters are randomly adjusted to point into an eight-dimensional formula H0-H9, and then the octonion Mandelbrot set is scanned to find an interesting area to zoom into. The ranges are reset, and the lighting is set for optimum viewing (if lighting is checked in the Random Batch window.)

2.27

Channel O2

Random Octonion2 (Channel O2 button)

A random octonion Julia fractal is generated. The essential octonion parameters are randomly adjusted to point into an eight-dimensional formula H0-H9, and then the octonion Mandelbrot set is scanned to find an interesting area to zoom into. The ranges are reset, and the lighting is set for optimum viewing.

This option uses the octonion formulas H0-H9, with random dimensional switching (one of OE-OK for Oi) to create octonion fractals that may extend to eight dimensions.

2.28

Channel QT

Random Quat-Trap (Channel QT button)

A random quat-trap fractal is generated using one of Paul Carlson's orbit trap methods and a quaternion image. A separate image is created for both quaternion and orbit-trap formulas, in figures 1 and 2, then the figures are merged and drawn as one image. See Orbit-Trap options for further details on quat-trap pictures.

2.29

Channel HF

Random 3D Julia Field (Channel HF button)

A 3D height field fractal is generated for a random Julia set. Like Random Julia, a set of formulas appropriate for 3D height fields is scanned to find an interesting Julia set, and then the parameters are adjusted to produce a height field image. The ranges are reset and the lighting is set for optimum viewing.

2.30

Channel LD

Random Landscape (Channel LD button)

A random 3D landscape (midpoint displacement fractal) is generated.

© 2003 - 2010 ... Mystic Fractal

2.31

Channel LS

Random Lystem (Channel LS button)

A random lsystem fractal is generated. An .ls file is selected at random from the startup directory, the palette randomized, and then a random amount of mutation is applied. The lighting is set for optimal viewing, and the figure rotated a random amount on the x y and z-axis.

2.32

Channel OR

Random Orbital (Channel OR button)

A 3D orbital fractal is generated by randomizing parameters for one of nine strange attractors.

Based on Sprott's method of using random affine transformations to find interesting strange attractors.

2.33

Channel NT

Random Newton/Halley (Channel NT button)

A random Julia fractal is generated using Newton's method or one of Halley's methods.

2.34

Channel HY

Random Hypercomplex (Channel HY button)

A random Julia fractal is generated with hypercomplex extensions.

2.35

Channel IF

IFS Slide Show (Channel IF button)

A random slide show of ifs images is displayed in the active view. The show is based on 11 different base forms that are randomized to create millions of variations. Press the 'home' key to see the base forms only. Press the 'end' key to display randomized images. Press the 'r' key to repeat the current form. Press the backspace key to display all forms. Press the escape key or left mouse button to end the show.

2.36

Random Render button

Random Render button

The rendering options for the current fractal are randomized. Does not affect formula or range variables. For quaternion and 3D height fields, a random

coloring filter is applied.

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Fractal ViZion Remote 12

13 Fractal Zplot Application Help

2.37

Save button

Save button

Use this button to save and name the active drawing. Fractal ViZion displays the Save As dialog box so you can name your drawing.

To save a drawing with its existing name and directory, use the File/Save command.

2.38

Load button

Load button

Use this button to open an existing data/image file in a new window. You can open multiple image files at once. Use the Window menu to switch among the multiple open images.

2.39

Bmp radio button

BMP radio button

Use this button to select the BMP format when loading and saving fractals. This is the default

Windows bitmap format, readable by most Windows programs that use image files. This is also the fastest method of loading and saving fractals, but requires more disk space, since no compression is used. Windows keeps track of the last six BMP files saved or loaded (displayed in the Files menu.)

2.40

Png radio button

PNG radio button

Use this button to select the PNG format when loading and saving fractals. This format uses medium compression without loss of image quality.

2.41

Jpg radio button

JPG radio button

Use this button to select the JPEG format when loading and saving fractals. This format uses moderate compression but with some loss of image quality. Preferable for posting to the net, since most browsers can display jpeg files. This option uses the jpeg library written by the Independent

JPEG Group.

2.42

||||| button

||||| button

Through a series of windows, this allows you to name and open an avi animation stream and choose a compression method. After choosing the frame rate (1-60) and using the file requester to name the file, you are given a choice of compression methods. You can also choose no compression for optimum view quality. (All compression methods degrade the original images, some more than

© 2003 - 2010 ... Mystic Fractal

Fractal ViZion Remote 14

others.) The first key frame in the stream is then drawn and written to the file.

Note: after the stream is opened, the size of the fractal that can be drawn is fixed at the size of the frame. No changes can be made to the size until the stream is closed.

2.43

> button

> button

With this option, frames are written to a stream based on the difference between the current key frame and the previous key frame. The first key frame is written when you open a stream. The next key frame is created each time you use this option. In between you can zoom or change Fvr variables as much as necessary. The stream is only written to when this option is used. The last key frame is automatically saved after the 'tween' series is written. The number of frames may range from 1-1500 frames between keys. With a frame number of 1 only the key frames are written. This allows animation to be created that incorporate all scalable variables in Fractal ViZion.

Use the Cancel button to exit this dialog without initializing a new series of frames.

Check the Log Scaling box if you want the frames to be written with logarithmic space between frames, else linear space is used. Useful when zooming, where frames would otherwise be packed together at the end of the frame series.

2.44

[] button

[ ] button

Closes any open avi stream file. You need to do this before viewing the file or creating a new avi file. The stream is also closed when you exit Fractal ViZion.

2.45

V button

V button

Opens an avi file for viewing. You can preview any multimedia file by clicking on its file name. A multimedia box will appear to the right of the file list. Click on okay to open the main view window.

There are buttons to Play a file forwards or Backwards, or forward automatically with Auto rewind/ repeat. Click on Slow to slow down a video. Each click on Slow halves the viewing speed. A click on Stop freezes viewing and restores the view speed to normal playback.

Use the Open button to view a different avi file. Use the Save button to save the file in a different compression format. You must use a different name to save the file than the name that was used to open it. Click on the left-mouse button or any key to abort a save operation.

Note: the view avi requester can be used to preview any multimedia file, including midi files.

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15 Fractal Zplot Application Help

3 File menu

File menu commands

The File menu offers the following commands:

New

Open

Close

Save

Creates a new drawing.

Opens an existing drawing.

Closes an opened drawing.

Saves an opened drawing using the same file name.

Save As

Saves an opened drawing to a specified file name.

Load Palettes [PL]

Load palettes file.

Save Palettes [PL]

Save palettes to file.

Load Palette [MAP] Load a Fractint map file.

Load Text [VZT]

Load text (platform-independent) data file.

Save Text [VZT]

Save data in text (platform-independent) format

Save As [ZP6] -- x64

Save in Fractal ViZion x64 format

Plot To File

Plot large bitmap images directly to png file.

Exit

Exits Fractal ViZion.

3.1

File New command

New command (File menu)

Use this command to create a new drawing window in Fractal ViZion. The image and data for the opening picture are used to create the new window.

You can open an existing data/image file with the

Open command

.

Shortcuts

Keys: CTRL+N

3.2

File Open command

Open command (File menu)

Use this command to open an existing data/image file in a new window. You can open multiple image files at once. Use the Window menu to switch among the multiple open images. See

Window 1, 2, ... command

.

You can create new images with the

New command .

Shortcuts

Toolbar:

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File menu 16

Keys: CTRL+O

3.2.1

File Open dialog box

File Open dialog box

The following options allow you to specify which file to open:

File Name

Type or select the filename you want to open. This box lists files with the extension you select in the List Files of Type box.

List Files of Type

Select the type of file you want to open.

Drives

Select the drive in which Fractal ViZion stores the file that you want to open.

Directories

Select the directory in which Fractal ViZion stores the file that you want to open.

Network...

Choose this button to connect to a network location, assigning it a new drive letter.

3.3

File Close command

Close command (File menu)

Use this command to close the window containing the active image. If you close a window without saving, you lose all changes made since the last time you saved it.

You can also close a drawing by using the Close icon on the drawing window, as shown below:

3.4

File Save command

Save command (File menu)

Use this command to save the active drawing to its current name and directory. When you save a drawing for the first time, Fractal ViZion displays the

Save As dialog box so you can name your

drawing. If you want to change the name and directory of an existing drawing before you save it, choose the

Save As command .

Shortcuts

Toolbar:

Keys: CTRL+S

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17 Fractal Zplot Application Help

3.5

File Save As command

Save As command (File menu)

Use this command to save and name the active drawing. Fractal ViZion displays the Save As dialog box so you can name your drawing.

To save a drawing with its existing name and directory, use the Save command .

3.5.1

File Save as dialog box

File Save As dialog box

The following options allow you to specify the name and location of the file you're about to save:

File Name

Type a new filename to save a drawing with a different name. Fractal ViZion adds the extension

.zp.

Drives

Select the drive in which you want to store the drawing.

Directories

Select the directory in which you want to store the drawing.

Network...

Choose this button to connect to a network location, assigning it a new drive letter.

3.6

File Load Palettes command

Load Palettes command (File menu)

Use this command to load a palette file [.pl]. The palette file contains 21 palettes created previously with Fractal ViZion (or another version of the program.)

3.7

File Save Palettes command

Save Palettes command (File menu)

Use this command to save all palettes for the current session in a palette file [.pl].

3.8

File Load Palette [MAP] command

Load Palette [MAP] command (File menu)

Use this command to load a Fractint-type map file. The palette in the map file replaces the currently selected palette.

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File menu 18

3.9

File Load Text [VZT) command

Load Text [VZT) command

Load text (platform-independent) data file. Useful for transferring data files between Fractal ViZion and iViZionaire.

3.10

File Save Text [VZT) command

Save Text [VZT) command

Save data in text (platform-independent) format. Useful for transferring data files between Fractal

ViZion and iViZionaire. This command is disabled for images that use the landscape, orbital, ifs, or lsystem types, or the inversion or torus options, to ensure a minimum of compatability between programs. Some differences between rendering styles may be observed, but image shapes should be maintained.

3.11

Save As [ZP6] -- x64

Save data in Fractal Zplot x64 compatible format. This allows you to transfer old zp data files to a file format that can be loaded by Fractal Zplot x64, the 64 bit version of Fractal Zplot.

3.12

Plot to file

Plot to File

Allows you to plot a large bitmap directly to a .png file without the added system requirements of keeping the whole bitmap in memory. The Target group sets the bitmap resolution (width 800 to

14400. Drawing aspect is that of the current image.) Click on Okay to set the target file name and start a new plot to file. Note: the 3200X2400 bitmap size is suitable for 8 1/2X11 printouts at 320-

720 dpi. The larger bitmap sizes are suitable for poster-size printouts. This option is not available with the merging options, or with anti-aliasing. Also, solid-guessing is disabled when using this option.

3.13

File 1, 2, 3, 4, 5, 6 command

1, 2, 3, 4, 5, 6 command (File menu)

Use the numbers and filenames listed at the bottom of the File menu to open the last six drawings you closed. Choose the number that corresponds with the drawing you want to open.

3.14

File Exit command

Exit command (File menu)

Use this command to end your Fractal ViZion session. You can also use the Close command on the application Control menu. Note: if you choose to exit while plotting, the program does not terminate, but stops the plotting so the program can be safely exited.

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19 Fractal Zplot Application Help

Shortcuts

Mouse: Double-click the application's Control menu button.

4

Keys:

Edit menu

ALT+F4

Edit menu commands

The Edit menu offers the following commands:

Undo

Copy

Clip

Undo last edit, action or zoom.

Copy the active view and put it on the Clipboard.

Define area of view and copy to clipboard.

Paste

Cubic Values

Insert Clipboard contents.

Edit cubic parameters.

Octonion Values

Edit octonion constants.

Quat-Trap

Edit quat-trap parameters.

RGB Thresholds

Set threshold values for color interpolation.

Text

Edit and add text to drawing.

Preferences

Startup preferences and defaults.

4.1

Edit Undo command

Undo command (Edit menu)

Use this command to undo the last action. An image can be continued after an undo, if continue was enabled before the last action. Color-cycling is disabled after using Undo, though.

Undo is disabled for bitmaps whose width is not a multiple of 8.

Shortcut

Keys: CTRL+Z

4.2

Edit Copy command

Copy command (Edit menu)

Use this command to copy the active view to the clipboard. The entire view is copied to the clipboard.

Shortcut

Keys: CTRL+C

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Edit menu 20

4.3

Edit Clip command

Clip command (Edit menu)

Use this command to copy a part of the active view to the clipboard. A zoom box is used to select the part to be copied. Click outside the view frame or press escape to exit this option.

Shortcut

Keys: CTRL+L

4.4

Edit Paste command

Paste command (Edit menu)

Use this command to paste from the clipboard. The clipboard must contain a bitmap. If the bitmap is larger than the view, it is clipped. The zoom cursor is used to set the left/top corner in the view where the bitmap will be pasted. Click outside the view frame or press escape to exit this option.

Shortcut

Keys: CTRL+V

4.5

Cubic Values Window

Cubic Values Window

To support the cubic Mandelbrot formulas (g0 thru g9), variables have been added to adjust z-origin and magnify the z-plane while calculating pixel depth. Normally you can keep origin set at (0,0,0) and z-mag at 1.0. But these can be useful to tweak in small increments when drawing Mandelbrot quaternions. Then a small change in z-origin can rotate details on a close-up slightly, so you can adjust the view accordingly. Each shift in z-origin will require re-zooming to get back to the area of interest. The z-mag variable makes the z-plane non-symmetrical; pushing up some details while other details recede. So it too is useful to change viewpoint. The affect on Julia quaternions is less noticeable for z-origin shifts. You can usually re-zoom to produce the same Julia image.

Cubic Mandelbrots quaternions use the S and Si variables to set the initial value and range of the 3rd dimension. Starting from an initial value of 'S', Si is normally set to 2*abs (S), when the Type is

Mandelbrot. Si can also be set to center the 3rd dimension on something besides 0.0. Si has no affect in 2-D hypercomplex mode. (In 2-D mode, the S variable acts as one of the two fixed dimensions, along with the 4th Center variable.)

The 4th Center points to the center of the fourth dimension (with quaternions/hypernions the 4th

Dimension variable is used.)

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21 Fractal Zplot Application Help

In addition, the Arg value has the following affect on cubic Mandelbrots:

0 -- compute M+, using Z for z space

1 -- compute M+, using greater of S or Z for z space

2 -- compute M-, using greater of S or Z for z space

3 -- compute M+ and M-, use lesser of M+/M- for pixel depth

4 -- compute M+ and M-, use greater of two for pixel depth

5 -- compute M+ and M-, use difference of two for pixel depth

6 -- compute M+ and M-, use sum of two for pixel depth

7 -- compute M+ and M-, use vector magnitude of two for pixel depth

8 -- compute M+ and M-, use intersection of two (CCL)

9 -- compute M+ and M-, use M+ or difference of two if M+>M-

Args 3-9 affect only cubic Mandelbrots.

For args 0-9, a second argument determines the plane that is used for the fourth dimension: b -- b-imag (default)

B -- b-real a -- a-imag

A -- a-real

A third argument also works for args 0-9: j -- alternate cubic Mandelbrot mapping

Therefore an arg value of '3B' uses the union of M+ and M- and b-real as the fourth dimension.

4.6

Formula Window

Formula/Type Window

Fun #1 and Fun #2 are combo controls for entering/selecting up to two formulas either from their drop-down lists or in the custom form of Z=AZ+BZ+c. Z is the complex variable or function, 'c' is the complex constant, and A and B are optional real constants. There are additionally a Type control, an Arg control and an Arg Limit control that determine how the above formulas are processed. Note: when the fractal type is quaternion, custom formulas are only processed through the Fun #1 or the Formula box. The Fun #2 combo only uses the built-in formulas in its drop-down list.

The Type control accepts a value of 0 to 11. For a value of 0, the first formula is always used and the second formula is ignored. For a value of 1, the second formula is processed and the first formula is ignored.

Type 1 is of use only if you are switching between two functions and don't want to reenter them each time you plot the other one.

For a value of 2, the first formula is processed if the real component of Z is greater than 0, else the second formula is used.

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Edit menu 22

For values of 3, the first formula is processed and its output becomes the input of the second formula, which is then processed.

Type 4 takes the average of Fun#1 and Fun#2.

Type 5 alternates between the two functions while iterating.

Type 6 takes the quotient of both functions.

Type 7 uses the lowest iterative results of both Fun#1 and Fun#2

Type 8 uses the highest iterative results of both Fun#1 and Fun#2

Type 9 uses the Formula box to enter up to 1000 characters per formula.

Text can be pasted from the clipboard to the formula box by using the keystrokes shift-insert. Text may be moved from box to box by using shift-delete to move it first to the clipboard.

Type 10 produces Escher-like tilings for Julia sets, as described in The Science of Fractal Images.

This uses both fun#1 and fun#2 in combination. Fun#1 sets the stage for the target set fun#2. To reproduce the images in The Science of Fractal Images, use z=z*z for fun#1 and p0 for fun#2.

Fun#1 may be any formula other than z=z*z, but may produce unsymmetrical tilings. Fun#2 can be any built-in formula. This type is mainly applicable to Julia-type sets, but may be used with limited effect with the MandelbrotP-type set. Use the Rise box in the Parameters window to set the degree of tiling desired. The higher the Rise number (1-235), the greater the density of the tiling. Note: for quaternion "Escher" fractals, the Rise number has no effect, and Fun#1 is processed without hypercomplex extensions. This results in a 3D tiling effect on the x/y axis that can be multiplied by increasing the number of terms in Fun #1, as in z=z*z*z.

Type 11 uses Fun#1 to produce "genetic" style fractals. Here you enter a character string or word that can contain any of 25 letters 'a' through 'y'. Each letter or "gene" references a built-in formula, as defined in the Genes window. For every iteration the program cycles through the character string and uses whatever formula is referenced by the characters. Notes: If Max. Iter is shorter than the gene string then only part of the string is used. For quaternions, which usually have a very short iteration cycle, changing Max. Iter can have a dramatic effect on the fractal output.

The f1-f4 combo boxes are used to designate the 'fn' user-definable part of a generalized function.

This can be one of the 41 function types listed (sin(w),cos(w) etc.) A few of the options are formulas themselves (L3 (w) the Legendre function, the gamma function and G(w)the Gaussian function), so quite complex (though mathematically unintelligible) formulas are possible.

The S control is used to enter the variable 's' used in many of the built-in functions.

The Si control is used to enter the variable 'si' the imaginary component for s in some of the built-in functions.

The Converge control is used to enter the convergence limit for Renormalization or Convergence

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23 Fractal Zplot Application Help

plots. This is a measure for how close z needs to approach an attractor to be mapped to that attractor color. A small value .0001-.000001 is normally used. Smaller values of the convergence variable produce a more accurate plot, while increasing the computation time somewhat. On some formulas it may be necessary to reduce this variable to its smallest value (.00000001) to eliminate some artifacts (spots) caused by non-convergence (at a higher limit value.)

For the random displacement routines, the S box is used to control the fractal dimension (valid inputs are 0-1.0). A lower number gives the fractal a higher dimension, and thus more ragged and choppy appearance. In 3D mode the magnify box works as for other 3D plots, except that a value of .4-1.0

is adequate for most plots. For 2D plots or backgrounds, the magnification factor is used as a color scalar. A value less than 1.0 reduces the number of colors by the same factor. This is necessary to create more realistic clouds. The color palette for 2D plots is rotated by the cutoff number. This allows some independent color control given the limitations of a palette-based system. The arg box is used to control shaping and fullness of the finished plot. The plot may include random additions at every point or just the end points. The shape may be scaled linearly or non-linearly. The following values for the arg box control random additions and shape:

0 -- random additions at end points only; linear shaping.

1 -- random additions at every point; linear shaping.

2 -- random additions at end points only; non-linear shaping.

3 -- random additions at every point; non-linear shaping.

About formula syntax: This applies if you elect to enter your own formula into one of the function boxes and use the parser to generate the plot. The use of parenthesis is necessary around complex exponents or variables. E.g.: '(z-i)^(1.5e)'. For a complete list of variables, operators and functions recognized by the parser, see

Parser Information

.

Up to 500 user-named-complex variables and constants may be included in a formula. A variable must begin with a letter and may contain numbers and letters only. A variable may be up to 9 characters long. A constant may be up to 20 digits long, including the decimal point. Fractal ViZion uses syntax similar to Fractint's formula style with an initialization section, followed by the main formula, and an optional bailout routine. Comments may be entered on the same lime with a preceding ';'. Some variables such as 'pixel' and p1 are named after Fractint's predefined variables.

These are provided to allow Fractal ViZion users to more easily convert Fractint formula types to

Fractal ViZion use. However, Fractal ViZion doesn't prompt you to enter values into p1 (the cr and ci boxes) or p2 (the s and si boxes) or p3 (the limit and converge boxes). Since p1 is used in the iteration process as 'c', p1 cannot be used as a variable independent of c. A ':' terminates the initialization section. Multiple phrases may be entered in the main formula or initialization sections on the same line by using the terminator ',' between phrases. Use ctrl-enter to terminate a line in the formula box. An optional bailout routine may be entered as a phrase at the end of the formula. If the bailout phrase equals a value other than TRUE during iteration, the iteration loop is exited. There are other flags such as Convergence and Biomorph, if set, that can force exit also.

The arg control is limited to 25 characters.

The Title text box is used with the hot key 'T' to annotate a picture with text. Use the Edit/Text

© 2003 - 2010 ... Mystic Fractal

Edit menu 24

command to change font, text color or format text into multiple lines. Text in this box is not saved in a picture's data file, but once entered the same text can be used over and over for different pictures.

Useful for adding copyright/author info to batches of pictures. Since the same title text may be used many times, it is shared among views and saved in the file "prefs.txt" in Fractal ViZion's startup directory.

Click on the Apply button to use the formulas currently displayed in the window. Use the Okay button to apply the formulas and close the window, or use Cancel to exit the window without making any changes or cancel current changes. If changes have been applied before clicking on

Cancel, then the formulas will revert to the state when the window was last opened. Note: commands exterior to this window (as in the Demo menu) may cause the window to be closed and reopened with new values. In this case Cancel only returns the formula values to the "new" values.

The Reset button returns all boxes and slider values to their original values when the window was opened.

The three buttons named Rand fun#1, Rand fun#2 and Rand f1-f4 are used to pick formulas/ functions at random. Clicking on Rand fun#1, a formula is chosen (from the 100 built-in formulas) for fun #1. Clicking on Rand fun#2, a formula is chosen for fun #2. Clicking on Rand f1-f4, functions are chosen for f1-f4.

Select FraSZle Formulas to switch to the FraSZle formula set.

4.7

Fractal Variables

Fractal Variables

The window opened varies with the fractal type selected, and contains all the major variables that

Fractal ViZion now scales between key frames of an avi stream.

4.7.1

Parameters Window

Parameters Window

There are edit controls for entering the complex constant (real and imaginary parts), and the min/max ranges for the real and imaginary window coordinates. Fractal Zplot uses three-corner plotting for easier rotating, so boxes are provided for Top Left, Top Right and Bottom Right real/imaginary coordinates. These reflect the current range values that may have been derived from zooming with the Zoom option. Slider-type controls affect the number of iterations (1-9999), the z-limit (1-9999).

Cj, ck, and hj are for entering hypercomplex parameters. Cx, cy, cz and cw are used as complex

C increments for Julia-Tower types. The Size slider controls the overall size of the picture. The Size slider sets the horizontal resolution, while the vertical resolution is then scaled according to the fullscreen VGA ratio, 4 to 3(1:1 if that aspect is selected through the Auto menu.) The Sector slider controls which of 4 sectors the picture will be drawn in, if the Size is less than or equal to (the fullscreen horizontal resolution)/2. Otherwise the picture is centered according to the full-screen dimensions. This allows you to show zooms of a particular function by using different sectors, or show the affect of different plotting options. Each sector is erased individually. Note: if you try to

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25 Fractal Zplot Application Help

continue a plot in a different sector than you started with, the plot will continue in the original sector.

The Thumbnail button next to the Size slider is used to set a thumbnail size quickly. The thumbnail size toggles between 1/4 and 1/8 of the horizontal screen resolution, e.g. 200X150 or 100X75 for an 800X600 screen.

The more iterations used, the longer it takes to plot a function, but more detail will be present. 10-

20 is sufficient for most biomorphs, while more iterations will be required for Mandelbrot and Julia sets, depending on the detail required.

Start color and end color boxes are provided to limit the palette colors that the current image uses.

Use start color of 0 and end color 235 to use the full palette. The color-cycling keys (arrows and enter/backspace) work for only those colors that the start/end colors designate.

Zooming is not available while plotting height fields in 3D.

The screen or sector is always cleared before beginning a new 2D plot, but not with 3D plots or 3D backgrounds. Use 'C' to clear the screen for a new 3D plot. Use 'B' to clear a 3D background.

The Reset button returns all boxes and slider values to their original values when the window was opened.

For 3D plots there are four additional edit boxes, the Magnification Factor, Y-Offset, Slope/Cutoff and Rise. The Magnification Factor controls the amount of Z-Axis depth for each point plotted.

This can range from 0 to 10000.0. Use less for smaller plots. The amount of magnification used depends on the smoothing factor, size of plot and iterations. Excess magnification can result in a plot being clipped off the screen. Note: for a magnification of 0.0, the curve is totally flat and the lower boundary is clipped to the line being drawn, instead of a level starting from the boundary of the plot sector. In order to create full-screen three-dimensional plots, a larger triangular plot is clipped to the sector's square limits. The Size slider must be set to at least double the sectors 2D horizontal resolution for a seamless 3D rendering. The 3D plot is drawn left to right, so that the lower edge of the plot can be set with the Y-Offset box. Potential is mapped with continuous color changes from the bottom of the plot to the starting line. The Y-offset should be adjusted so that the starting line is nearly invisible, with as smallest a section shown as possible, to minimize odd color changes at the start of the plot. For some views, it is impossible to avoid a clipped-polygon effect if the potential are high at the start of the function's min/max zones. You need to zoom out somewhat, to retain the

2D image area in full-screen 3D. Also, a horizon is established on all 3D plots at approximately 1/3 down from the top of the plot. This works by clipping all points above the horizon, but retaining the height of all points at or below the horizon.

The Y-Offset controls centering of the plot up and down. The plot is automatically centered left to right. Use a small positive or negative offset to start (under 0.5 or zero), then go from there. A positive y-offset moves the plot down on the screen. This value will change with the size of the plot or with increases in magnification. Also, less Y-Offset is generally required on Low plots than High plots. The slope factor is used to determine how steep points near the upper limit of escape times appear on screen. Start with a value from 1 to 2 with Log Plots. A lower limit of 1.0 is imposed for the smoothing factor. Continuous potential plots usually use a slope value of 2.0. A higher

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Edit menu 26

magnification factor is necessary to bring out detail on close-ups.

Since the magnification needed for extreme close-ups increases exponentially with the iterations used, the magnification variable also represents an exponentially increasing factor: 'magnify^(log

(iterations))'.

For 3D (height field) plots, the Rise box is used to specify a height for the set potential. Normally the set is mapped at zero or the highest potential (# of colors-1.) You can change this to create

Mandelbrot lakes or sheer cliffs. Using a Rise value of less than 0 automatically maps the set potential at the highest potential. The rise value also sets the color index for the Mandelbrot lake with ray-traced plots.

For 2D plots, the Cutoff box acts as a palette multiplier or divider, depending on whether the value entered is less than or greater than 1.0. The palette color is divided by the Cutoff to speed up or slow down color changes. Cutoff values are limited to a low minimum of .001. For level curves and the add and multiply color-scaling options, use a negative cutoff value to maintain a smooth palette.

This ensures that the multiplier is used before the (floating-point) palette values are converted to

(integer) palette indexes.

For random displacement plots, the Rise value determines the minimum height plotted. Heights below the Rise value are treated as 0 altitude. For 2D plots, pixels at minimum height are not plotted.

The Rotate box is used to rotate the picture by any degree. This rotates all three points that define the Z-plane. The rotational angle is reset to zero after each use.

Related Topic:

Quaternion describes the Quaternion generator's data-collection window.

4.7.2

Quaternion Window

Quaternion Window

This is the data-collection window for Fractal Zplot's Quaternion generator. Minx, maxx, miny and maxy are the spatial variables for framing the quaternion object. These are usually updated automatically when you use the zoom box. Min Z and Max Z define the three-dimensional space that is used to map the quaternion image. Normally Min Z is the negative of Max Z, but Min Z can be adjusted in the positive direction to shear off the front of the quaternion object. This has the effect of exposing the insides of a quaternion. Const1-4 are cr, ci, cj and ck respectively. Maxiter is the same as iterations in the Parameter's window. Three rotate variables determine the 3D angle of rotation. Step and Fine are pitch adjustments that bear on the quality of the plot at an expense of lengthier calculations.

The Bailout variable can range from .000001 to 65000. For most escape-time formulas that have an attractive point at infinity, the bailout is set to a value 4 or greater. A larger value tends to make the figure smoother and fuller with some loss of detail. When the variable is set to a value less than

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27 Fractal Zplot Application Help

one, then a bailout routine geared to convergent formulas such as built-in formulas that use Newton's method. The bailout uses Alessandro Rosa's "cutoff rate" routine to limit the basins of attraction (to viewable 3D areas) for formulas that use Newton's or Halley's method, etc. This method also works for escape-time formulas, though slower and requiring more iterations to bring out detail.

When you click on Okay, the quaternion generator looks at the Fun#1 gadget in the New Formula window. If this contains a preset variable that function is iterated for its escape time, then the results are ray-traced in any 3D object that may be created. If the Type is set to 9, any custom formula

(Fractint-style: as in 'z=z^3+c#') in the Formula box is used. Not all functions may produce a usable

3D object with this method, but it's interesting to experiment with. The Quaternion option works for both Mandelbrot and Julia sets, depending on which type is selected.

The Slice variables may be altered to display different 3-D planes of the quad set.

The Plane variables B (Back), F (Front), and P (Position) allow you to flatten part of the quaternion figure. For normal plotting, these variables default to 0. To have any effect on the image, the Back variable must be different in value from the Front variable. The difference between back and front variables determines where the image is flattened. These variables are limited to +/-9.99, with normal values being in the range –z to +z. The position variable sets the color of the flattened plane.

4.7.3

Axiom Window

Axiom Window

Enter the basic angle, 0.5-360. This angle will be used for any turning command that doesn't use an additional argument. Enter an axiom, up to 255 characters in the axiom box. Enter the initial width of the line, (>=1.) Angle and width variables can be non-integer. A line with width greater than one causes the turtle to draw a tubular line in 3D space. See the 's' rule (Demo/Lsystem info) in the extended rule set for info on changing the line shape. Click on Apply to apply the current axiom to the image. Click on Okay to set the new axiom into memory and close the window. Click on

Sample or Random Sample to use one of the pre-existing lsystems in the startup directory. Click on

Rules to edit the rules associated with the current axiom.

4.7.4

Rules Window

LSystem Rules

The top row contains the command line buttons, and the input boxes below are used to enter production rules and labels.

The input boxes:

'Rule' -- enter up to 255 characters for each production rule.

'Label' -- the label for the production rule. Can be any ASCII character. If the character has no keyboard equivalent, you can enter it as its decimal number (0-255.) This may or may not produce

© 2003 - 2010 ... Mystic Fractal

Edit menu 28

a visible character when drawn in the rules list. Using shift-delete and shift-insert for these characters enables them to be copied and pasted into any rule.

The command line buttons:

'Add' -- adds whatever production rule has been entered into the string gadgets, if the label does not duplicate a label already in use. Up to 255 rules can be entered. The window below the string gadgets displays the current production rules in use. Use the slider device to scroll through the rules, if more than nineteen rules are used.

'#' & 'Recall' are used in conjunction to edit a previously added rule. Enter the number of the rule, as displayed below, into the # box and click on Recall. The rule is entered into the input boxes.

(Alternately, you may click on a displayed rule with the cursor and left-mouse button, and it will be recalled.) This enables 'Change' and 'Delete' to change or delete the rule. An extra step here prevents changing or deleting the wrong rule by mistake. Hint: you can also change the rule and then Add it in the same operation, if the label is changed. Use 'Cancel' to change another rule, if you don't really want to change or delete the rule you Recalled. (# box is disabled during a Recall operation.)

'Cancel' -- cancels the recall operation.

'Insert' -- inserts a rule before the numbered rule as entered in the # box. You can insert a rule after recalling and changing it, too. You can't insert a rule with a label already in use. You must add at least one rule before you can insert a rule.

'Okay' -- terminates the edit-rule session and closes the edit rule window.

4.7.5

Orbit Function

Orbit Function

Function is the edit control for entering the two-letter code for one of 17 built-in orbital functions. A

quick reference to the built-in formulas is found here

.

N is the number of iterations used to produce the fractal. The demo menu defaults to 1000000 for exploratory purposes. Increase to its maximum value (999999999) for a smooth finished fractal.

Change the angle of rotation with the three rotational variables. Options for the Flame-type function

(R6) are provided as a set of check boxes.

Click on Apply to start a new plot, or Okay to apply and close the window.

After the fractal is drawn you may want to try to export it as a model for other 3D graphics programs. The Steps variable controls the resolution of the export model. The Export Orbital command (File menu) only works with some orbital fractals, mainly those that have a broad surface area at the center.

The Reset button returns all edit boxes to their original values when the window was last opened.

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29 Fractal Zplot Application Help

4.7.5.1

Orbits Quick Reference

Orbits Quick Reference

p0: -- Sprott strange attractor #1 p1: -- Sprott strange attractor #2 p2: -- Sprott strange attractor #3 p3: -- Sprott strange attractor #4 p4: -- Sprott strange attractor #5 p5: -- Latoocarfian p6: -- Latoocarfian Mutation Alpha p7: -- Latoocarfian Mutation Beta p8: -- Latoocarfian Mutation Ceta p9: -- 3D Attractor (C. Pickover) r0: -- Lorenz 3D Attractor (E. Lorenz) r1: -- Million-Point Sculptures (C. Pickover) r2: -- Quadratic by Julian C. Sprott r3: -- Modified Quadratic by Julian C. Sprott r4: -- Chaotic Iterated Map r5: -- Chaotic Iterated Map #2 r6: -- Flame #1(variation of Scott Drave's Flame method)

4.8

Ray-Tracing Window

Ray-Tracing Window

The Light Point variables (lightx thru lightz) determine the direction of the light source used in the raytracing algorithm. The ViewPoint represents the angle at which the object is ray-traced, which can affect Phong highlights greatly. This has no effect on the camera view.

The Lighting variables shininess, highlight, gamma and ambient are used to adjust ambient light and highlights. The ranges for these variables appear beside their label. Decreasing the shininess value increases light reflected by the object and the apparent sheen on the object's surface. The ambient value controls the amount of ambient light that illuminates the object. The highlight value increases or decreases the specular (Phong) highlighting, while the gamma value increases or decreases the intensity of the light source's illumination. Once a plot is started, the lighting variables and light point can be changed without redrawing the object.

Click the Apply button to redisplay a plot after changing the lighting variables or light point. Click the Okay button to close the Ray-Tracing Window, applying new settings, if the variables were modified. Click on Cancel to revert to the state that existed when the ray-tracing window was opened. Click on Defaults to set the lighting and viewpoint variables to the built-in defaults for these variables.

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Edit menu 30

4.9

Octonion Values Window

Octonion Values Window

Octonions (built-in functions H0-H9) have a form of xr+xi+xj+xk+xE+xI+xJ+xK. Additional options are entered via the Arg box in the

Edit/Formula window

. Use the Randomize button to set a random value for octonion planes E-K and complex constants cj-cK.

4.10

Edit Palette

Edit Palettes

Use the palette editor to modify the palette(s) in use.

It is important to realize that palettes are software-simulated in Fractal ViZion (since 24-bit color supports no hardware palettes), so color-cycling and palette switching are not fast operations as with a 256-color system that supports palettes.

There are copy and spread options to smooth or customize the existing palettes in Fractal ViZion.

You can then save all the palettes in a .pl file, or by saving the entire function and bitmap (v1.08+ saves all the palettes in the data file.)

Colors are shown in 8 groups of 32 colors. This makes it easy to create divide-by-8, divide-by-4 and divide-by-2 palettes with 256 colors (Use a start color of 0 and an end color of 255.) With

Fractal ViZion, a palette is actually 65280 colors, with each succeeding color (except the last) followed by 255 colors that are evenly spread from one color to the next.

Use the RGB-slider controls to edit any color in the palette. Select Copy to copy any color to another spot in the palette. Select Spread to define a smooth spread of colors from the current spot to another spot in the palette. Copy and Spread take effect immediately when you select another spot with the mouse button. You can cancel the operation with the Cancel button. In Fractal

ViZion, colors do not cycle smoothly when you adjust the RGB/HSV sliders. This would be too slow with true color. The Map button is used to map color changes to an image after you are done adjusting the sliders. In the HSV mode, color spreads are based on HSV values instead of RGB values, which in some cases results in brighter color spreads.

Right-click on any point on the main window and the palette color for that pixel will be displayed in the palette editor. You can use any of the color-cycling keys (after clicking on the main window) to see the effects of the cycling in the palette editor window. Note: color cycling and color-selectionfrom-pixel only works when the image has been drawn in the current session. If you load a preexisting image file, you must redraw it to cycle colors, etc. Anti-aliasing, 3D height fields, undoing an action and hsv filtering also disable color cycling.

Use Reset to reset the colors of the palette in use, to where it was before it was cycled or modified.

Note: if you change palettes with one of the function keys, any modifications to a previous palette

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31 Fractal Zplot Application Help

are unaffected by the Reset button.

Use Reverse to reverse the order of the colors in the palette. This affects only those colors in the start-color to end-color range. Useful for reversing divide-by-eight palettes, etc., for orbit-trap pictures that require a reversed palette.

Use Neg to create a palette that is the complement of the current palette.

Use Smooth to create a random palette with smooth color spreads. Use Scramble to create a palette with random color indexes.

Use SRG to switch the red and green components of all palette colors.

Use SRB to switch the red and blue components of all palette colors. SRB and SRG are disabled in

HSV mode. You can use these buttons to form eight different palettes by repeatedly switching red, green and blue components.

Use the Random palette button to randomize the current palette. The Randomize variables, rmin, rmax, bmin, bmax, gmin, and gmax act as limits that are applied after the palette after initial randomizing, to make the palette conform to the desired spectrum of colors.

Note: unless you click on Reset before exiting the editor, changes are permanent to the palette edited, no matter which way you close the editor (Okay button or close box.)

4.10.1 Reverse button

Reverse button

Use Reverse to reverse the order of the colors in the palette. This affects only those colors in the start-color to end-color range. Useful for reversing divide-by-eight palettes, etc., for orbit-trap pictures that require a reversed palette.

4.10.2 Neg Button

Neg button

Use Neg to create a palette that is the complement of the current palette.

4.10.3 Map Button

Map button

In Fractal ViZion, colors do not cycle smoothly when you adjust the RGB/HSV sliders. This would be too slow with true color. The Map button is used to map color changes to an image after you are done adjusting the sliders.

4.10.4 H/R Button

H/R button

Change from HSV to RGB mode or back. In the HSV mode, color spreads are based on HSV values instead of RGB values, which in some cases results in brighter color spreads.

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4.10.5 Spread Button

Spread button

Select Spread to define a smooth spread of colors from the current spot to another spot in the palette.

4.10.6 Copy Button

Copy button

Select Copy to copy any color to another spot in the palette.

4.10.7 SRG Button

SRG button

Use SRG to switch the red and green components of all palette colors. RGB mode only.

4.10.8 SRB Button

SRB button

Use SRG to switch the red and blue components of all palette colors. RGB mode only.

4.10.9 Okay Button

Okay button

Click on Okay to exit the palette editor, applying unmapped color changes to picture (if colorcycling is enabled.)

4.10.10 Reset Button

Reset button

Use Reset to reset the colors of the palette in use, to where it was before it was cycled or modified.

Note: if you change palettes with one of the function keys, any modifications to a previous palette are unaffected by the Reset button.

4.10.11 Cancel Button

Cancel button

You can cancel a copy or spread operation with the Cancel button.

4.10.12 Red Slider

Red slider

Use the RGB/HSV-slider controls to edit any color in the palette.

4.10.12.1 Red edit box

Red edit box

Shows red/hue value of selected color index.

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33 Fractal Zplot Application Help

4.10.13 Green Slider

Green slider

Use the RGB/HSV-slider controls to edit any color in the palette.

4.10.13.1 Green edit box

Green edit box

Shows green/saturation value of selected color index.

4.10.14 Blue Slider

Blue slider

Use the RGB/HSV-slider controls to edit any color in the palette.

4.10.14.1 Blue edit box

Blue edit box

Shows blue/value magnitude of selected color index.

4.10.15 Smooth Button

Smooth palette button

Use to create a random palette with smooth color spreads.

4.10.16 Scramble

Scramble

Use to create a palette with random color indexes.

4.11

Edit Quat-trap

Quat-Trap (Edit menu)

With this window you can edit and display the two separate functions that make up the quat-trap fractal. You can change the orbit-trap width, complex c or type, then display the orbit-trap with the

Draw Trap button. You can change the quaternion's function or complex c, then display the quaternion with the Draw Quat button. Use Apply or Okay to display the composite fractal. See

Orbit-Trap

options for further details on quat-trap pictures.

4.12

Size

Size command

This allows you to set the drawing area for a picture, independent of the Windows screen size. It also shows which size is currently in use. The aspect for the drawing is based on the ratio of X

(horizontal width) to Y (vertical height.) The custom setting allows for any size/aspect that system

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Edit menu 34

memory will permit. Videos are limited to the standard 4/3 vga aspect or 1/1. Midi output is limited to images with the standard 4/3 aspect. The minimum size for an image is 40X30. Note: if the image is less than 100 width, the aspect must be 4/3 for solid-guessing to work properly.

4.13

RGB Thresholds

RGB Thresholds (Edit menu)

When Fractal ViZion plots pixels using a simulated palette, the colors are normally scaled

(interpolated) to fit between one of 236 color indexes. These 236 colors can be edited by the palette editor. Sometimes you don't want an intermediate color to be plotted, if adjacent color indexes differ a lot.

There are two tests that can be made to determine the degree of color difference between adjacent color indexes. The first is RGB, where the difference in individual components is tested. The thresholds are set so that color indexes that differ in values exceeding the individual RGB thresholds will bypass the interpolation process. The values can be set from 0 to 255, with 255 the default. At

255, all pixels are interpolated except the first and last colors. At 0, no pixels are interpolated.

Test 2 measures the difference in the sum of the RGB values of adjacent colors. If the sum is greater than a preset limit (0-765), no interpolation is done.

These two tests can be combined with the Both option.

4.14

Edit Text command

Text (Edit menu)

Allows you to edit text and font and apply it to a drawing. Select the font button to set the font style, size and color. In the text window click on Okay to add a line of text to the current image. (You can add multiple lines of text too, up to 80 characters.) The cursor will change to a crosshair.

Position the cursor where you want the text to start and left-click the mouse. Note: font and title text are saved in the file "prefs.txt" in Fractal ViZion's startup directory. Title text can also be edited (as a single line only) in the Edit/Formula window.

4.15

Preferences

Preferences (Edit menu)

Each time you use the Reset command, Fractal ViZion restores data variables to built-in defaults.

The Set Defaults button allows you to change some of the data variable defaults to whatever the current settings are. Some of the customizable variables include step, fine, formula, viewpoint, lighting, rotational angles, Phong and x/y space. (Iterations, Type, Constants, and a few other variables are excluded to maintain compatibility with the 'G' command.) The new Reset defaults are

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5

35 Fractal Zplot Application Help

saved in the file "prefs.txt" when you close the program (if the Defaults check box is selected.) The check boxes in the group "Save on Program Close" allow you to change the default startup mode of a few Auto options, such as Auto Redraw, and the Random Setup variables. By keeping the boxes selected, Fractal ViZion saves the last changes you make to these options. If you want to go back to the initial settings (the way Fractal ViZion was packaged originally) you can click on the Reset

Defaults button. This restores the data, Auto variables and random setup defaults.

Use the Reset Dialog Positions button to reset all non-modal dialog positions to x/y positions that will fit within a 640X480 screen. Sometimes when you switch screen resolutions to a lower resolution there might be dialogs that are off the screen and thus are inaccessible when reopened.

This can happen too if the Registry key for the dialog position becomes corrupt. (The program keeps track of all non-modal dialogs' last positions in the Software portion of the Registry.) Close all open dialog windows you want to reset before using this command, or open a New draw window before using the command.

Use the Default Directories tab to change the default directories for saved and loaded items in

Fractal ViZion. Click on the "…" button next to each default directory box, and use the folder requester to pick a different directory, or create a new directory with the Make New Folder button.

The default directories are saved at program close in the Registry and reloaded when you next open

Fractal ViZion.

Fine-Tuning menu

Fine-Tuning menu commands

TheFine-Tuning menu offers the following commands:

Auto Sound Alert

Enable or turn off sound alerts.

Auto Time

Auto Random

Auto Redraw

Auto Clear

Auto Remote

Show time used to plot image.

Randomize new fractal type

Redraw image on command.

Clear drawing area before new plot.

Open remote automatically at startup.

The following commands can be used via the Color menu:

Color-Scaling menu

Palette menu

Divide By One Palette No split palette.

Divide By Two Palette Split palette into two sections.

Divide By Four Palette Split palette into four sections.

Divide By Eight Palette

Split palette into eight sections.

The following commands can be used via the 2D Render Menu:

Anti-Alias

Biomorph

Use anti-aliasing, with 1X4 or 1X2 super-sampling.

Escape on any part of z.

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Fine-Tuning menu 36

Bioconvergence

Biomorph Off

Newton->

Renormalization

Convergence

Relates Biomorph method to convergence.

Reset biomorph flag.

Define Newton method or clear Newton flag.

Renormalization.

Color converging points with level set colors.

Default Function

Level Curve->

Use default function type

Set level curve or reset level curve flag.

Decomposition->

Binary or continuous decomposition.

Decomposition Off

Reset decomposition flag.

Spin

Increment C by scaled factor of cx at every iteration.

Torus

Use torus method.

Torus Off

Reset torus flag.

The following commands can be used via the 3D Render Menu:

Texture Scale

Atan Coloring

Bof60 Coloring

Set scaling factor for texture.

Use Atan algorithm for coloring.

Use Bof60 algorithm for coloring.

Potential Coloring

Surface Filter

Add Noise

Color by magnitude of z.

Define surface filter.

Add noise to coloring.

Factors

Edit noise factors.

Reset Noise Seed

Re-seed random noise generator.

Coloring Filter

Define coloring filter.

Invert

Invert Off

Switch->

Filter

HSV Filters

Orbit Traps

Invert image around circle.

Reset inversion flag.

Switch z components or z for c.

Choose an optional tail-end filter.

Define HSV filters.

Set orbit trapping method.

5.1

Auto Alert command

Auto Sound Alert command (Fine-Tuning menu)

With this command enabled (on by default), the user is notified by a sound clip when a drawing is completed or user-canceled. By disabling this command the completion exclamation is suppressed and also any alert that contains a message box. Note: some sound clips are automatically generated by Windows, or there is no text alert for a given error condition. In these cases the sound alert is unaffected by the Auto Alert command.

5.2

Auto Time command

Auto Time command (Fine-Tuning menu)

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37 Fractal Zplot Application Help

With this command enabled (on by default), the time that an image takes to plot is displayed when the plot is complete. Fractal ViZion saves the condition of this option at session's end, so if you disable it and close the program, the option will be disabled when you restart Fractal ViZion.

5.3

Auto Random

When this option is selected (default on), for every new fractal (created using the New button on the

Remote or on opening the program) the fractal type is randomized. Deselect this option if you want to select your own fractal type.

5.4

Redraw command

Auto Redraw command (Fine-Tuning menu)

With this command disabled (on by default), redraw does not occur except when the Draw command is executed, or Continue. Most of the time you want to see the results of changing a parameter or mapping option, so redraw occurs automatically with parameter or mapping changes.

Sometimes you want to change more than one parameter before redrawing the image, or you may want to *continue* a mid-point displacement plot (after loading the file, this is the only way to recreate the plot.) So you need to turn this option off then.

5.5

Auto Clear command

Auto Clear command (Fine-Tuning menu)

With this command enabled (on by default), the drawing area is cleared before starting a new plot.

You can turn off this option when you want to see the effect of minor changes to parameters, as they affect the plot pixel by pixel, or when setting up a multiple-layered fractal, as in a 3D landscape.

You can use the shift-c command ( hot keys

) to clear the drawing area at any time.

5.6

Auto Remote command

Auto Remote command (Fine-Tuning menu)

With this command enabled (on by default), the remote is opened immediately at program startup.

Handy if you find the remote useful and don't want to click on the toolbar button each time the program starts up.

5.7

Color menu

Color menu commands

The Color menu offers the following commands:

Color-Scaling menu

Palette menu

Divide By One Palette

Divide By Two Palette

Divide By Four Palette

No split palette.

Split palette into two sections.

Split palette into four sections.

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Fine-Tuning menu 38

Divide By Eight Palette

Split palette into eight sections.

5.7.1

Color-Scaling menu

Color-Scaling menu commands

The Color-Scaling menu (in Color menu) offers the following commands:

Escape->

Level

Continuous Potential

Use Level

Background

Set Only

Graded Palette->

Use Palette 1 ...

Log High

Log Low

Escape-time color scaling.

Color scaling based on log(z).

Color scaling based on continuous potential.

Use level curve option for all coloring.

Set external points to background color.

Plot points in complex set only.

Use non-repeating vs modulus palette.

Use palette 1 for background filter.

Logarithmic mapping of level potential.

Logarithmic mapping of reversed potential.

Continuous Potential High

Linear mapping of level potential.

Continuous Potential Low

Linear mapping of reversed level potential.

Ray Trace

Ray trace 3D plot.

Sea Level

Set sea level at ground zero.

5.7.1.1

Escape

Escape

Five options are included that color a point based on its escape time (when it blows up.)

The Iteration option uses only the point's escape time.

Escape-time coloring.

The Iteration+ option uses the sum of a point's escape time and the value chosen (which can be picked from a menu that mirrors the Map menu.) A window is opened to set a q factor (1-200), which scales the sum value.

The Iteration* option uses the product of a point's escape time and the value chosen. .) A window is opened to set a q factor (1-200), which scales the product value.

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39 Fractal Zplot Application Help

The Angle option use the absolute value of a point's exit angle (theta.) This is the atan method in

Fractint.

The Angle-Iteration option uses the angle formed by the difference between a point's last two exit values and subtracts the point's escape time. Using the Angle-Only option on the Decomposition menu, escape times are not subtracted from the difference angle. This is Paul Carlson's atan method.

5.7.1.2

Level

Level

A point is colored based on its logarithmic escape. A window is opened to set a q factor, which controls the smoothness of picture color. A higher q factor results in grainier pictures and excess detail. Too low a q factor results in loss of colors and detail.

Level coloring.

5.7.1.3

Continuous Potential

Continuous Potential

A point is colored based on its continuous potential (when it blows up.) A window is opened to set a q factor, which controls the smoothness of picture color. A higher q factor results in grainier pictures and excess detail. Too low a q factor results in loss of colors and detail.

Continuous-Potential coloring.

5.7.1.4

Use Level Curve

Use Level Curve

All points are colored according to the choice selected from the Level-Curve Flag option. Defaults to Linear Map #4 if no Level Curve is checked. This works with Decomposition and other methods that would normally not use Level-Curve shading.

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5.7.1.5

Set Only

Set Only

The Set Only flag plots all external points in the background color.

5.7.1.6

Background

Background

An external point is colored with the background color. This works like the Set Only flag, except with decomposition plots and Biomorph/Epsilon plots. Normally, when a point is decomposed, its escape time or level color is added to its arg (exit angle) to determine its final coloring. With

Background color-scaling, only a point's arg determines its color.

With Biomorph/Epsilon plots, all external points are colored with the background color and all

Biomorph/Epsilon points are colored with the set color.

5.7.1.7

Graded Palette

Graded Palette

All points are colored with a non-repeating graded palette versus the default repeating(modulus) color scaling. Has no effect on the Background or Use Level Curve options, or when you use the cutoff value in the Parameters window as a color multiplier. When used with the Escape/Iteration coloring mode and a negative cutoff value, iterations are interpolated to reduce banding in escapetime pictures. Three options are provided for smoothing. The Interpolated version works for most escape-time formulas except convergent types (Newton and renormalization.) The Mandelbrot version is based on Linas Vepstas' log log algorithm, and is designed mainly for formulas that use z^2 as their main focus, such as z^2+c. The Exponential smoothing method is based on Ron

Barnett's algorithm, and works for both escape time and convergent-type fractals.

Level-graded coloring.

5.7.1.8

Use Palette

Use Palette 1 for Background Filter

This allows you to use palette 1 with a background filter, instead of the default palette, to add highlights to a picture. When Background filter is checked in the Filter window and a filter is selected, FV uses palette 1 for any color indexes modified by the filter. This option is not available with solid-guessing.

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41 Fractal Zplot Application Help

5.7.1.9

Log High

Log High

Maps hill or mountain-type plots with a logarithmic slope based on iterations (escape time of Z.)

Logarithmic slopes smooth out fast changes in point potentials. User specifies the level of the set point with the Rise box (0 to [# colors-1]).

5.7.1.10 Log Low

Log Low

Maps valley or lake-type plots with a logarithmic slope based on iterations (escape time of Z.)

Logarithmic slopes smooth out fast changes in point potentials. User specifies the level of the set point with the Rise box (0 to [# colors-1]).

5.7.1.11 Continuous Potential High

Continuous Potential High

Maps hill or mountain-type plots with a logarithmic slope based on potential (the absolute value of Z when the point escapes.) This option usually produces smoother contours with the Mandelbrot set than the Log options. User specifies the level of the set point with the Rise box (0--[# colors-1]).

5.7.1.12 Continuous Potential Low

Continuous Potential Low

Maps valley or lake-type plots with a logarithmic slope based on potential (the absolute value of Z when the point escapes.) This option usually produces smoother contours with the Mandelbrot set than the Log options. User specifies the level of the set point with the Rise box (0--[# colors-1]).

5.7.1.13 Ray Trace

Ray Trace

Add a light source, with Phong highlights, to 3d plots. Color palettes should be continuous (dark to light to dark) to take best advantage of this option. The light source parameters may be altered in the Edit Quaternion window. Starting with version 1.22, this option is the default for all 3D fractal types. New: if you prefer to generate 3D plots, such as quaternions, without ray-tracing, deselect this flag before drawing the plot. The image draw will be the pre-image, using the existing palette, as is, without ray-tracing (Phong and shading.)

5.7.1.14 Use Sea level

Use Sea Level

Imposes a water level at zero potential for 3D plots. All points below zero potential are mapped at zero to create lakes or oceans around 'land' areas.

5.7.2

Palette menu

Palette menu commands

The Palette menu (in Color menu) offers the following commands:

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Fine-Tuning menu 42

Palette #1-21

Use one of 21 palettes.

5.7.2.1

Palette 1-21 command

Palette command (Palette menu)

Switch to palette #.

5.7.3

Color Divpal1 command

Divide by One Palette (Color menu)

Palette is not split before applying to pixel.

5.7.4

Color Divpal2 command

Divide by Two Palette (Color menu)

Palette is split into two parts before applying to pixel.

5.7.5

Color Divpal4 command

Divide by Four Palette (Color menu)

Palette is split into four parts before applying to pixel.

5.7.6

Color Divpal8 command

Divide by Eight Palette (Color menu)

Palette is split into eight parts before applying to pixel.

5.8

2D Render menu

2D Render menu commands

The 2D Render menu offers the following commands:

Anti-Alias

Biomorph

Bioconvergence

Biomorph Off

Newton->

Renormalization

Convergence

Default Function

Use anti-aliasing, with 1X4 or 1X2 super-sampling.

Escape on any part of z.

Relates Biomorph method to convergence.

Reset biomorph flag.

Define Newton method or clear Newton flag.

Renormalization.

Color converging points with level set colors.

Use default function type

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43 Fractal Zplot Application Help

Level Curve->

Set level curve or reset level curve flag.

Decomposition->

Binary or continuous decomposition.

Decomposition Off

Reset decomposition flag.

Spin

Torus

Increment C by scaled factor of cx at every iteration.

Use torus method.

Torus Off

Reset torus flag.

.

5.8.1

Anti-Alias

Anti-Alias

Applies a 2 to 1 or 4 to 1 averaging filter to every pixel plotted, to reduce jaggies and other highfrequency noise. This increases the processing time 4 to 8 times, so is mainly a final rendering method, not for general development use. Not available for most 3D-type fractals (for quaternions, only single-pass mode is supported -- or without ray-tracing, solid-guessing can be used), plot-tofile, midpoint-displacement, 3D backgrounds or with the tesseral solid-guessing method. Note: because of the lengthy time required for applying the anti-aliasing filter, and because anti-aliasing calculates different smoothing colors each time the palette is changed, all color-cycling and paletteswitching hot keys are disabled with the anti-alias flag set.

5.8.2

Biomorph

Biomorph

Biomorphs test the real Z and imaginary Z values after breaking the iteration loop. If the absolute value of either is less than the preset zlimit, the point is mapped as part of the set. This method produces biological-like structures in the complex plane. Normally the biomorph tendrils are colored in the set color (the color reserved for non-divergent or inner points.) With the Set Only flag on, the tendrils are colored according to the color-scaling option used (other external points are colored in the background color.) A window is opened each time this option is selected to set a color for the area that falls within the biomorph trap. This can be 0-235.

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Fine-Tuning menu 44

5.8.3

Bioconvergence

Bioconvergence

This option relates the Biomorph method to convergence for convergent-type fractals (Newton,

Renormalization and Convergence.) A pseudo-biomorphic algorithm is applied to converging points.

5.8.4

Biomorph Off

Biomorph Off

Turns off the biomorph flag, including bioconvergence too. Alternatively you can enter -1 in the

Biomorph window to turn off the bio-flag.

5.8.5

Newton Set

Newton Set

The Newton flag is used to map the zeros of a particular function after the Newton transformation has been applied to the function. The program doesn't make the transformation (z-(f(z)/f'(z)), where

' stands for d/dx), but it does allow you to map up to 6 attractors. Each time the Newton flag is set, a window is opened to allow you to enter up to 6 attractors (or repellers) of the function.

This flag is mutually exclusive with the Boundary Scan, Convergence and Renormalization flags, and automatically excludes all points that don't converge to one of the attractors set, within the preset number of iterations. The points that converge are colored with one of up-to-6 possible color spreads (the built-in functions may allow more colors) evenly spaced in the current palette, according to the root they converge to and the time it takes to converge. The non-converging points are mapped with the set color or their level set color (with a level flag set) after the maximum number of iterations. Non-converging points show up typically as round areas or spots.

Generally, a limit of 50 iterations gives optimum results. The Newton transformation is normally used with Julia sets, as the attractors (solutions of the formula) can be calculated beforehand. It's also possible to explore the Mandelbrot set applied to Newton's method, but only with some of the builtin formulas mentioned above. It this case, the solutions of the formula for every point on the screen have to be calculated separately, which the program does in a dedicated routine.

5.8.6

Newton Off

Newton Off

Turns off the Newton flag, otherwise this option is disabled.

5.8.7

Renormalize

Renormalization

The Renormalization flag uses a hierarchical lattice transformation to map magnetic phases, with either the Julia set or Mandelbrot set as the iterated function. (Consult The Beauty of Fractals by

Pietgen and Richter for appropriate formulas to use.) Basically, the default-mapping algorithm checks orbits for convergence to 1 or infinity, and scales these points in different colors. This flag is

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45 Fractal Zplot Application Help

mutually exclusive with the Newton, Convergence and Boundary Scan flags. The default method of display actually only checks if z passes through 1. This is similar to the epsilon-cross mapping method. For the original renormalization formulas, there is a strong orbital attraction to 1. For other functions, this mapping produces unusual effects (with obscure mathematical foundations.) For the built-in functions, the convergence tests (type 2 or 3 and 6-8 display types) are the same as the ones used with the Newton flag. So either method produces similar results with the same formula. The differences are worth playing with, though.

Original picture

With renormalization

5.8.8

Convergence

Convergence

With the Convergence flag set, the program does a convergence/periodic check on all points. This is similar to the convergence checks done with Newton and Renormalization, but also the orbits of each point are saved to determine if the orbit repeats. When an orbit repeats, the iteration loop is broken and the point colored according to its break time. Depending on the iteration limit, the last

200 points of each orbit are tracked for this check. This flag is mutually exclusive with the Newton,

Renormalization, and Boundary Scan flags. May use Newton display methods 3 and 6-8(alternate convergence tests), by setting the arg gadget to these values.

5.8.9

Default Function

Default Function

When this option is enabled (off by default), convergent functions are iterated according to their original type. Fractal ViZion allows treating a renormalization curve as a Newton curve, or vice versa, but the governing flag must be set through the flags menu. The Default Function option allows a built-in function to work as a Newton or renormalization curve without those flags being set.

Newton functions work only as Newtons and likewise for renormalization formulas. Function types

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Fine-Tuning menu 46

that use two built-in functions can distinguish between convergent and non-convergent formulas and use the suitable escape or convergent checking for each formula.

5.8.10 Level Curve

Level Curve

Level-curves map the set points based on how small the value of Z gets. This allows the inside of the complex set to be color-scaled. Log Map #1 produces colored bands on the inside of the complex set. Points are mapped according to what the value of z is at final iteration. Small Log #2 and Linear Map #5 produce circular patterns inside the complex set. Points are mapped according to the smallest value z gets during iteration. Indexed Log #3 and Indexed Linear #6 are mapped according to the time it takes z to reach its smallest value. Level curves 2,3(and 5,6) are described more fully in The Beauty of Fractals. Linear Map #4 is mapped like Log Map #1 (with the mapped value of the function at its final iteration applied to the color palette) and produces 3D-like effects with the Epsilon-Cross method. The Log methods use a log palette, while method #4-6 use linear palettes. This option can override (or may be overridden by) many of the options in the Color-

Scaling menu. Decomposition doesn't use Level Curve shading, unless you select the Use Level

Curve option.

Log Map #1

Small Log #2

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47 Fractal Zplot Application Help

Indexed Log #3

Bubble #7 uses Paul Carlson's contour-mapping method to produce 3D-like bubble pictures. The method is very sensitive to which formula is used, working best with the basic Mandelbrot set z^2+c and the like. Color-mapping should be set to Use Level Curve. This is a trial and error method that uses two other variables to produce the final effect, magnify and cutoff, as entered in the Parameters window. Magnify is used to screen unwanted background contours in the plot, while cutoff is used to fill out the color palette. Magnify should be a low value, usually less than .1, to eliminate the contours that usually appear in escape-type Mandelbrot/Julia sets. If it is too large the bubbles will be too crowded, while too small a value will cause the bubbles to disappear. Cutoff needs to be a small negative value, usually equal to the magnify value times the number of color splits. E.G., for a magnify value of .1 you should use a cutoff value of -.8 for a divide-by-eight palette. An incorrect cutoff value will cause the colors to overlap in the bubbles. For split palette pictures, the colors are divided according to their level index, as in Indexed Linear #6. The color ranges should be graded from light to dark to highlight the bubble centers.

5.8.11 Decomposition

Decomposition

When a Decomposition flag is set, you have the option of performing either a binary or continuous decomposition. Toggle the External/Internal option for either an external or internal decomposition.

The Angle-Only option excludes the Coloring-scaling options from consideration when plotting points derived from decomposition. It also excludes escape times in connection with the Angle-

Iteration option on the Color-Scaling menu. An external decomposition decomposes points that are outside the complex set. An internal decomposition decomposes the complex set. For Mandelbrot/

Julia curves, z-arg is broken into two parts for a binary decomposition. For Newton/

Renormalization curves, the binary decomposition is also related to the number of solutions a formula has, if it supports mapping option 1. Continuous decomposition breaks z-arg into n parts, where n=angles (2-256), as set in the Continuous Decomposition window.

Note: With the graded-palette option checked, the decomposition option is extended for extra smoothness in Fractal ViZion. The number of angles is internally multiplied by 236 to track the decomposition angle more closely.

(Consult The Beauty of Fractals by Peitgen & Richter for a mathematical explanation of decomposition.) When Biomorph or Epsilon is decomposed, the tendrils or hairs are decomposed as external points. Use the Set Only flag to emphasize the tendrils and hairs when external decomposition is used.)

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Fine-Tuning menu 48

Binary Decomposition

Continuous External

Continuous Internal

5.8.12 Decomposition Off

Decomposition Off

Turns off all decomposition flags and resets the Internal/Eternal option to Eternal.

5.8.13 Spin

Spin

When the Spin flag is set, the complex constant is incremented by a scaled factor of cx(cx*c/ iterations) at every step of iteration. Unavailable for Julia Tower types.

5.8.14 Torus

Torus

Pixels are mapped around a torus, and then expanded to fit the drawing area. A generalized form of

Earl Hinrichs' torus method, variables are provided for center x and center y to define the c and z

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49 Fractal Zplot Application Help

radii and may both equal 0.0. Results will vary with the formula used, but resembles the warping effect found in hypercomplex images. Two versions of this method are provided: the Pixel method which uses pixel values to map the torus to the fractal space, and the Two-Pi method which uses an initial rectangle 2 pi by 2 pi to map the torus to a fractal image. With the Two-Pi method, when you zoom the rectangle's size and starting points are changed to match the zooming area. The rectangle's coordinates are saved with the fractal. If you turn off the torus flag after zooming and then reinitialize the torus flag, the rectangle reverts to a 2X2 area, so the image will change accordingly. Rotating is not supported for the Two-Pi method, but does work in a limited way with the Pixel method.

5.8.15 Torus Off

Torus Off

Turns off the torus flag. Alternatively you can enter a negative value to turn off this flag.

5.8.16 Map menu

Map menu commands

The Map menu offers the following commands:

Z-Real

Z-Imag

Abs(Z-Real)

Abs(Z-Imag)

Mapping based on real part of z only.

Mapping based on imaginary part of z only.

Mapping based on absolute value of real part of z.

Mapping based on absolute value of imaginary part of z.

Z-Real + Z-Imag

Mapping based on sum of parts of z.

Abs(Z-Real)+Abs(Z-Imag)

Mapping based on absolute value of parts of z.

>Abs(Z-Real) or Abs(Z-Imag)

Mapping based on highest absolute value of parts of z.

<Abs(Z-Real) or Abs(Z-Imag

Mapping based on lowest absolute value of parts of z.

Abs(Z)

Mapping based on absolute value of z.

5.8.16.1 Z-Real

Z-Real

Map based on the real part of the complex number Z; used to map exponential Julia sets, etc.

5.8.16.2 Z-Imag

Z-Imag

Map based on the imaginary part of the complex number Z.

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5.8.16.3 Abs(Z-Real)

Abs(Z-Real)

Map based on the absolute value of the real part of the complex number Z; used to map exponential

Julia sets, etc.

5.8.16.4 Abs(Z-Imag)

Abs(Z-Imag)

Map based on the absolute value of the imaginary part of the complex number Z.

5.8.16.5 Z-Real+Z-Imag

Z-Real + Z-Imag

Map based on the sum of the real part and the imaginary part of the complex number Z. Changes the way banding appears in complex mappings.

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51 Fractal Zplot Application Help

5.8.16.6 Abs(Z-Real)+Abs(Z-Imag)

Abs(Z-Real) + Abs(Z-Imag)

Map based on the absolute value of the real part plus the absolute value of the imaginary part of the complex number Z. Changes the way banding appears in complex mappings.

5.8.16.7 >Abs(Z-Real) or Abs(Z-Imag)

>Abs(Z-Real) or Abs(Z-Imag)

Map based on the greater of the absolute value of the real part or the imaginary part of the complex number Z. Works like a logical 'or', where either part of z must exceed zlimit to break the iteration loop. Changes the way banding appears in complex mappings.

5.8.16.8 <Abs(Z-Real) or Abs(Z-Imag)

<Abs(Z-Real) or Abs(Z-Imag)

Map based on the lesser of the absolute value of the real part or the imaginary part of the complex number Z. Works like a logical 'and', where both parts of z must exceed zlimit to break the iteration loop. Changes the way banding appears in complex mappings.

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5.8.16.9 Abs(Z)

Abs(Z)

Map based on the absolute value of the complex number Z (traditionally calculated by taking the square root of the sum of the squares of the real and imaginary parts of Z, but Fractal ViZion uses only the 'sum'(modulus of z) for break-point tests.) The standard method of mapping Julia and

Mandelbrot sets.

5.9

3D Render menu

The 3D Render menu offers the following commands:

Texture Scale

Set scaling factor for texture.

Atan Coloring

Bof60 Coloring

Use Atan algorithm for coloring.

Use Bof60 algorithm for coloring.

Potential Coloring

Color by magnitude of z.

Surface Filter

Add Noise

Factors

Define surface filter.

Add noise to coloring.

Edit noise factors.

Reset Noise Seed

Re-seed random noise generator.

Coloring Filter

Define coloring filter.

5.9.1

Texture Scale

Texture Scale

Opens a window to edit texture scale factors. The higher the scale factors, the more repetitive the texture becomes. You can adjust the factors to make the texture asymmetrical on the x, y or z-axis.

Scale A is used to adjust the texture scale for the atan and Bof60 coloring options.

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53 Fractal Zplot Application Help

5.9.2

Atan Coloring

Atan Coloring

Uses an Atan algorithm by David Makin to color a quaternion image.

5.9.3

Bof60 Coloring

Bof60 Coloring

A variation of the Bof60 algorithm found in the classic Pietgen/Ricter text, The Beauty of Fractals, adapted by David Makin to color a quaternion image.

5.9.4

Potential Coloring

Potential Coloring

The magnitude of z (at the quaternion border) is used to color the (quaternion) image.

5.9.5

Surface Filter

Surface Filter

Here you define a quaternion surface filter based on a real function. Like a coloring filter, except that the formula is used to warp the quaternion's shape. The Magnify variable is used to control the intensity of the filter. Use the Preview button to see what the filter looks like with x and y ranges of -

2pi to 2pi.

Use the Random Filter button to generate a random surface filter. The best surface filters will use the z value and one or both of the other variables (x or y.)

5.9.6

Add Noise

Add Noise

Add noise to (quaternion) image texture. A variation of Perlin's noise algorithm is used to add natural randomness to an image's coloring.

5.9.7

Factors

Factors

Edit noise factors. The blend variable determines how much noise is added to an image. The higher the blend, the more pronounced the noise appears. This also tends to darken an image, which can be compensated for by decreasing Gamma. The Grain variable determines the frequency of the noise. The higher the grain, the noisier the image appears. You can adjust how the noise maps to an image by changing the scale factors. Higher scale factors make the image noisier on the respective axis (x, y and z.)

The Surface Warp variable allows you to apply the same noise to a (quaternion) figure's shape also, like a surface filter. Small values are best for creating realistic surface variations, like stone and wood grain.

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Fine-Tuning menu 54

5.9.8

Reset Noise Seed

Reset Noise Seed

The random noise generator is re-seeded. Use this to create variations on the noise texture.

5.9.9

Coloring Filter

Coloring Filter

Here you define an hsv filter based on a real function. A generalization of Earl Hinrichs' sine-wave coloring method, the function can be any formula, up to 80 characters, that uses the z components x and y. X and y are the real and imaginary parts of the last z value in the iteration loop. Sample function: sin(x+y)+cos(x*x). The Magnify slider is used to control the intensity of the filter (in conjunction with the Magnify variable in the HSV filter window.) Use the Preview button to see what the filter looks like with x and y ranges of -2pi to 2pi.

Note: the coloring filter can also be used with quaternions, ray-traced orbital fractals and 3D height fields (when the Ray Trace option is selected) to map palette indexes or spreads to the x/y ranges of these fractal types. The variable 'z' is equated to the zbuffer value in this case, so may also be included in the formula. Quaternions and ray-traced fractals normally use palette index one (the second index, zero being reserved for the background color) for their predominant color, with pixel intensities/colors affected by the lighting variables. When the coloring filter formula is defined, up to

235 colors can be used (the full palette) to create mixed textures.

The trig and exponential functions translated include sine (sin), arc sine (asn), cosine (cos), arc cosine(acs), tangent (tan), hyperbolic tangent (th), hyperbolic sine (sh), hyperbolic cosine (ch), log

(log), natural log (ln), power (pow), arc tangent (atn), absolute value (abs), exponential (exp) and square root (sqr.)

The math functions are *(multiply),-(subtract),/(divide), and +(add).

The constants are PI and E (ln (1)), plus any floating-point number up to 9 digits (including the decimal point).

The power function (x to the y power) is entered in standard notation: x^y', with optional parenthesis necessary around complex exponents or variables.

Note: Range limits exist for arguments to these functions: exp, arc sine, hyperbolic sine, arc cosine, hyperbolic cosine, arc tangent, and hyperbolic tangent (+/-100.0 for the exponential , +/-200.0 for hyperbolic functions, +/-1.0 for the arc functions), the log functions (must be >0) and the power function (x must be integral and non-zero when y<0, and 0^0 is undefined). Square root is undefined for x<0. No filtering is done when these limits are exceeded.

Syntax for an acceptable formula is AS([XY])+bs([xy])...

.up to 80 characters per formula. Algebraic notation is supported to a limited degree. E.G. you can enter a variable as 2x^2, instead of 2*x*x.

A and B are optional constants.

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55 Fractal Zplot Application Help

S is an optional trig function (1 to three letters: 1 will work for sine, cosine and tangent, but use the above abbreviations for the other functions. X and Y are the standard variables. The '+' could be any of the math functions.

The parser interprets up to 10 levels of parenthesis. Use parenthesis to separate complex expressions. Use parenthesis to embed trig functions within other trig functions, etc.

5.10

Orbit traps

Orbit Traps

This includes methods that trap the orbit of a point if it comes in range of a pre-specified area or areas.

The Epsilon-Cross method colors points only if the absolute value of Z-real or Z-imaginary is less than or equal to Epsilon (a small value.) Other points are mapped at the time they blow up

(exceed the zlimit.) This produces hair-like structures that branch wildly from the complex set boundaries. For the Epsilon-Inside option, the epsilon method is applied only to points included in the set. For the Epsilon-Outside option, the epsilon method is applied only to points outside the set.

The Globe method uses a circular area around the origin to map a point's orbits. This produces sphere-like structures.

The Ring method uses an area formed by two circles around the origin to map a point's orbits. This produces ring-like structures.

The Four-Circles method (Paul Carlson) uses four circular areas to map a point's orbit. This produces sphere-like structures.

The Square method uses an area formed by two squares around the origin to map a point's orbits. This produces ring-like structures with right angles.

The Petal method (Paul Carlson) also uses four trap areas to form flower-like patterns.

Epsilon2 is used to create windows into the stalks. The default value is 0.0, which produces solid stalks.

A window is opened to enter a value for Epsilon and Epsilon2, which are used to define the size of the trap areas (.001-2.0 and 0.0-epsilon.) The exclude box is used to exclude the first # iterations (0-99) from orbit trapping.

To produce the maximum 3-D effects (as Phil Pickard and Paul Carlson do) with these options, Level Curve #4 must be set, and the Cutoff value (in the Parameters window) should equal the negation of the epsilon value (-epsilon.) You'll need to set up a special palette with a number of color ranges that matches the split-palette number if set. Built-in examples d3-d6 illustrate how to set up 3d-like fractals.

To automate the process of producing Paul Carlson's 3-D like fractals, a check box has been added to this window for 'Carlson extensions'. This sets the Background and Set Only flags as well as the Level Curve #4 and the cutoff value, and sets the Map to <Abs(Z-Real) or Abs(Z-

Imag). An exclude value of 2 is also used. The Map and exclude values are extra parameters that

Paul uses in some of his formulas, and may be omitted in other cases. This is easily done by first

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Fine-Tuning menu 56

enabling the Carlson extensions by checking the box and clicking on Okay, then opening the window again and un-checking the box and changing the appropriate variables/flags.

5.10.1 Orbit trap values

Orbit Trap Values

Enter a value for Epsilon and Epsilon2, which are used to define the size of the orbit trap areas (.001-2.0 and 0.0-epsilon.) The exclude box is used to exclude the first # iterations (0-99) from orbit trapping.

Click on Apply to apply changes without closing the window. Click on Okay to close the window and apply changes, if any. Click on Cancel to exit the window without changing parameters.

Epsilon2 is used to create windows into the stalks. The default value is 0.0, which produces solid stalks. Epsilon2 has no effect on the Petal method.

Check Carlson extensions to apply additional options that are used in Paul Carlson's 3D-like orbittrap methods.

5.11

Draw command

Draw command

Use this command to draw or redraw the image for the current fractal variables. Clicking inside the draw window with the left-mouse button stops all plotting. Use the Continue command to restart

plotting from the current column. When the lsystem option is selected, a custom window

is opened

to enter lsystem draw options.

5.11.1 Draw Lsystem command

Draw (Lsystem) command

The lsystem draw window has edit boxes to set projection angles, roll pitch and yaw. The angles have a range of -360 to 360 degrees. The initial turtle heading is 0,1,0 -- which moves the turtle up in the direction of the y-axis when the projection angles are all 0. The view may be improved when some of the projection angles are non-zero, or to rotate an object done with LParser to an appropriate viewing angle.

The skewing angles affect the line styles when the line width is greater than 1. These can warp the lines(tubular or poly) to a non-uniform width.

The light point is similar to that used when Fractal ViZion draws 3D quaternions.

The Lighting variables shininess, highlight, gamma and ambient are used to adjust ambient light and

Phong highlights. The ranges for these variables appear beside their label. Decreasing the shininess value increases light reflected by the quaternion and the apparent sheen on the quaternion's surface.

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57 Fractal Zplot Application Help

The ambient value controls the amount of ambient light that illuminates the quaternion. The highlight value increases or decreases the specular (Phong) highlighting, while the gamma correction increases or decreases the intensity of the light source's illumination.

Note: once a plot has been drawn, the lighting variables and light point can be changed without redrawing the lsystem. Click the light bulb button to redisplay a picture after changing the lighting variables or light point.

You enter the level of recursion into the level box. The maximum level of recursion is limited to 50.

Use the scale slider to set the size of the drawing, .05 to 1.0. Each lsystem is pre-scaled prior to drawing, to approximate the size of the draw window. To speed up the pre-scaling, the width of the line drawn isn't used to map image limits. So it's possible an object may extend off the window's edges, when a line width is greater than 1 and the scale is 1. In this case, use a scale less than 1 to

"shrink fit" the drawing.

Click on Draw to recompute and draw the current axiom.

Select Wire Frame to draw a wire frame model of the current axiom/rules. This is useful to speed up drawing while prototyping an lsystem.

Enter additional Steps (default 1), to increase the resolution of solid rendering. Each additional step increases rendering time proportionally, but can increase the smoothness of certain area fills, as in polygon rendering of large objects.

Enter a non-zero mutation value to mutate the current lsystem. This follows the same mutation rules as LParser. Small values (less than 5) can produce extremely lengthy mutations. Use Fixed

Mutation to limit the size of the mutated lsystem to the original size. In this case the no rules are appended or inserted. Only angle and movement commands are switched.

Select the extended set box to enable the extended rule set. Most input files made for LParser will work with the extended set enabled, but the extended rules may not work if the command is overridden by a production rule.

5.12

Continue Draw

Continue Draw

Continues a plot that was aborted early. The plot is restarted at the beginning of the last row drawn.

Continue is disabled when an Image/Merge option is selected.

5.13

Full Screen

Full Screen (Image menu)

Displays the entire plot, expanding or shrinking the image to fit in a maximized window without title bar, scroll bars or menu bar. At all other times, part of the picture is hidden by the inclusion of the title bar, toolbar, scroll bars and menu bar.

To exit full-screen mode, press any key or click the left-

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Fine-Tuning menu 58

mouse button.

5.14

Mandelbrot

Mandelbrot

Mandelbrots base their mapping on varying inputs of complex C, which corresponds to the min/max values set in the Parameters window. With Mandelbrot0, Cr and Ci represent the initial value of Z before the first iteration. This is normally zero, but can be changed to produce non-symmetrical

Mandelbrots, or Mandelbrots based on formulas whose initial value of Z must be non-zero to generate anything (such as the trig-based Mandelbulb formulas.)

Mandelbrot set

5.15

Julia

Julia

Julia sets normally have a fixed complex C, with varying inputs of Z, which corresponds to the min/ max values set in the Parameters window. This option, without the Bound flag set, generates the socalled 'filled-in' Julia set, which includes non-escaping points as well as the Julia set.

Julia from Mandelbrot

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59 Fractal Zplot Application Help

Julia set

5.16

Invert

Invert

The Invert flag inverts the plane around a circle. A window is opened that allows the user to specify the circle's radius and center coordinates. Select Auto Coords to let Fractal ViZion calculate the center coordinates and circle radius. Using Auto Coords, the new radius and center coordinates are calculated when the picture is next drawn. You can zoom on an inverted picture as long as radius and center coordinates remain the same. Use the Perspective box to alter the X/Y symmetry of the inversion. A smaller Perspective value (less than 1.0) stretches the inversion in the vertical direction.

Original picture

Inverted

5.17

Invert Off

Invert Off

Turns off the inversion flag. Alternatively you can set the inversion radius to 0.0 to turn off inversion.

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Fine-Tuning menu 60

5.18

Switch

Switch

When a Switch flag is set, you have the option of switching the real and imaginary parts of Z, or switching Z for C. The real part of Z is exchanged with the imaginary part of Z after each iteration.

Using this technique with the Mandelbrot set produces a tricorn-like plot. When Z is switched for

C, normally you get Mandelbrots from Julia sets and vice versa. This option also offers an inverse to the Julia Tower, with glimpses into the elusive Mandel Tower, a sort of gateway to infinity.

5.19

Filter

Filter

Based on Stephen C. Ferguson's filter algorithms in his program Iterations, this option allows you to choose one of 29 tail-end filters to apply to any 2D plot. The name of the filter corresponds roughly to its effect on the basic Mandelbrot-squared set. The effect will vary with the formula and fractal type chosen. This overrides the Background option on the Color-Scaling menu. Useful to add detail to orbit-trap pictures, as well as perk up any otherwise ordinary picture. Filters 27-29 are generalized filters that use fn4 and fn3 (in the edit/formula window) for expanded scope. The "cross" filters use the epsilon variable from the orbit-trap window. (To set epsilon's value without applying an orbit-trap, first change the value of epsilon to the value desired, and set the orbit-trap flag. Then turn off the orbit-trap flag. Epsilon value remains unchanged when the orbit-trap flag is turned off.)

The Magnify variable is used to intensify or de-intensify the effect of the filter. This value can range from 1-500 nominally. The Add Offset box is checked when you want the filter to add an offset to the color value normally plotted. The Exclude Background box works like the Add Offset box, except that background pixels are unfiltered. With the Replace All box checked, the filter totally replaces the normal color value, which can lead to very different color-rendering. With the

Background Only box checked, only pixels which would normally be colored with the background color (index 0) are filtered.

5.20

HSV Filters

HSV Filters

This option allows you to choose up to 3 hsv filters to apply to any 2D plot. These effectively extend the range of the palette used, or reassign colors independent of the limits of the palette. The filters may be assigned to modify hue, saturation or value component of the pixel's normal color.

The filters may be Iteration (time-based), Z-Potential (last z), Exit Angle (polar-based), Level Index

(time/magnitude based), Coloring Filter (formula/magnitude based), Z-Potential (smallest z), or Orbit

Trap. Formula-type orbit traps default to Epsilon Cross for this option.

The Magnify variable is used to intensify or de-intensify the effect of the filter. Begin with 0.5 and increase or decrease as necessary.

The Start slider defines the origin of the filter so that it acts as a decrement or increment for various filter values. For the Iteration filter the values correspond to 0-Max Iterations. Start values range

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61 Fractal Zplot Application Help

from 0-100 percent of the nominal filter range. For the Iteration filter, any Start value greater than 0 would make all Iterations less than the Start value decrements. The Exit Angle normally ranges from

-PI to PI, so a Start value of 50 would make all angles less than 0 decrements. The Z-Potential filter has a range of 0-zlimit^2, for values that do not escape the iteration loop. For escape values, a modulus is used to bring the filter within hsv parameters. For the Level Index filter and Coloring filter, the Start slider represents a decrement/increment of -1 to 1.

The Add Offset box is checked when you want the filter to add an offset to the color value normally plotted. With the Replace All box checked, the filter uses only the offset for the color value, which can lead to very different color rendering. This differs somewhat from how the Replace All box in the Filter's windows works. Here the offset is still loosely tied to the pixel's normal color. With the

Background Only flag checked, only pixels which would normally be colored with the background color (index 0) are filtered.

Notes: The Level Index filter was designed specifically for modifying a pixel's hue, though may produce interesting results applied to a pixel's saturation or value component. Hsv filtering is not available with solid-guessing. Use the Coloring Filter window to define a coloring formula. Sample coloring formula: sin(x*x)+cos(y*y).

5.21

Pilot

Pilot

Opens the Pilot window to adjust key parameters, rotate, zoom and redraw the figure interactively.

The current image is reduced to one quarter normal for faster redraw. Each click on a Pilot button increments or decrements a parameter. The Speed slider controls the rate at which the buttons operate (default is 10.)

6

Press the space bar or Click on Ok to open a new window and draw the altered image full-size.

Press Esc or click on Cancel to exit this mode without opening a new window. Note: when using this option while an AVI stream is open, a new window isn't opened, but the altered figure is drawn in the current draw window, the changed parameters replacing the previous ones.

View menu

View menu commands

The View menu offers the following commands:

Toolbar

Shows or hides the toolbar.

Status Bar

Shows or hides the status bar.

Phoenix

Phoenix Curve.

Cycle

Cycle colors.

Symmetry-> Horizontal, vertical or XY symmetry.

Zoom

Zoom into rectangle.

New View on Zoom

New view on zoom.

Reset->

Reset coordinates, current figure or all figures

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View menu 62

6.1

View Toolbar command

Toolbar command (View menu)

Use this command to display and hide the Toolbar, which includes buttons for some of the most common commands in Fractal ViZion, such as File Open. A check mark appears next to the menu item when the Toolbar is displayed.

See

Toolbar

for help on using the toolbar.

6.1.1

toolbar

Toolbar

The toolbar is displayed across the top of the application window, below the menu bar. The toolbar provides quick mouse access to many tools used in Fractal ViZion,

To hide or display the Toolbar, choose Toolbar from the View menu (ALT, V, T).

Click To

Open the remote which contains shortcut buttons for many common tasks and options in

Fractal ViZion

Use Pilot to rotate, zoom and adjust key parameters of 3D figures

Open an existing drawing. Fractal ViZion displays the Open dialog box, in which you can locate and open the desired file.

Save the active drawing or template with a new name. Fractal ViZion displays the Save

As dialog box.

Draw Mandelbrot set

Draw Julia set

Zoom into rectangle.

Set image size.

Edit palette.

Edit formula/type data.

Edit fractal parameters.

Draw image from current parameters.

Continue drawing.

Reset coordinates.

Show picture full-screen.

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63 Fractal Zplot Application Help

Display info about Fractal ViZion.

6.2

View Status Bar Command

Status Bar command (View menu)

Use this command to display and hide the Status Bar, which describes the action to be executed by the selected menu item or depressed toolbar button, and keyboard latch state. A check mark appears next to the menu item when the Status Bar is displayed.

See

Status Bar

for help on using the status bar.

6.2.1

status bar

Status Bar

The status bar is displayed at the bottom of the Fractal ViZion window. To display or hide the status bar, use the Status Bar command in the View menu.

The left area of the status bar describes actions of menu items as you use the arrow keys to navigate through menus. This area similarly shows messages that describe the actions of toolbar buttons as you depress them, before releasing them. If after viewing the description of the toolbar button command you wish not to execute the command, then release the mouse button while the pointer is off the toolbar button.

The right areas of the status bar indicate which of the following keys are latched down:

Indicator

CAP

NUM

SCRL

6.3

Phoenix

Description

The Caps Lock key is latched down.

The Num Lock key is latched down.

The Scroll Lock key is latched down.

Phoenix

The Phoenix flag rotates the planes, so that the imaginary plane is mapped horizontally and the real plane is mapped vertically.

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This option is normally used for mapping Phoenix curves (Shigehiro Ushiki), which are Julia-related curves based on the formula f(z+1)=z²+p+qz. 'p' and 'q' are constants, and the 'z' term of 'qz' is actually the value of z^n-1, or the previous value of z before the current iteration. 'zn' is reserved by

Fractal ViZion to represent this value, while the complex constant set in the Parameters window becomes 'p' and 'q'. The real part of the complex constant is 'p' and the imaginary part of the constant is 'q' (when the Phoenix option is chosen).

If the Phoenix flag is used with the Mandelbrot option, 'j' and 'k' should be used as the constants, since the complex constants p and q are already used as the starting value of 'z0'.

6.4

Cycle command

Cycle command (Color menu)

Use this command to cycle colors when not plotting. Works with any coloring mode, but not with hsv filtering or anti-aliasing. Undoing an action disables the cycle command until the image is redrawn.

6.5

Symmetry

Symmetry

This produces a mirror image from left to right (vertical) or top to bottom (horizontal) or both (xy).

You can zoom with symmetry, but the results will be uncertain if the zoom box is off-center on the window or if rotation is used. Symmetry has no effect when used with the Tesseral option, 3D, or random fractals.

Vertical symmetry

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65 Fractal Zplot Application Help

Horizontal symmetry

XY symmetry

6.6

Zoom

Zoom

Turns on zoom mode, so that detail of the current plot may be magnified. Alternatively, just click inside any drawing window, move the mouse, and the zoom box will appear. Using the mouse, move the zoom box over the portion of the plot you wish to magnify. Hold the left mouse button to shrink the box or the right button to enlarge it. Use the left and right arrow keys to rotate the box counter-clockwise or clockwise. Use the up and down arrow keys to squash or expand the box, changing the aspect of the image. You start a zoom by pressing the space bar. You abort a zoom by clicking outside the main window or in the title bar, or by pressing the escape key. The program will begin a new plot at the new coordinates. You may zoom in by defining a box inside the current drawing area. You zoom out by drawing a box outside the current drawing area. The outer zoom limits are between -1000 and 1000. The precision is that of double precision (64 bits)

Note: Zooming in a three-dimensional plot is not supported, except for quaternion plots, nor is zooming on a random (midpoint displacement) fractal. Rotating a quaternion while zooming is possible, but inexact. Depending on the rotational angles set in the quaternion window, you may end up rotating on the z-axis, which may rapidly result in overshooting the area of zoom. So avoid large rotations combined with high levels of zoom. It is better to rotate at low zoom levels, and then zoom without rotating, for close-ups. If you change screen resolutions, you must redraw the bitmap image for a function before you can accurately zoom on it.

Zooming with orbital-type fractals turns on magnify mode, so that details of the current screen may be examined. The selected area is magnified to fill the current sector or screen. The x and y ranges

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do not change though, so the magnified area may not show up all at once, since other areas of the function may have to be recalculated first. The magnify mode can only be used on one area of a screen at a time. Magnifying is then disabled until a new map is generated though the New button in the Edit/Orbit Function window. You can increase the resolution of a magnified map, though, via the iterations slider. Use Okay to redraw the map at increased resolution. Rotating is not supported with orbital fractals.

6.7

New View on Zoom command

New view on zoom (View menu)

With this option enabled, a new window is opened with each zoom, instead of the zoom box area replacing the original image. Ignored in avi mode.

6.8

Reset

Reset

Reset the current figure or all figures to an empty Mandelbrot. All functions in the New Formula data are blanked. All options on the Flags menu are reset to their default settings. The Print

Function Data ignores any reset figures.

The Ranges Only command resets only the real Z and imaginary Z ranges in the Parameters window

(to +/-2.0 and +/-1.5.) No other menus or variables are affected. This is useful in conjunction with the "P" command to generate and view Julia sets. After setting the complex-C variable via shift-P

(Caps Lock off), you need to reset the Z ranges to see the entire Julia set after zooming into a

Mandelbrot set. The Reset All option resets all figures.

6.9

Solid Guessing

Solid Guessing

In the solid-guessing plotting mode, the program guesses at colors that lie inside rectangular areas of the plot. It first computes all the perimeter pixels of a rectangle, and checks if all the pixels have the same color. If so, all the pixels inside the rectangle are colored the same and no further calculations are done on that rectangle. Otherwise the rectangle is broken into four parts and the above procedure is repeated for each part. If any of the perimeter pixels are different at this point, all the remaining pixels in the smaller rectangle are computed. The screen is updated in groups of 16 lines.

This method can be much faster than the default single-pass mode that Fractal ViZion starts in. This is especially true for plots that have large areas of a single color. For very intricate plots that have little open space, the solid-guessing mode can still be 15-20% faster.

The solid-guessing mode can fail for some plots that have small areas of one color that are islands inside the rectangular areas under test.

Solid-guessing is not available for lsystems, orbital fractals, 3D height fields or midpoint displacement.

© 2003 - 2010 ... Mystic Fractal

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67 Fractal Zplot Application Help

Window menu

Window menu commands

The Window menu offers the following commands, which enable you to arrange multiple images in the application window:

Cascade

Tile

Arrange Icons

Arranges windows in an overlapped fashion.

Arranges windows in non-overlapped tiles.

Arranges icons of closed windows.

Size Desktop

Size drawing area to window frame.

Window 1, 2, ...

Goes to specified window.

7.1

Cascade

Cascade command (Window menu)

Use this command to arrange multiple opened windows in an overlapped fashion.

7.2

Tile

Tile command (Window menu)

Use this command to arrange multiple opened windows in a non-overlapped fashion.

7.3

Arrange Icons

Window Arrange Icons Command

Use this command to arrange the icons for minimized windows at the bottom of the main window. If there is an open drawing window at the bottom of the main window, then some or all of the icons may not be visible because they will be underneath this drawing window.

7.4

Size DeskTop

Window Size DeskTop Command

Use this command to size the active drawing window to its frame size. Use after Tile command to reduce white space around a drawing that is smaller than screen size.

7.5

1, 2, ...

1, 2, ... command (Window menu)

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8

Fractal ViZion displays a list of currently open drawing windows at the bottom of the Window menu. A check mark appears in front of the drawing name of the active window. Choose a drawing from this list to make its window active.

Help menu

Help menu commands

The Help menu offers the following commands, which provide you assistance with this application:

Getting Started

Index

Hot Keys

Tutorial for new users of Fractal ViZion.

Offers you an index to topics on which you can get help.

Quick reference to Fractal ViZion's hot keys.

Parser Info

Quick reference to Fractal ViZion's parser variables and functions.

Built-in Formulas

Quick reference to Fractal ViZion's built-in formulas.

Bibliography

Lsystem info

Sources for fractal information and complex numbers.

Quick reference to Fractal ViZion's lsystem (description and syntax).

About Fractal ViZion

Displays the version number and author info for this application.

8.1

Getting Started

Welcome to Fractal ViZion!

This is a short tutorial that will cover some of the basic commands to make the most of Fractal

ViZion. For help on any menu command, press shift-F1 while the command is highlighted.

When Fractal ViZion is first started, a random fractal is automatically generated. There are nineteen channels of randomly generated fractals that can be accessed via Fractal ViZion's remote. Use the toolbar icon to hide/show the remote, or Ctrl-R. Since most fractals FV is capable of generated are configured automatically, including a random palette, I will touch briefly on some of the things a new user might want to explore to enhance the fractals created thus. In the end, the enhancements and touchups are what will give your fractals their unique personality.

The most basic changes to a fractal are coloring and framing. Use the palette editor to modify colors on a fractal. In some cases, the fractal will need to be redrawn to show the color changes, while other fractals that can be color-cycled will show the color changes immediately. You can zoom on most fractals by clicking in the draw window and using the mouse buttons to enlarge/shrink the zoom box, and use the left/right arrow keys to rotate the zoom box. Press the space bar to redraw the fractal using the new zoom coordinates, or click on the window's frame or title bar to abort a zoom.

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69 Fractal Zplot Application Help

Note: zooming is not available on lsystem or landscape fractals, and rotating may be disabled when some types of symmetry are used.

Additional changes to the fractal may be made via the FVR button on the FV remote. Most fractal types have additional variables that can be varied for animation purposes or for general esthetics.

Some specialized filters can be applied to the fractal via the Fine-Tuning menu, and symmetry can be altered via the View menu.

This completes the Getting Started tutorial. Be sure to read the

hot keys ,

palette editor and

remote

sections for additional info. The

Bibliography lists additional reference material for a better

understanding of the fractal types and functions contained in Fractal ViZion.

8.2

Index

Index command (Help menu)

Use this command to display the opening screen of Help. From the opening screen, you can jump to step-by-step instructions for using Fractal ViZion and various types of reference information.

Once you open Help, you can click the Contents button whenever you want to return to the opening screen.

8.3

Hot Keys

Hot keys

Ctrl+F1-Ctrl+F9,Ctrl+F11, Ctrl+0-Ctrl+9 --- change to one of 21 color palettes -- useable during plotting.

Ctrl+F12 holds the palette of the most recently loaded function.

Tab --- Replaces the currently-selected palette with the palette in F11.

Useful when you want to make a palette file(.pl) from the palettes in a lot of individual bitmap files.

Use the copy data and paste data commands to move the palette from another drawing window into

F11. Select the palette(Ctrl+F1-Ctrl+F9,Ctrl+F12,Ctrl+0-Ctrl+9) you want to move Ctrl+F11 into, then press Tab.

Hint: you can copy data from and to the same window, to move the currently selected palette into

Ctrl+F11.

up arrow --- forward cycle colors one step, including set color -- useable during plotting.

down arrow --- back cycle colors one step, including set color -- useable during plotting.

Shift-B -- erases the 3D background, leaving the 3D plot intact.

Shift-C -- clear the screen to the current background color.

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Shift-P --- grab point from Mandelbrot set(real and imaginary parts) and put values in complex constant. Cursor changes to a cross-hatch, which you position over the area of the Mandelbrot set of interest. Then click the left-mouse button to transfer the pixel's coordinates to the c constant.

Click outside window or in window frame to exit routine without "grabbing" a point.

Shift-J -- like "P", except that a Julia set is drawn immediately in sector 2 at size 100 and with iterations of 100. This is a fast exploratory routine for finding interesting Julia sets that can also be used where the Mandelbrot set is discontinuous as in the Phoenix formula. Unlike the "P" command,

"grabbing,"(and drawing) continues until you click on the window's frame. Note: when you exit the J command, once you find an interesting Julia set; another window is opened with the Julia type set.

The parameters in the original window revert to their original Mandelbrot settings.

Shift-G -- like "J", except that a quaternion set is generated in sector 2, using an iteration count of 10 and other parameters are changed temporarily to suit quaternion plots. (Hj is set to 2.0, the rotational variables are reset and the light source is set to -1, 1 and -3.) Quaternion math is used where possible. Once you find an interesting quaternion set using "G", like the J command another window is opened that sets the fractal parameters to those in the exploratory qjulia window (fast qjulia mode.) The parameters in the exploratory window revert to their original Mandelbrot settings.

Shift-W -- like "G" except

this modifies the cj and ck elements of the complex constant to locate a quaternion Julia set.

Shift-S -- a combination of the "G" and "W" commands. The "W" command is implemented with the right mouse button.

Shift L: -- Like 'G' except this modifies the ci and cj elements of the complex constant to locate a quaternion Julia set. This complements the loxodromic formula (L0) that requires a non-zero cj for best results. Cr should be close to zero to maintain symmetry in the loxodromic figure.

Shift-U -- like "J" and "G" except that you choose the target type and beginning parameters. This is useful for exploring cubic Mandelbrot sets, insofar as you can scan the complex c planes for interesting variations. Also works for other multi-dimensional Mandelbrot formulas that are dependent on the c planes. Hint: turn off the Image/Auto-Redraw option after drawing the mapping picture. You can then set the target set (Mandelbrot, quaternion, etc.) before using this key sequence, without erasing the mapping picture.

Shift-D -- use the mouse to examine pixel depth in a drawing. By clicking with the left-mouse button on any area in the current fractal, the pixel's depth and bailout values are displayed in the status bar, along with the pixel's coordinates. Useful to locate Mandelbrot islands while zooming on an image.

Click on the title bar or press Esc to exit this command.

Shift-F -- generate a Julia set from a formula's MandelbrotP space. Random points in a formula's current Mandelbrot space are scanned for an interesting Julia.

Shift-Z -- zoom in/out coordinates. Like the menu command except does not immediately redraw the picture. This allows you to zoom into another screen sector without erasing the previous picture.

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71 Fractal Zplot Application Help

Shift-T -- annotate a picture with text. Cursor changes to a crosshatch, which you position over the area where you want the text to start. Then click the left-mouse button to transfer any text (from the

Edit/Text window) to the picture. Can be used with Undo. Use the Edit/Text command to change font, text color or format text into multiple lines. This is useful for adding copyright/author info to a finished picture.

8.4

Parser

Parser Information

Functions (capital letters are optional, and parenthesis are necessary around complex expressions)

The following information takes the form "standard function" ---"form used by Fractal ViZion to represent standard function".

sine z --- sin(z) or SIN(Z) ; where Z can be any complex expression hyperbolic sine z --- sinh(z) or SINH(Z) arcsine z --- asin(z) or ASIN(Z) cosine z --- cos(z) or COS(Z) hyperbolic cosine z --- cosh(z) or COSH(Z) arccosine z --- acos(z) or ACOS(Z) tangent z --- tan(z) or TAN(Z) hyperbolic tangent z --- tanh(z) or TANH(Z) arctangent z --- atan(z_) or ATAN(Z) cotangent z --- cotan(z) or COTAN(Z) arccotangent z --- acotan(z) or ACOTAN(Z) e^z --- exp(z) or EXP(z) -- the exponential function natural log of z --- log(z) or LOG(Z) absolute value of z --- abs(z) or ABS(Z) square root of z --- sqrt(z) or SQRT(Z) z squared --- sqr(z) or SQR(Z) real part of z --- real(z) or REAL(Z) imaginary part of z --- imag(z) or IMAG(Z) modulus of z --- mod(z) or MOD(Z) or |z| -- (x*x + y*y) conjugate of z -- conj(z) or CONJ(z) -- (x-yi) flip(z) --- flip(z) or FLIP(Z) -- exchange real and imaginary parts of z (y+xi) polar angle of z -- theta(z) if/then/endif – if(argument), then (phrase) endif -- if argument is true then do phrase else skip phrase

('then' tag is optional, but a comma should follow argument or put 'if(argument)' on separate line) if/then/else/endif - if(argument), then (phrase) else (phrase) endif -- if argument is true then do phrase else skip phrase and do alternate phrase('then' tag is optional, but a comma should follow argument or put 'if(argument)' on separate line)

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Note: if/then/endif and if/then/else/endif loops can be nested only when endifs follow each other at the end of the loops. For example: if(argument) if(argument) then (phrase) endif endif.

Math operators

+ --- addition

- --- subtraction

* --- multiplication

/ --- division

^ --- power function

< --- less than

<= --- less than or equal to

> --- greater than

>= --- greater than or equal to

!= --- not equal to

== --- equal to

|| --- logical or (if arg1 is TRUE(1) or arg2 is TRUE)

&& --- logical and (if arg1 is TRUE and arg2 is TRUE)

Constants and variables

complex constant --- c# or C#, read/write.

complex conjugate --- cc# or CC#, read-only.

convergence limit --- cl# or CL# -- the constant entered in the Converge gadget, read-only.

cr -- the constant entered in the cr box in the Parameters window(use j# for parser) ci -- the constant entered in the ci box in the Parameters window(use k# for parser) e --- e or E -- 1e^1 -- 2.71828, read/write.

i --- i or I -- square root of -1,read/write.

iteration --- iter# -- iteration loop counter j --- j# or J# -- real part of the complex constant, read-only.

k --- k# or K# -- coefficient of the imaginary part of the complex constant, read-only.

Note: j and k are the actual values of the complex constant terms as they are used in the iteration process, so will vary when the Mandelbrot option is used.

m --- m# or M# or pixel --a complex variable mapped to the pixel location as defined by the z coordinates entered in the Parameters window, read/write.

maxit -- the maximum number of iterations, as set in the Parameters window, read only p --- p# or P# -- real constant used in phoenix maps; uses the real part of the complex constant when the Phoenix option is chosen, read-only.

p1 – the complex constant entered in the cr and ci gadgets, read-only.

pi --- pi or PI -- 3.14159, read/write.

q --- q# or Q# -- real constant used in phoenix maps; uses the imaginary part of the complex constant when the Phoenix option is chosen, read-only x --- x# or X# -- real part of Z, read/write.

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73 Fractal Zplot Application Help

y --- y# or Y# -- coefficient of the imaginary part of Z, read/write.

z --- z or Z -- function value at any stage of the iteration process, read/write.

zn# or ZN# -- the value of z at the previous stage of iteration, read-only.

8.5

Built-in Formulas

Built-in Formulas (enter the following prefix into the Function #1 or Function #2 edit boxes) a0 -- spiral network -- C. Pickover. a1 -- z-(1/z+c) -- try with renormalization applied. Sequel to t9. a2 -- fn(z)-(fn(z)+c) -- generalized form of a1.

a3 -- alternate Newton/Halley map of sinz-c. see t3 variant. a4 -- user-defined complex set: fn(z)+fn(z)+c.

a5 -- hypercomplex Newton/Halley map of h^3+c.

a6 -- Hypercomplex Newton/Halley map of fn(h)+c.

a7 -- user-defined complex set: fn(z)+fn(c).

a8 -- fn(z)+zn+c -- from Fractal Creations.

a9 -- q^3+c -- cubic Quaternion set.

b0 -- alternate Newton/Halley map of exp(z)-c. as for t3 variant.

b1 -- alternate Newton/Halley map of log(z)-c.

b2 -- Newton/Halley map of phoenix curve.

b3 -- cfn(z)+zn -- user-defined complex formula.

b4 -- fn(z)+kzn+j -- generalized phoenix curve formula.

b5 -- fn(z) a preformula for use with type 3 composite fractals. uses limit gadget to select function.

b6 -- Newton/Halley map of fn(z)+fn(z)+c.

b7 -- Newton/Halley map of cfn(z).

b8 -- fn(z)*fn(z)+c.

b9 -- Newton/Halley map of foggy coastline #1.

c0 -- Newton/Halley map of foggy coastline #2.

c1 -- Newton/Halley map of fn(fn(z))+c.

c2 -- cfn'(z), where fn'(z)=first derivative of user-defined function.

c3 -- fn(z)+fn'(z)+c.

c4 -- fn'(z)+fn(c).

c5 -- fn(fn'(z))).

c6 -- first order gamma function: (z/e)^z*sqr(2*pi*z)+c.

c7 -- Newton/Halley map of fifth degree Legendre polynomial: 1/8(63z^5-70z^3+15z; display method 2 default.

c8 -- (z^2+e^(-z))/(z+1): second-order convergence formula for finding root of ze^z-1=0.

c9 -- Newton/Halley map of fn(z)*fn(z)+c.

d0 -- z^s/limit+c: anti-derivative of z^n; s may be complex using the si variable as its imaginary component.

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Help menu 74

d1 -- Sterling expansion of gamma function: (z/e)^z*sqr(2*pi/z)+c.

d2 -- Newton map of fn'(z)-fn(z)+c: generalized first degree Laguerre polynomial. Newton map only. (Note 13).

d3 -- fn(1/(fn(z)+c)).

d4 -- z^2-c; where zreal=abs(zreal)(Paul Carlson's "alien" Julia set). d5 -- z^2; where zreal=abs(zreal)-cr, zimag=zimag-ci (Paul Carlson Julia set).

d6 -- cfn(z)+c. d7 -- Newton's method applied to (x^3+y^2-cr=0 and y^3-x^2+ci=0). Newton map only. from

Sylvie Gallet and Fract19.par.

d8 -- Newton's method applied to fn1(x)+fn2(y)-cr=0 and fn3-fn4+ci=0 (Note 15) Newton map only.

d9 -- Bill13 from Bill Rossi via the Internet.

e0 -- solves Newton/Halley transformation of (z+j)(z+k)(z^2+1) for either Julia or Mandelbrot set. e1 -- solves Newton/Halley transformation of (z+j)(z^2+z+k) for Mandelbrot and Julia set. e2 -- solves Newton/Halley transformation of (z-1)(z^2+z+c) for either Julia or Mandelbrot set. e3 -- solves Newton/Halley transformation of (z+j)(z+k)(z+1) for either Julia or Mandelbrot set. e4 -- Chaos Game Julia IFS (M. Barnsley). e5 -- snowflake Julia IFS (as described in Fractals Everywhere by M. Barnsley).

e6 -- solves Newton/Halley transformation of logz-c. e7 -- solves Newton/Halley transformation of exp(z)-c. e8 -- solves Newton/Halley transformation of (z-c)(z+1)(z-1) for Mandelbrot or Julia set. e9 -- solves Newton/Halley transformation of (z-c)(z+c)(z^2+c^2) --- z^4-c^4. f0 -- generalized form of Earl Hinrichs' sophomore sine function(ssin) -- limit*fn(z)+s+si, where fn(z)

=(fn1(x),fn2(y)).

f1 -- c^2/(1-cz^2) -- variant of p4.

f2 -- Gallet-4-01, from Sylvie Gallet's extensive Internet collection (Note 16).

f3 -- Gallet-4-02, from Sylvie Gallet.

f4 -- Gallet-6-01, from Sylvie Gallet.

f5 -- Gallet-6-02, from Sylvie Gallet.

f6 -- Gallet-6-03, from Sylvie Gallet.

f7 -- Gallet-6-04, from Sylvie Gallet.

f8 -- Gallet-6-05, from Sylvie Gallet.

f9 -- Gallet-7-01, from Sylvie Gallet.

g0 -- z^3-3c^2z+s -- cubic Mandelbrot (Note 17) g1 -- z^3-3sz+c -- alternate cubic Mandelbrot g2 -- z^3-3c^2z^2+s -- cubic Mandelbrot variant g3 -- z^3-3sz^2+c -- alternate cubic Mandelbrot variant g4 -- z^3-3c^2/z+s -- cubic Mandelbrot variant g5 -- z^3-3s/z+c -- alternate cubic Mandelbrot variant g6 -- z^3-3c^3*z+s -- cubic Mandelbrot variant g7 -- z^3-3s/z^2+c -- alternate cubic Mandelbrot variant g8 -- z^3-3c^2+z+s -- cubic Mandelbrot variant

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75 Fractal Zplot Application Help

g9 -- z^3-3s+z+c -- alternate cubic Mandelbrot variant h0 -- (O*C^-1)(C*O)+c -- octonion set (Note 4) h1 -- cO*CC(1-C*O) -- octonion set h2 -- O*C(O-CC*O)+c -- octonion set h3 -- cO^2+O^2*CC-c -- octonion set h4 -- O^2*CC+O^2*C+c -- octonion set h5 -- O^3*CC+c -- octonion set h6 -- O^3*CC+O^3*C+c -- octonion set h7 -- O^4*CC+c -- octonion set h8 -- cO^3*CC+c -- octonion set h9 -- O^2*CC+O^3*C+c -- octonion set i0 -- 2*z*c#*cos(pi/z) -- Godwin Vickers i1 -- 2*z*c#*sin(pi/z) -- Godwin Vickers i2 -- 2*z*c#*tan(pi/z) -- Godwin Vickers i3 -- 1/z^3+sin(z)*c^2 i4 -- cosh(z)/c*z+c^2 i5 -- exp(z)/z^3-c i6 -- cosh(z)*z^2-c^2 i7 -- 1/z^2-cz-c i8 -- tan(z)-czc i9 -- (tan(z)-1/z^3)/c^2 j0 -- cosh(z)*cos(z)+1/c j1 -- z^4/(z^3-c^2) j2 -- 1/z^2-tan(z)+c j3 -- tan(z)/z^3+c j4 -- 1/z^3-cz+1/c j5 -- z^3+tan(z)*c j6 -- 1/z^3-tan(z)-c j7 -- tan(z)-sin(z)+c j8 -- 1/z^2-tan(z)+1/c j9 -- z^4+sin(z)/c k0 -- Mandelbrot set(sine variation) k1 -- z^1.5+c -- Godwin Vickers k2 -- (z^2-z^(2-s))/s+c -- Escher set by Roger Bagula k3 -- z^2+z/(|z|+c) -- Roger Bagula k4 -- quantum set -- S.M. Ulam k5 -- prey predator #1 -- Roger Bagula k6 -- prey predator #2 -- Roger Bagula k7 -- Klein group #1 -- Roger Bagula k8 -- Klein group #2 -- Roger Bagula k9 -- Klein group #3 -- Roger Bagula

© 2003 - 2010 ... Mystic Fractal

L0 -- Loxodromic by Thomas Kromer (fixed type)

L1 -- squared loxodrome

L2 -- Gedatou by Thomas Kroner (fixed type)

L3 -- Ventri by Thomas Kroner (fixed type)

L4 -- squared gedatou

L5 -- fn(z)-cfn(z) (Note 5)

L6 -- fn(z)+fn(z)+c"

L7 -- cfn(z)+c"

L8 -- fn(z)+cfn(z)+1"

L9 -- fn(z)+c m0 -- fn(fn(z))+c m1 -- fn(z)+fn(c) m2 -- fn(z)+zn+c m3 -- cfn(z)+zn m4 -- fn(z)+kzn+j -- generalized phoenix curve m5 -- fn(z)*fn(z)+c m6 -- fn(1/(fn(z)+c)) m7 -- (1/fn(z))^2+c m8 -- (1/fn(z))^3+c m9 -- fn(z)/(1-fn(z))+c n0 -- Sinus by Thomas Kromer (fixed type) n1 -- Sinus #2 by Thomas Kromer (fixed type) (Note 18) n2 -- Rings of Fire by Thomas Kromer (fixed type) (Note 18) n3 -- Teres by Thomas Kromer (fixed type) n4 -- z^2+y+c n5 -- z^2+y[n+1]+c n6 -- z^2+zi+c n7 -- z^2+zi[n+1]+c n8 -- zr^2+3zi+c n9 -- zr^2+4zi+c o0 -- z^2+timewave[n]+c -- Note 7 o1 -- Loxodromic #2 by Thomas Kromer o2 -- Loxodromic #3 by Thomas Kromer o3 -- Sinus #4 by Thomas Kromer o4 -- Sinus #5 by Thomas Kromer o5 -- Marmor by Thomas Kromer o6 -- Armor by Thomas Kromer o7 -- Three-Parameter Julia set by Kenneth Gustavsson o8 -- Three-Parameter Julia set #2 by Kenneth Gustavsson o9 -- Three-Parameter Cubic Julia set -- Kenneth Gustavsson

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77 Fractal Zplot Application Help

p0 -- z^2+c --- the standard Mandelbrot or Julia set. p1 -- cz(1-z) --- the self-squared dragon set. p2 -- c(z-1/z) --- alternate Mandelbrot or Julia set. p3 -- cz^2-1 --- alternate Mandelbrot or Julia set. p4 -- c^2/(c+z^2) --- alternate Mandelbrot or Julia set. p5 -- z^3+c --- cubic Mandelbrot or Julia set. p6 -- ((z^2+c-1)/(2z+c-2))^2 -- renormalization formula #1 for x-plane or q-plane pictures (Note

9). p7 -- z^2+j+kzn --- Phoenix curve (Ushiki). Uses Fractint extensions for degree(>2 or <-3.).

p8 -- Julia/Mandelbrot set (modified from M. Barnsley) (Note 2).

p9 -- fn(z)-cfn(z) -- generalized frothy basin (J. Alexander.) (Note 7).

q0 -- (2z^3+c)/(3z^2) q1 -- (2z^3+c)/(3z^2+1) q2 -- Newton's method applied to exp(z)-c q3 -- Newton method applied to sin^2z-c q4 -- Newton's method applied to sin3z-c q5 -- Newton's metod applied to sinz+cosz-c q6 -- Newton's method applied to exp(z)sinz-c q7 -- Newton's method applioed to exp(sinz)-c q8 -- Newton's method applied to fn(z)+c q9 -- Newton's method applied to fn(z)+fn(z)+c r0 -- Newton/Halley map of z^3+conj(z)c -- exploratory function based on modified frothy basin.

r1 -- z^z+z^s+c --- Biomorphs, etc. ; s may be complex using the si variable as its imaginary component.

r2 -- z^s-z+c --- Biomorphs, etc.; s may be complex using the si variable as its imaginary component.

r3 -- fn(z)+exp(z)+c -- Biomorphs, etc. r4 -- solves Newton/Halley transformation of (z^2-c)(z+1). (Notes 3,4,5,11). r5 -- cfn(z) -- transcendental Julia curve, etc. r6 -- cexp(z) -- exponential Julia curve, etc. with additional plane checking when real value of Z exceeds 50. If cos(imag-Z) >= 0, point is considered part of Julia set. r7 -- fn(z)+cfn(z)+1 -- generalized form of t9.

r8 -- foggy coastline #1 Mandelbrot IFS (M. Barnsley) (Note 14).

r9 -- foggy coastline #2 Mandelbrot IFS (M. Barnsley) (Note 14).

s0 -- solves Newton/Halley transformation of sinz-c. s1 -- sexpz+c -- transcendental Mandelbrot or Julia set. s2 -- c(1+z^2)^2/(z^2-1) -- alternate Mandelbrot/Julia set. s3 -- solves Newton/Halley transform of tan(z)-c .

s4 -- IFS (x=sy+j,y=-sx+k (x>0); else x=sy-j,y=-sx-k (modified from M. Barnsley) (An alternate version of this formula is executed when the limit value is non-integral.) s5 -- solves Newton/Halley transform of z^s-1 (Julia set only). s6 -- composite function cz-c/z & z^2+c (C. Pickover).

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s7 -- transcendental function fn(z)+c .

s8 -- ((z^3+3(c-1)z+(c-1)(c-2))/(3z^2+3(c-2)z+c^2-3c+3))^2 -- renormalization formula #2 for xplane or q-plane pictures .

s9 -- Newton/Halley map of z(z^limit-1) (Julia set only). t0 -- Newton/Halley map of z(z^limit-c) (Julia or Mandelbrot set ; display method 2 default; limit>=1.0 ; use 1.0 for initial z with Mandelbrot0 type, or use MandelbrotP type.). t1 -- Newton/Halley map of Chebyshev function cos(n*arccos x). t2 -- Newton/Halley map of Hermite polynomial: 16x^4-48x^2+12.

t3 -- alternate Newton/Halley map of tanz-c (Julia or Mandelbrot set.) the twist is in the second derivative of the Halley type. t4 -- Newton/Halley map of z^limit-c (Julia or Mandelbrot set ; display method 2 default; limit>=1.0

; use 1.0 for initial z with Mandelbrot0 type, or use MandelbrotP type.). t5 -- fn(fn(z))+c -- user-defined complex set. When the first function is z^2 and the second function is conj(z), this becomes the z-conjugate set, zz^2+c, the tricorn set. (Note 12).

t6 -- Volterra-Lotka equations discretized by modified Huen method (from The Beauty of Fractals).

t7 -- c^z+c -- tetration of z. t8 -- q^2+c -- Quaternion set (from Computer, Pattern, Chaos and Beauty) (Note 8).

t9 -- z+cz+1 --try with Newton's method applied. A buggy algorithm found this one. u0 -- Mandelbulb (sine-based non-trig version) -- Note21 u1 -- Mandelbulb (cosine-based non-trig version) u2 -- Mandelbulb (sine-based trig version) u3 -- Mandelbulb (cosine-based trig version) u4 -- Tricorn Mandelbulb (sine-based non-trig version) -- Note 22 u5 -- Tricorn Mandelbulb (cosine-based non-trig version) u6 -- Tricorn Mandelbulb (sine-based trig version) u7 -- Tricorn Mandelbulb (cosine-based trig version)

FraSZle/QuaSZ formulas a0 -- spiral network -- C. Pickover a1 -- z+z^3/c+c a2 -- tan(z)-(z^2+c) a3 -- z^2+arcsinz-c a4 -- cos(z)+csc(z)+c a5 -- cos(z^3)+c a6 -- z^7+c a7 -- sin(z)+c^3 a8 -- z^3+zn+c a9 -- Quaternion set -- q^3+c^3 b0 -- 1/z^2+exp(z)-c b1 -- 1/z^3+logz-c b2 -- z^3+j+kzn b3 -- cz^2+zn+c

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b4 -- sin(z)+kzn+j b5 -- vers(z)+c b6 -- vers(z)+covers(z)+c b7 -- c*vers(z)+c b8 -- vers(z)*covers(z)+c b9 -- (foggy coastline #1)^2+c -- Barnsley c0 -- (foggy coastline #2)^2+c -- Barnsley c1 -- sin(vers)+c c2 -- csec(z^2)+c c3 -- z^3+sech(z^2)+c c4 -- c*z^(c-1)+z^2 c5 -- cos(-1/w^2)+c c6 -- (z/e)^z*sqr(2*pi*z)+c c7 -- 1/8(63z^5-70z^3+15z)+c c8 -- (z^2+e^(-z))/(z+1)+c c9 -- z^2*exp(z)+c d0 -- (z^3)/c+c d1 -- z^3*sqr(2*pi/z)+c d2 -- z^2-csc(h)cot(h)+c d3 -- (1/(sin(z)+c))^3 d4 -- z^2-c ; where x=abs(real(z)), y=imag(z) -- Paul Carlson d5 -- z^2; where x=abs(real(z))-cr, y=imag(z)-ci -- Paul Carlson d6 -- c*cos(z)+c d7 -- z*cos(z)+c d8 -- z*2+z/sin(z)+c d9 -- z^2+z^pi+c e0 -- (z+j)(z+k)(z^2+1)+c e1 -- (z+j)(z^2+z+k)+c e2 -- (z-1)(z^2+z+c) e3 -- (z+j)(z+k)(z+1)+c e4 -- chaos formula -- Barnsley e5 -- snowflake IFS -- Barnsley e6 -- cos(z)+c e7 -- z^3+exp(z)+c e8 -- (z-c)(z+1)(z-1)+c e9 -- z^4-c^4 f0 -- z^2+z^3+sin(c)+cos(c)+c f1 -- z^2*(1-cz^2)+c f2 -- 1/(z*z/c)+c f3 -- 1/(z*z)-z+c f4 -- (1+c)/(z*z)-z

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f5 -- (1+c)/(z*z/c)+c f6 -- (1/c)/(z*z/c)+c f7 -- 1/((z*z)/c)+(c/(1/z-c)) f8 -- 1/(z*z*z/c)+c f9 -- 1/(z*z*z)-z+c g0 -- z^3-3c^2z+s -- cubic Mandelbrot (Note 3) g1 -- z^3-3sz+c -- alternate cubic Mandelbrot g2 -- z^3-3c^2z^2+s -- cubic Mandelbrot variant g3 -- z^3-3sz^2+c -- alternate cubic Mandelbrot variant g4 -- z^3-3c^2/z+s -- cubic Mandelbrot variant g5 -- z^3-3s/z+c -- alternate cubic Mandelbrot variant g6 -- z^3-3c^3*z+s -- cubic Mandelbrot variant g7 -- z^3-3s/z^2+c -- alternate cubic Mandelbrot variant g8 -- z^3-3c^2+z+s -- cubic Mandelbrot variant g9 -- z^3-3s+z+c -- alternate cubic Mandelbrot variant h0 -- (O*C^-1)(C*O)+c -- octonion set (Note 4) h1 -- cO*CC(1-C*O) -- octonion set h2 -- O*C(O-CC*O)+c -- octonion set h3 -- cO^2+O^2*CC-c -- octonion set h4 -- O^2*CC+O^2*C+c -- octonion set h5 -- O^3*CC+c -- octonion set h6 -- O^3*CC+O^3*C+c -- octonion set h7 -- O^4*CC+c -- octonion set h8 -- cO^3*CC+c -- octonion set h9 -- O^2*CC+O^3*C+c -- octonion set i0 -- 2*z*c#*cos(pi/z) -- Godwin Vickers i1 -- 2*z*c#*sin(pi/z) -- Godwin Vickers i2 -- 2*z*c#*tan(pi/z) -- Godwin Vickers i3 -- 1/z^3+sin(z)*c^2 i4 -- cosh(z)/c*z+c^2 i5 -- exp(z)/z^3-c i6 -- cosh(z)*z^2-c^2 i7 -- 1/z^2-cz-c i8 -- tan(z)-czc i9 -- (tan(z)-1/z^3)/c^2 j0 -- cosh(z)*cos(z)+1/c j1 -- z^4/(z^3-c^2) j2 -- 1/z^2-tan(z)+c j3 -- tan(z)/z^3+c j4 -- 1/z^3-cz+1/c j5 -- z^3+tan(z)*c

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81 Fractal Zplot Application Help

j6 -- 1/z^3-tan(z)-c j7 -- tan(z)-sin(z)+c j8 -- 1/z^2-tan(z)+1/c j9 -- z^4+sin(z)/c k0 -- Mandelbrot set(sine variation) k1 -- z^1.5+c -- Godwin Vickers k2 -- (z^2-z^(2-s))/s+c -- Escher set by Roger Bagula k3 -- z^2+z/(|z|+c) -- Roger Bagula k4 -- quantum set -- S.M. Ulam k5 -- prey predator #1 -- Roger Bagula k6 -- prey predator #2 -- Roger Bagula k7 -- Klein group #1 -- Roger Bagula k8 -- Klein group #2 -- Roger Bagula k9 -- Klein group #3 -- Roger Bagula

L0 -- Loxodromic by Thomas Kromer (fixed type)

L1 -- squared loxodrome

L2 -- Gedatou by Thomas Kroner (fixed type)

L3 -- Ventri by Thomas Kroner (fixed type)

L4 -- squared gedatou

L5 -- fn(z)-cfn(z) (Note 5)

L6 -- fn(z)+fn(z)+c"

L7 -- cfn(z)+c"

L8 -- fn(z)+cfn(z)+1

L9 -- fn(z)+c m0 -- fn(fn(z))+c m1 -- fn(z)+fn(c) m2 -- fn(z)+zn+c m3 -- cfn(z)+zn m4 -- fn(z)+kzn+j -- generalized phoenix curve m5 -- fn(z)*fn(z)+c m6 -- fn(1/(fn(z)+c)) m7 -- (1/fn(z))^2+c m8 -- (1/fn(z))^3+c m9 -- fn(z)/(1-fn(z))+c n0 -- Sinus by Thomas Kromer (fixed type) n1 -- Sinus #2 by Thomas Kromer (fixed type) (Note 6) n2 -- Rings of Fire by Thomas Kromer (fixed type) (Note 6) n3 -- Teres by Thomas Kromer (fixed type) n4 -- z^2+y+c n5 -- z^2+y[n+1]+c n6 -- z^2+zi+c

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n7 -- z^2+zi[n+1]+c n8 -- zr^2+3zi+c n9 -- zr^2+4zi+c o0 -- z^2+timewave[n]+c -- Note 7 o1 -- Loxodromic #2 by Thomas Kromer o2 -- Loxodromic #3 by Thomas Kromer o3 -- Sinus #4 by Thomas Kromer o4 -- Sinus #5 by Thomas Kromer o5 -- Marmor by Thomas Kromer o6 -- Armor by Thomas Kromer o7 -- Three-Parameter Julia set by Kenneth Gustavsson o8 -- Three-Parameter Julia set #2 by Kenneth Gustavsson o9 -- Three-Parameter Cubic Julia set -- Kenneth Gustavsson p0 -- z^2+c (Note 2) p1 -- cz(1-z) p2 -- z(z-1/z)+c p3 -- cz^2-c p4 -- z^2+cz^2+c p5 -- z^3+c p6 -- ((z^2)*(2z+c))^2+c p7 -- z^2+j+kzn p8 -- (x^2-y^2-j,2xy-k) when x>0; else (x^2-y^2-c+jx,2xy+kx-k) -- Barnsley (Note 1) p9 -- z^2-cz^3+c q0 -- (2z^3+c)/(3z^2) q1 -- (2z^3+c)/(3z^2+1) q2 -- Newton's method applied to exp(z)-c q3 -- Newton method applied to sin^2z-c q4 -- Newton's method applied to sin3z-c q5 -- Newton's metod applied to sinz+cosz-c q6 -- Newton's method applied to exp(z)sinz-c q7 -- Newton's method applioed to exp(sinz)-c q8 -- Newton's method applied to fn(z)+c q9 -- Newton's method applied to fn(z)+fn(z)+c r0 -- z^3+conj(z)c+c r1 -- z^z+z^3+c r2 -- z^3-z+c r3 -- z^2+exp(z)+c r4 -- (z^2-c)(z+1) r5 -- cz^3+c r6 -- z^2+cexp(z)+c r7 -- sin(z)+cz^2+c

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r8 -- (z-1)/c when x>=0; else (z+1)/cc -- Barnsley r9 -- (z-1)/c when kx-jy>=0; else (z+1)/c -- Barnsley s0 -- sin(z)-c2 s1 -- z^4+exp(z)+c s2 -- (c(z^2+1)^2)/(z^2-1) s3 -- z^2+tan(z)+c s4 -- strange attractor IFS (s=1.4142) -- Barnsley s5 -- z^5-c^4 s6 -- composite function cz-c/z && z^2+c s7 -- 1/z^2+c s8 -- (z^3+3(c-1)z+(c-1)(c-2) s9 -- z(z^5+c^2) t0 -- z(z^6+c^2) t1 -- z^7-z^5+z^3-z+c t2 -- z^4-z^2+c t3 -- z^3+tanz+c t4 -- z^2+z^c+c t5 -- z^3+c/z+c t6 -- discretizes Volterra-Lotka equations via modified Heun algorithm t7 -- z^3+c^z+c t8 -- Quaternion set -- q^2+c^2 t9 -- z^2+cz^2+c u0 -- Mandelbulb (sine-based non-trig version) -- Note 21 u1 -- Mandelbulb (cosine-based non-trig version) u2 -- Mandelbulb (sine-based trig version) u3 -- Mandelbulb (cosine-based trig version) u4 -- Tricorn Mandelbulb (sine-based non-trig version) -- Note 22 u5 -- Tricorn Mandelbulb (cosine-based non-trig version) u6 -- Tricorn Mandelbulb (sine-based trig version) u7 -- Tricorn Mandelbulb (cosine-based trig version)

Note 1: all pertinent menu flags must be set for built-in functions to work as described.

Note 2: For further info on Michael Barnsley's formulas, see his "Fractals Everywhere".

Note 3: Halley map requires the Newton flag to be set. This is another numerical approximation method for finding complex roots. For all Newton/Halley functions, the Newton map is the default.

The Halley option is specified through the Arg Gadget, the second character being set to 'h', after the display method(1-9). E.g. '1hr' would designate a relaxed Halley map with display method 1.

Note 4: Halley and Newton maps can use one of nine display methods:

#1 (the default mode, except for functions using sinz, expz, logz and tanz, or fn(z), which default to method 2, and don't use methods 1,4 or 5): ---colors represent the root(the zero) which a point

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converges to.

#2 (if the Arg Gadget is set to 2, or for functions of sinz, tanz, logz, expz, and fn(z)): ---colors represent the number of iterations a point takes to converge.

#3 (if the Arg Gadget is set to 3) -- colors represent the number of iterations a point takes to converge according to an alternate formula described by C. Pickover in Computers, Pattern, Chaos and Beauty.

#4 (if the Arg Gadget is set to 4) -- a merging of methods 1 and 3. After the point converges according to the alternate formula #3, its roots are colored according to #1.

#5 (if the Arg Gadget is set to 5) -- a variation of method 1, with double-convergence checking inside the loop.

#6, #7 and #8-- alternate convergent formulas.

#9 -- a variation of method 3, with double-convergence checking.

Note 5: An additional third argument that affects the convergence speed of Newton/Halley maps may be one of the following six methods:

'r': relaxed Newton method uses the formula z = z-sf(z)/f'(z).

'm': modified Newton transformation uses the formula: z-(f(z)/(f'(z)+si)). Note: Si here references the s variable * i, not the complex variable s+si.

'd': relaxed modified Newton method uses the formula: z-(sf(z)/f'(z)+si)).

'p': premodified Newton transform uses the formula: sz-(f(z)/f'(z)).

'c': complex Newton transform uses the formula : z-(f(z)/(f'(z)+c)), where c is the complex constant.

'n': Nova variation by Paul Derbyshire, z-(f(z)/(f'(z))+c.

The s constant is entered via the S gadget.

Note 7: the term 'fn(w)' represents any one of 47 user-defined functions chosen through the f1-f4 gadgets:

0: sin(w).

4: tan(w).

1: sinh(w).

2: cos(w).

5: tanh(w).

6: exp(w).

3: cosh(w).

7: ln(w).

8: w^c 9: w^z. 10: 1/w.

12: w^3.

13: abs(w).

11: w^2.

14: sqrt(w).

15: w.

16: conj(w).

17: csc(w).

18: csch(w).

19: sec(w).

20: sech(w).

21: cot(w).

22: coth(w).

23: cw.

24: 1.

25: arsin(w).

26: arcsinh(w).

27: arccos(w). 28: arccosh(w).29: arctan(w).

30: arctanh(w). 31: arccot(w). 32: arccoth(w).

33: vers(w). 34: covers(w). 35: L

3

(w): 3rd degree Laguerre polynomial.

(w): first order gamma function.

37: G(w): Gaussian probability function -- (1/sqr(2pi))*e^(.5w^2).

36: gamma

38: c^(s+si).

39: zero.

42: wy+wx*i(flip).

40: w^(s+si). 41: |(wx)|+|(wy)|*i(abs).

43: conj(cos(w))--cosxx.

44: theta(w) -- polar angle(w).

45: real(w).

46: imag(w).

When only fun#1 or fun#2 is used and a single user-defined function is involved, the function is taken from f1. When two user-defined functions appear in a function, the f2 gadget supplies the second

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function type, except as noted below. For Newton/Halley maps involving z^z, the first derivative is defined as z^z*(1+ln(z)). An alternate derivative formula (z*z^(z-1)) is used when a non-integral value is entered as an arg limit (e.g.: 0.1). (This produces interesting effects, though mathematically inaccurate.) For plots that use both fun#1 and fun#2(type 2 or 3, etc), fun#1 takes its functions from f1 and f2 and fun#2 takes its functions from f3 and f4.

Note 8: The quaternion and hypercomplex functions use the complex c gadgets to input cr, ci, cj and ck. These may be zero when generating a Mandelbrot-like set of these functions. Julia sets may then be mapped by grabbing points (cr,ci) from interesting areas near this set. Cj and ck must be entered manually for Julia sets. The hj and hk gadgets are used to input the z and w coefficients of the j and k planes. Use small amounts to start for these variables (0-1.0.) Values of 0 for hj, hk, cj and ck result in a two-dimensional slice that matches the standard (non-hypercomplex) type.

Higher values of z and w (as well as cj and ck) produce more pronounced asymmetry in the complex mapping.

Note 9: Renormalization functions use the Arg Gadget for plotting options (1-4,6-8) as follows:

0 or 1: default renormal, with anti-ferromagnetic points mapped only for Julia sets.

Paramagnetic points (those converging to 1) and ferromagnetic points (those escaping to infinity) are mapped for both Mandelbrot and Julia sets.

2: anti-ferromagnetic points are mapped for the Mandelbrot set. This is actually a level-set mapping for points that do not escape to infinity or converge to 1.

3: uses an alternate convergence formula for paramagnetic and anti-ferromagnetic points.

4: a combination of methods 0 and 3, with characteristics of both methods appearing in plot.

6-8: alternate convergence methods, same as those used with Newton/Halley maps

An optional argument for renormalization 'n' follows the convergence method. This is an alternate bailout method for ferromagnetic points.

Note 10: Most of the built-in functions (except for real Newtons and the Gallet formulas) have hypercomplex extensions when values of cj, ck, hj or hk are non-zero.

Note 11: Hypercomplex Newton/Halley maps use only type 2 and type 3 convergence tests.

Note 12: The default version of hypercomplex conjugate is defined as conjugate(h)=hr-hi-hj+hk. A variant of the hypercomplex conjugate uses an arg limit with a non-integral value (e.g.: 2.1.) This makes all imaginary components of h negative, such that conjugate(h)=hr-hi-hj-hk.

Note 13: The formula for a first degree Laguerre polynomial is e^t(d/dt(t/e^t))=d/dt(t)-t.

Note 14: With a non-integral value entered in the limit gadget, an alternate (Fractint) version of this formula is executed.

Note 15: For real Newtons, the function selected from the f1-f4 boxes is a real function. For a real conjugate, the negation of the real term is used.

Note 16: formulas by Sylvie Gallet have been modified to allow both Mandelbrot and Julia sets to

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be drawn from them. Except for Gallet-6-02, the bailout is set with the built-in variable zlimit in the

Parameters window. Gallet-6-02 uses the complex constant p3 (limit and converge) for bailout.

Note 17: For the traditional cubic Mandelbrot (example in The Science of Fractal Images), hj, hk, cj and ck (hypercomplex components of z and c) should be set to zero when used with the quaternion type.

Cubic Mandelbrots quaternions use the S and Si variables (in the New Formula window) to set the initial value and range of the 3rd dimension. Starting from an initial value of 'S', Si is normally set to

2*abs (S), when the Type is MandelbrotP or Julia Tower. Si can also be set to center the 3rd dimension on something besides 0.0. Si has no affect on Mandelbrot0 or Julia types, height fields or in 2D mode. (In 2D mode, the S variable acts as one of the two fixed dimensions, along with the

Limit variable.)

The Limit variable (NF window) points to the fourth dimension (in the Quaternion window the 4th

Dim. variable points to hk.)

In addition, the Arg value (entered in NF window) has the following affect on cubic Mandelbrots:

0 -- compute M+, using hj for z space

1 -- compute M+, using greater of S or hj for z space

2 -- compute M-, using greater of S or hj for z space

3 -- compute M+ and M-, use lesser of M+/M- for pixel depth

4 -- compute M+ and M-, use greater of two for pixel depth

5 -- compute M+ and M-, use difference of two for pixel depth

6 -- compute M+ and M-, use sum of two for pixel depth

7 -- compute M+ and M-, use vector magnitude of two for pixel depth

8 -- compute M+ and M-, use intersection of two (CCL)

9 -- compute M+ and M-, use M+ or difference of two if M+>M-

Args 3-9 affect only quaternion-type cubic Mandelbrots, while args 0-2 can be used in 2D mode, or with height fields.

For args 0-9, a second argument determines the plane that is used for the fourth dimension: b -- b-imag(default)

B -- b-real a -- a-imag

A -- a-real

A third argument also works for args 0-9: j -- alternate cubic Mandelbrot mapping(for compatibility with alpha versions of Fractal Zplot 1.19)

Therefore an arg value of '3B' uses the union of M+ and M- and b-real as the fourth dimension.

Note 18: For the loxodromic functions, Sinus #2 and Rings of Fire, the Arg Limit variable (in the

Edit Formula/Type window) is used as an additional ingredient. Try values -1.5 to 1.5 for Sinus #2 and 1.5 or PI/2 for Rings of Fire.

Note 19: Octonions have a form of xr+xi+xj+xk+xE+xI+xJ+xK. For these formulas C is the octonion constant (1,1,1,1,1,1,1,1) and CC is the octonion conjugate (1,-1,-1,-1,-1,-1,-1,-1).

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87 Fractal Zplot Application Help

Additional options are entered via the Arg box in the Quaternion editor window. To rotate the extra four-octonion dimensions (E-K) use the following syntax:

0I -- rotate OE-OK to OI-OE

0J --

0K -- rotate OE-OK to OJ-OI rotate OE-OK to OK-OJ

To normalize the C and CC constants: n -- normalize C and CC (default is un-normalized)

Alternate octonion initialization: c -set OE-OK to .01 at beginning of each iteration

With octonions, you have your choice of two different algebraic systems (depending on whether the

Type is set to Quaternion or Hypernion.) Hyper-octonions use alternate definitions of the basic octonion multiplication tables. This is similar to the difference between hypercomplex quaternions

(hypernions) and quaternions. The algebra for octonion and hyper-octonions differ in how they conform to (or fail in) the associative and commutative laws.

Note 20: The timewave array is derived from the formula of Matthew Watkins and Peter Meyer's translation to 'c'. In the Julia set version the array acts as a continuously varying complex constant of the form, "tc1=timearray[(2^iteration)%384]/25, tci= timearray[(2^iteration)%384+1]/25, etc. In the Mandelbrot version, the array acts as an increment to zreal and zimag, with initialization being in the same form as the Julia set. The divisor "25" was chosen strictly for esthetics, as this value enhances the openness of the timewave fractal.

Note 21: These formulas were designed by Paul Nylander and David White, in their search for the true "3D Mandelbrot." The non-trig versions are limited to exponents -2 to -8 and 2 to 8, inclusive.

Use the trig-based formulas for exponents beyond -8 and 8 or non-integral exponents such as 2.5.

The non-trig versions are approximately 3-4 time faster. Enter the exponent in the 's' box. Using negative exponents with the Mandelbulb formulas produces inverted fractals, with a large hole in the center of the 3D figure. They are very sensitive to bailout magnitude. Use a low value such as 4 for best results. Since most of the detail in inverted 3D fractals is inside the "shell" the Dive function is also useful to reveal hidden detail. The Mandelbulb formulas use only one type of 3D algebra based on trigonometric functions, so the other Types such as quaternion are N/A. To produce the 3D version of Mandelbulb, select Quaternion, Hypernion or Complexified from the Type menu.

Note 22: These formulas are based on the Mandelbulb formulas using negative exponents.

However, the coding was modified to produce a variation that closely resembles the Tricorn formula: z^2+c, where the imaginary part of z is multiplied by -1. The Tricorn formula is also known as the three-corner-hat formula... The exponent is limited to the integral range 2-8 in the non-trig formulas. The trig-based formulas support positive non-integral exponents only. Enter the exponent in the 's' box.

8.6

Lsystem info

Lsystem info

Fractal ViZion contains an extended version of the 0L-system (string rewriting) described in The

Science of Fractal Images (edited by Pietgen and Saupe.) The basic algorithm has been expanded

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Help menu 88

and modified extensively, and now supports the 3-D syntax of Laurens Lapre's LParser. In addition an expanded set of rules that allow looping and 3D formula mapping (lissajous curves) have been added. To enable the program to run at higher depth levels without extra megabytes of memory, an alternate method of string interpretation has been implemented, using recursion. String size is kept to a minimum, while the axiom is interpreted character by character.

For those who are not familiar with L-systems, a discussion of their basic features and how they differ from other turtle graphics is a good starting point. Lsystems have as their base an axiom, which is a string of character commands, or a single character. The character can be a command in itself, such as 'F' for forward, or it may be a production rule, which is a string of commands also.

The axiom and production rules are created and edited with separate editors. When the axiom is

'run', the axiom is scanned for commands and/or rules. Recursion is used to substitute where necessary a production rule string for a single character, to the depth specified at runtime. In 0Lsystems, quite long character strings can be built this way and then executed. Once the string is built, it is then stripped of non-commands and executed without displaying the commands on the screen.

The size of the drawing object is thus established and it is then scaled and drawn on the screen. The first significant difference from other turtle graphics systems is now apparent: self-scaling. The object

(s) will never overrun the screen boundaries. Objects can be displayed to the limits of the window, regardless of window size. Since object size is always relative to the window size, a scaling factor is used to create similar objects of different size. Other turtle systems would vary the length of an object's sides.

L-systems have a stack available to keep track of turtle position and variables whenever it is necessary to return to a particular point in the drawing. This is useful for producing tree structures and other branching patterns. Fractal ViZion puts on its stack the turtle position, headings, line style, and color information.

The third significant difference between L-systems and other turtle graphics interpreters is in its handling of recursion. In L-systems, recursion is a built-in feature. You only need to specify the level of recursion, and the lsystem does its string interpretation to that level. Fractal ViZion uses recursion to call its own interpretation routines, and thus eliminates the need for lengthy strings to interpret. The price for the above built-in features is speed. Lsystems can never be as fast drawing a dragon curve, for instance, as a dedicated dragon-drawing program. On the other hand, fractal generators have trouble matching the speed of L-systems where branching patterns are involved. There is no need to recurse backwards to reach a branching node in L-systems: just pop the stack, and you're there.

Fractal ViZion is capable of rendering polygons in three dimensions, but cannot match the speed of a dedicated 3-D modeler. Hidden-line removal is done via a z-buffer, with simple ray-tracing to enhance the view. L-systems offer the user a chance to experiment with a very different approach to graphics/fractal production.

Syntax for Fractal ViZion's Lsystem (default and extended)

The default syntax follows Laurens Lapre's LParser implementation. The initial heading is (0,1,0), which will move the turtle up in the direction of the y-axis, when no rotation is applied to the axis (or a rotation of 0,0,0). All turning commands are three-dimensional. The following is extracted from

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89 Fractal Zplot Application Help

Laurens Lapre's LParser package.

---------------------------------------------------------------------------

Turtle Orientation commands

---------------------------------------------------------------------------

+ turn left around up vector

+(x) turn x left around up vector

- turn right around up vector

-(x) turn x right around up vector

& pitch down around left vector

&(x) pitch x down around left vector

^ pitch up around left vector

^(x) pitch x up around left vector

< roll left (counter clockwise) around forward vector

<(x) roll x left around forward vector

> roll right (clockwise) around forward vector

>(x) roll x right around forward vector

---------------------------------------------------------------------------

Special Orientation commands

---------------------------------------------------------------------------

| turn 180 deg around up vector

% roll 180 deg around forward vector

$ roll until horizontal

~ turn/pitch/roll in a random direction

~(x) " in a random direction with a maximum of x degrees

t correction for gravity with 0.2

t(x) correction for gravity with x

---------------------------------------------------------------------------

Movement commands when {} active

---------------------------------------------------------------------------

F move forward and draw full length record vertex

F(x) move x forward and draw record vertex

Z move forward and draw half length record vertex

Z(x) move x forward and draw record vertex

f move forward with full length record vertex

f(x) move x forward record vertex

z move forward with half length record vertex

z(x) move x forward record vertex

g move forward with full length don't record vertex

g(x) move x forward don't record vertex

. don't move record vertex

---------------------------------------------------------------------------

Structure commands

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Help menu 90

---------------------------------------------------------------------------

[ push current state

] pop current state

{ start polygon shape

} end polygon shape

---------------------------------------------------------------------------

Inc/Dec commands

---------------------------------------------------------------------------

" increment length with 1.1

' decrement length with 0.9

"(x) multiply length with x also '(x)

; increment angle with 1.1

: decrement angle with 0.9

:(x) multiply angle with x also ;(x)

? increment thickness with 1.4

! decrement thickness with 0.7

?(x) multiply thickness with x also !(x)

---------------------------------------------------------------------------

Additional commands

---------------------------------------------------------------------------

c increment color index

c(x) set color index to x

---------------------------------------------------------------------------

Extended command syntax: r(n) -- repeat the execution of the following commands by the number specified by the following character or characters. The default is 1, or one repetition. For example: 'r(5)' would repeat a loop

5 times.

rv -- repeat the execution of the following commands until the current heading is reached again.

Used to create turtle graphics subroutines that stop when the the total turning is a multiple of 360 degrees.

rp -- repeat the execution of the following commands until the current position(x,y,z) is reached again. Used for mapping lissajous curves.

@ -- end of repeat loop. Repeat loops can be nested to a level of 10. You should begin and end a repeat loop inside of the same production rule or axiom, to avoid potential problems with the interpretation of that loop. Since recursion is used to interpret a production rule character by character, the '@' can only send the interpreter back to the beginning of that loop. Trying to get back to an axiom's repeat loop using a '@' from a production rule will trigger a syntax error.

's(n)' -- number of sides in line. 0 sides is default for a tubular line. s(3) sets a wedge-shaped line,

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91 Fractal Zplot Application Help

while s(4) sets a cubic or square line shape. Increasing the number of sides makes the line more rounded, but increases draw time. The sides command only works when the line width is greater than 1. (A narrower line can be rounded with fewer sides, to speed up drawing.)

3D curve(lissajous) commands:

F(Ln) -- draw segment of 3D curve, centered around current position.

f(Ln),g(Ln) -- move as segment of 3D curve, no drawing.

Z(Ln) -- draw half segment or n-length segment of 3D curve.

z(Ln) -- move as half-segment or n-length segment of 3D curve, no drawing.

+(Ln) -- rotates 3D curve's x and z (heading) vectors by n degrees.

-(Ln) -- rotates 3D curve's x and z (heading) vectors by -n degrees.

&(Ln) -- rotates 3D curve's x and y (heading) vectors by n degrees.

^(Ln) -- rotates 3D curve's x and y (heading) vectors by -n degrees.

<(Ln) -- rotates 3D curve's y and z (heading) vectors by n degrees.

>(Ln) -- rotates 3D curve's y and z (heading) vectors by -n degrees.

b(n) -- set boundary limit for 3D curve (default is 0). Used preceeding a 'rp' command, to allow the repeat loop more flexibility in deciding when the beginning postition has been reached. Some 3D curves require a loosening of the boundary for the rp loop to work. Higher boundary values make the rp loop more effective, though too high a value can result in premature bailout. To eliminate endless rp loops, a failsafe bailout is set at 50000 reps.

i(n) -- skip n segments in 3D curve (default 0.) Use to speed up drawing by decreasing segment resolution. Sharp angles may appear as disks rather than a continuous form when segments are skipped.

l(n) -- set starting angle for 3D curve (default is 0.) Using 'l' and r(n) different part of the 3D curve can be drawn. The 'rp' command automatically sets l to 0.

P -- set center position for 3D curve. Uses the current turtle position as a center point for 3D curves x{formula} -- set x formula for 3D curve. Can use any trig function, plus the angles, 'x', 'y' and 'z' and the loop variable 'd'. The default x formula is "sin(x*d)*cos(y*d)".

y{formula} -- set y formula for 3D curve. Can use any trig function, plus the angles, 'x', 'y' and 'z' and the loop variable 'd'. The default y formula is

"sin(x*d)*sin(y*d)" w{formula} -- set z formula for 3D curve. Can use any trig function, plus the angles, 'x', 'y' and 'z' and the loop variable 'd'. The default z formula is "cos(x*d)".

u(n) -- sets the x angle for 3D curve.

v(n) -- sets the y angle for 3D curve.

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Help menu 92

k(n) -- sets the z angle for 3D curve.

Notes:

The color index is initialized at 2. Each index of the current palette is used by the ray-tracer as a separate palette, so up to 236 different objects can be modeled using up to 236 different colors.

This works best when the index is a dark color. The color will be spread from dark to light when it is used during final ray-tracing.

The angles used in turning commands may be negative, as +(-25).

When F or a like drawing/movement command is followed by an argument, as in F(5), the turtle draws the absolute distance specified, in turtle units. This may be different from FFFFF, since the initial length of the draw vector is 100. An equivalent (and faster) draw command for FFFFF would therefore be F(500), assuming the length vector was not changed by one of the inc/dec commands.

When the length vector of a line is decreased or increased, the width of the line is proportionately changed also (to retain compatibility with LParser files.)

Under the extended rule set, when line width is changed during drawing, Fractal ViZion attempts to smooth out width changes by introducing a gradient, or gradual width change, while drawing. If two or more consecutive commands are introduced that change the line width before drawing, only the last width change will have this gradient.

Syntax errors and assignment errors that occur during runtime are displayed in a message box, and the run is halted. The message box contains the characters that caused the syntax error, to aid troubleshooting.

8.7

Bibliography

Bibliography

Complex Mathematics

Churchill, Ruel.V. and Brown, James Ward: "Complex Variables and Applications", Fifth Edition,

McGraw-Hill Publishing Company, New York, 1990.

Korn, Granino A. and Korn, Theresa M.: "Manual of Mathematics, McGraw-Hill Publishing

Company, New York, 1967.

Fractal Theory

Barnsley, Michael: "Fractals Everywhere", Academic Press, Inc., 1988.

Devaney, Robert L.: "Chaos, Fractals, and Dynamics", Addison-Westley Publishing Company,

Menlo Park, California, 1990.

Mandelbrot, Benoit B.: "The Fractal Geometry of Nature", W.H.Freeman and Company, New

York, 1983.

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93 Fractal Zplot Application Help

Peitgen, H.-O. and Richter, P.H.: "The Beauty of Fractals", Springer-Verlag, Berlin Heidelberg,

1986.

Formulas and Algorithms

Burington, Richard Stevens: "Handbook of Mathematical Tables and Formulas", McGraw-Hill

Publishing Company, New York, 1973.

Kellison, Stephen G.: "Fundamentals of Numerical Analysis", Richard D. Irwin,Inc. Homewood,

Illinois, 1975.

Peitgen, Heinz-Otto and Saupe, Deitmar: "The Science of Fractal Images", Springer-Verlag, New

York, 1988.

Pickover, Clifford A.: "Computers, Pattern, Chaos and Beauty", St. Martin's Press, New York,

1990.

Stevens, Roger T.: "Fractal Programming in C", M&T Publishing, Inc.,Redwood City, California,

1989.

Wegner, Tim, Tyler,Bert, Peterson, Mark and Branderhorst, Pieter: "Fractals for Windows", Waite

Group Press, Corte Madera, CA, 1992.

Wegner, Tim and Tyler, Bert: "Fractal Creations", Second Edition, Waite Group Press, Corte

Madera, CA, 1993.

Whipkey, Kenneth L. and Whipkey, Mary Nell: "The Power of Calculus", John Wiley & Sons,

New York, 1986.

8.8

Open Avi Stream

Open Avi Stream…

Through a series of windows, this allows you to name and open an avi animation stream and choose a compression method. After using the file requester to name the file, you are given a choice of compression methods. The compression methods include Intel Indeo Video®, Microsoft Video 1 and Cinepak Codec by Radius. (All compression methods degrade the original images, some more than others.) The first key frame in the stream is then drawn and written to the file.

Notes: after the stream is opened, the size of the fractal that can be drawn is fixed at the size of the frame. No changes can be made to the size until the stream is closed. New: If you open a video stream after setting up a batch mode (Demo menu), then the frames will be written as a series of bmp, obj or wrl files, depending on whether AVI Object or AVI WRL is also checked.

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Help menu 94

8.9

Write Frames

Write Frames…

With this option, frames are written to a stream based on the difference between the current key frame and the previous key frame. The first key frame is written when you open a stream. The next key frame is created each time you use this option. In between you can zoom or change Avi variables as much as necessary. The stream is only written to when this option is used. The last key frame is automatically saved after the 'tween' series is written. The number of frames may range from 1-1500 frames between keys. With a frame number of 1 only the key frames are written. This allows animation to be created that incorporates all scalable variables in Fractal ViZion.

Use the Cancel button to exit this dialog without initializing a new series of frames.

Check the Log Scaling box if you want the frames to be written with logarithmic space between frames, else linear space is used. Useful when zooming, where frames would otherwise be packed together at the end of the frame series.

Notes: key frames are saved in parameter files (.zp), with filenames of "bvf_image#_title.zp", where

'#' is the number of the keyframe and 'title' is the name of the working fractal file.

8.10

Close Avi Stream

Close Avi Stream

Closes any open avi stream file. You need to do this before viewing the file or creating a new avi file. The stream is also closed when you exit Fractal ViZion.

8.11

View Avi

View Avi…

Opens an avi file for viewing. You can preview any multimedia file by clicking on its file name. A multimedia box will appear to the right of the file list. Click on okay to open the main view window.

There are buttons to Play a file forwards or Backwards, or forward automatically with Auto rewind/ repeat. Click on Slow to slow down a video. Each click on Slow halves the viewing speed. A click on Stop freezes viewing and restores the view speed to normal playback.

Use the Open button to view a different avi file. Use the Save button to save the file in a different compression format. You must use a different name to save the file than the name that was used to open it. Click on the left-mouse button or any key to abort a save operation.

Note: the view avi requester can be used to preview any multimedia file, including midi files.

8.12

Batch Mode

Batch mode

Here you set parameters for batching and saving random-generated images to disk. When the

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95 Fractal Zplot Application Help

Repetitions value is non-zero, up to 1000 random images can be generated and saved to disk. Use a unique Filename to prevent batch files from overwriting existing image files. The Scan Limit directs the program on how many scans it makes through each formula before it skips to a new formula (if an interesting Julia fractal hasn't been found.) For lsystems, the scan limit is a limit on how many mutations on the rules may be done before drawing. There are check boxes that affect only random quaternions or lsystems also.

New: If you select this option before opening a video stream, then instead of an AVI stream, the program initializes a set of bmp image files. Each 'frame' is written to the directory specified in the

Demo/Batch mode window. Each frame is numbered with a postfix to the Batch-mode name from

0000 to 9999, e.g. 'quat0001.bmp'.

There are radio boxes that allow you to customize how random variables and menu options are processed to create new fractals:

Formula -- (default on) check to randomize built-in formula used

Lighting -- (default off) check to set default lighting

Symmetry -- (default off) check to randomize symmetry used

Rotation -- (default on) check to randomize camera angles

Coloring -- (default off) check to reset coloring parameters

Iteration -- (default off) check to randomize iterations

Z-Space -- (default on) check to set default z-space

Constants -- (default on) check to randomize the complex constants cj-cK

Flame -- (default off) check to randomize options for the flame orbital fractal window

Genetic -- (default off) check to randomize genetic word strings and 'genes' in Formula

Torus -- (default off) check to randomize torus option

Orbit Trap -- (default off) check to randomize orbit traps

Filters -- (default off) check to randomize filters

Biomorph -- (default off) check to randomize biomorph option

Converge -- (default off) check to randomize convergence options (Newton,

Renormalization or Converge)

Levelcurve -- (default off) check to randomize level curves

Decomposition -- (default off) check to randomize decomposition option

Escape -- (default off) check to randomize escape coloring methods

S/Arglimit -- (default off) check to randomize 's' variable and arglimit variable

Invert -- (default off) check to randomize invert option

For quaternions you have the option of selecting only quaternion types, only hypernion types

(hypercomplex) or a mixture of quaternion/hypernion types. For lsystems, you can set the mutation mode to Fixed, to allow a limited number of mutation factors. This results in an lsystem that remains the same number of lines as the original (un-mutated) lsystem. The base lsystem can be chosen at random, or only the current lsystem is mutated over and over (Use the Edit/Axiom window to set the current lsystem.) Each random lsystem starts with the base lsystem, to keep some consistency in the

© 2003 - 2010 ... Mystic Fractal

Help menu 96

mutated image. You can also control the size of the lsystem (small mutations can create very large lsystems, which can bog down the batch mode) by setting the Max Lines variable to a smaller value

(default is 25000.) A good value to use is 5000. Up to 5000 lines of the lsystem will be drawn before moving on to the next lsystem in the batch. Note: the max lines variable isn't saved with the image's data file, so the image can be recreated with any suitable lines limit, when not in batch mode.

Many lsystems don't exceed 5000 lines.

8.13

About Fractal ViZion

About Fractal ViZion

>>>>> Fractal ViZion™ v3.056 ©1998 - 2010 by Terry W. Gintz

Fractal ViZion graphs formulas based on 4-D complex number planes. It is based on the latest version of Fractal Zplot, so contains many advanced features. Fractal ViZion currently supports the

Mandelbrot set, Julia sets, and Phoenix curves, with millions of mapping variations. Also included are midpoint displacement routines to enhance 3D plots. 3D plot types now include quaternion, hypernion, orbital fractals and lsystems, in addition to the original height-field type. The complex math functions supported include sin(z), sinh(z), z^z, e^z, z^n, sqrt(z), cos(z), cosh(z), tan(z), tanh

(z), log(z), ln(z), n^z and others, including the gamma and Laguerre functions.

Up to two formulas for z using the above functions may be plotted, using traditional rules for generating Mandelbrot sets (Benoit B. Mandelbrot) and Julia sets (G. Julia.) Also, there are mapping options that use non-traditional methods, such as the epsilon-cross method (Clifford A.

Pickover), renormalization and IFS (Michael Barnsley).

Since the formula parser is an interpreter, with its inherent lack of speed, 208 'popular' and unusual formulas have been hard-coded to reduce graphing time by 40 to 60 percent (plus the FraSZle formula set too!) Several of the built-in formulas are capable of producing graphs that the program could not generate through the parser alone, such as applying the Mandelbrot set to Newton's method for solving quadratic equations. Also included in the built-in functions are the Quaternion and hypercomplex sets of four dimensions. Hypercomplex extensions (as described in Fractal

Creations) have been incorporated into most of the Mandelbrot and Julia functions except for the

Quaternion set (this is itself a special 4D version of the Mandelbrot set z^2+c.), and the real Newton formulas and formulas f3-f9 by Sylvie Gallet.

Also new to Fractal ViZion are the Mandelbulb formulas by Paul Nylander and Daniel White.

Thanks guys for all the fish!

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97 Fractal Zplot Application Help

Fractal ViZion requires a true-color video adapter for best results.

Memory requirements for Fractal ViZion vary with the size of the drawing area Fractal ViZion opens on, ranging from approximately 3 megabytes memory for a 640X480 area to 48 megabytes for a

2048X1536 area. Special routines have been added to reduce memory requirements for large bitmaps (up to 14400X10800) by writing these directly to a file instead of using a memory bitmap.

Acknowledgements: many thanks to Paul Carlson for providing me his algorithms for 3D-like fractals, and allowing me to incorporate his ideas into Fractal ViZion. Also, special thanks to Ron

Barnett for his help in setting up the animation routines, to Earl Hinrichs for sharing his unique programming methods on the fractal art and programmer's lists, to Frode Gill for his quaternion and ray-tracing algorithms, to Dirk Meyer for his Phong-shading algorithm, to Piet van Oostrum for his

T2MF, without which the midi support would still be a far off dream, and to David Makin for sharing his ideas on quaternion colorings and 3D insights. Thanks also to Francois Guibert for his

Perlin Noise example. A special mention for Stig Pettersson, for his ground breaking work in cubic

Mandelbrots, giving me clues to an historic fractal world I hardly knew existed. The multiwindowing interface in Fractal ViZion is courtesy of that extraordinary and prolific fractal programmer, Steven C. Ferguson, whose filters I have also included in FV. Steve's contributions to the look and feel of Fractal ViZion and previous versions of my program have had a deep impact on my fractal imaging experiments.

For a short history of the programs leading up to this program, see

Chronology .

8.13.1 Chronology

Chronology

History of this program:

In September 1989, I first had the idea for a fractal program that allowed plotting all complex functions and formulas while attending a course on College Algebra at Lane College in Eugene,

Oregon. In November 1989, ZPlot 1.0 was done. This Amiga program supported up to 32 colors,

640X400 resolution, and included about 30 built-in formulas and a simple formula parser.

May 1990 -- ZPlot 1.3d -- added 3D projections for all formulas in the form of height fields.

May 1991 -- ZPlot 2.0 -- first 236-color version of ZPlot for Windows 3.0.

May 1995 -- ZPlot 3.1 -- ZPlot for Windows 3.1 -- 60 built-in formulas. Added hypercomplex support for most built-in formulas.

May 1997 -- ZPlot 24.02 -- first true color version of ZPlot -- 91 built-in formulas. Included support for 3D quaternion plots, Fractint par/frm files, Steve Ferguson's filters, anti-aliasing and

Paul Carlson's orbit-trap routines.

June 1997 -- ZPlot 24.03 -- added Earl Hinrichs Torus method.

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Help menu 98

July 1997 -- ZPlot 24.08 -- added HSV filtering.

December 1997 -- Fractal Elite 1.14 -- 100 built-in formulas; added avi and midi support.

March 1998 --Split Fractal Elite into two programs, Dreamer and Medusa(multimedia.)

April 1998 -- Dofo 1.0 -- supports new Ferguson/Gintz plug-in spec.

June 1998 -- Dofo-Zon -- redesigned multi-window interface by Steve Ferguson, and includes

Steve's 2D coloring methods.

August 1998 --Dofo-Zon Elite -- combination of Fractal Elite and Dofo-Zon

October 1998 -- Dofo-Zon Elite v1.07 -- added orbital fractals and IFS slide show.

November 1998 -- Dofo-Zon Elite v1.08 -- added lsystems.

April 1999 -- Split Dofo-Zon Elite into two programs: Fractal Zplot using built-in formulas and rendering methods, and Dofo-Zon to support only plug-in formulas and rendering methods.

May 1999 -- Fractal Zplot 1.18 -- added Phong highlights, color-formula mapping and random fractal methods.

June 1999 -- completed Fractal ViZion -- first version with automatic selection of variables/options for all fractal types.

July 1999 -- Fractal Zplot 1.19 -- added cubic Mandelbrot support to quaternion option; first pc fractal program to render true 3D Mandelbrots.

September 2000 -- Fractal Zplot 1.22 -- added support for full-screen AVI video, and extended quaternion design options.

October 2000 -- QuaSZ (Quaternion System Z) 1.00 -- stand alone quaternion/hypernion/cubic

Mandelbrot generator

November 2000 -- Added octonion fractals to QuaSZ 1.01.

March 2001 -- Cubics 1.0 -- my first totally-3D fractal generator.

May 2001 -- QuaSZ 1.03 -- added Perlin noise and improved texture mapping so texture tracks with animations.

June 2001 -- Fractal Zplot 1.23 -- added Perlin noise and quat-trap method.

July 2001 -- QuaSZ 1.05 -- improved performance by converting many often-used dialogs to nonmodal type.

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99 Fractal Zplot Application Help

October 2001 -- FraSZle 1.0, QuaSZ formula and algebra compatible version of Fractal Zplot

November 2001 -- DynaMaSZ 1.0, the world's first Dynamic Matrix Systems fractal generator

January 2002 -- MiSZle 1.1 -- generalized fractal generator with matrix algebra extensions

May 2002 -- DynaMaSZ SE 1.04 (unreleased version)-- scientific edition of DMZ, includes support for user-variable matrix dimensions (3X3 to 12X12)

January 2003 -- Pod ME 1.0 -- first stand-alone 3-D loxodromic generator, Hydra 1.0 -- first 3-D generator with user-defined quad types and Fractal Projector a Fractal ViZion-like version of DMZ

SE limited to 3X3 matrices

May 2003 -- QuaSZ 3.052 -- added genetic-style function type and increased built-in formulas to

180. Other additions since July 2001: generalized coloring, support for external coloring and formula libraries, and Thomas Kroner's loxodromic functions.

May 2003 -- FraSZle and Fractal Zplot 3.052 -- added random 3D orbital fractals, new 3D export methods, upgraded most frequently-used dialogs to non-modal type and added genetic-style function type. FZ now based on FraSZle except for built-in formula list and Newton support.

© 2003 - 2010 ... Mystic Fractal

Index

- 3 -

3d: continuous potential high 41

3d: continuous potential low 41

3d: log high 41

3d: log low 41

3d: ray trace 41

3d: use sea level 41

- B -

break: bioconvergence 44 break: biomorph 43 break: biomorph off 44 break: convergence 45 break: decomposition 47 break: decomposition off 48 break: default function 45 break: newton 44 break: orbit traps 55 break: renormalization 44 button: [ ] 14 button: ||||| 13 button: > 14 button: abort 7 button: batch 6 button: bmp 13 button: channel 10 12 button: channel 11 12 button: channel 12 12 button: channel c1 10 button: channel c2 10 button: channel cj 10 button: channel es 9 button: channel hf 11 button: channel J1 8 button: channel J2 9 button: channel LD 11 button: channel LS 12 button: channel o1 11 button: channel o2 11 button: channel OR 12 button: channel Q1 9

© 2003 - 2010 ... Mystic Fractal

button: channel Q2 10 button: channel qt 11 button: channel SB 9 button: color 6 button: draw 7 button: formula 8 button: fvr 6 button: help 8 button: jpg 13 button: load 13 button: new 6 button: png 13 button: rand rend 12 button: rend 7 button: save 13 button: scan 7 button: size 6 button: undo 6 button: V 14 button: view 7 button:Light 8

- C -

color: blue edit box 33 color: blue slider 33 color: cancel button 32 color: copy button 32 color: divide by eight palette 42 color: divide by four palette 42 color: divide by one palette 42 color: divide by two palette 42 color: edit palette 30 color: green edit box 33 color: green slider 33 color: h/r button 31 color: map button 31 color: neg button 31 color: okay button 32 color: pixel 64 color: rand button 33 color: red edit box 32 color: red slider 32 color: reset button 32 color: reverse button 31 color: spread button 32 color: srb button 32 color: srg button 32

Index

100

101 Fractal Zplot Application Help

colorscaling: background 40 colorscaling: continuous potential 39 colorscaling: escape 38 colorscaling: graded palette 40 colorscaling: level 39 colorscaling: set only 40 colorscaling: use level curve 39 colorscaling: use palette1 40

- D -

demo: batch mode 94

- E -

edit: clip 20 edit: copy 19 edit: cubic values 20 edit: formula 21 edit: fractal variables 24 edit: lsystem axiom 27 edit: lsystem rules 27 edit: octonion parameters 30 edit: orbits 28 edit: parameters 24, 26 edit: paste 20 edit: preferences 34 edit: quattrap 33 edit: ray-tracing variables 29 edit: rgb thresholds 34 edit: size 33 edit: text 34 edit: undo 19 exit 18

- F -

files: load palette 17 files: load palettes 17 files: managing 15, 16, 17, 18 files: save palettes 17

- H -

help: about fractal zplot 96 help: bibliography 92 help: built-in formulas 73 help: channels 5 help: chronololgy 97 help: hot keys 69 help: parser info 71 help: remote 5 help: tutorial 68

- I -

image: auto alert 36 image: auto remote 37 image: auto time 36 image: clear 37 image: continue draw 57 image: draw 56 image: draw lsystem 56 image: pilot 61 image: redraw 37 image: reset 66 image: show picture 57 image:new view on zoom 66

- M -

map: <abs(z-real) or abs(z-imag) 51 map: >abs(z-real) or abs(z-imag) 51 map: abs(z) 52 map: abs(z-imag) 50 map: abs(z-real) 50 map: abs(z-real)+abs(z-imag) 51 map: z-imag 49 map: z-real 49 map: z-real+z-imag 50

- P -

palettes: selecting 42 pixel: invert 59 pixel: invert off 59 pixel: phoenix 63 pixel: solid guessing 66 pixel: symmetry 64 pixel: torus 48 pixel: torus off 49

© 2003 - 2010 ... Mystic Fractal

- R -

render: add noise 53 render: anti-alias 43 render: atan coloring 53 render: bof60 coloring 53 render: coloring filter 54 render: factors 53 render: filter 60 render: hsv filters 60 render: level curve 46 render: orbit trap values 56 render: potential coloring 53 render: reset noise seed 54 render: spin 48 render: surface filter 53 render: switch 60 render: texture scale 52

- S -

status bar 63

- T -

toolbar 62 type: julia 58 type: mandelbrot 58

- V -

video: close avi stream 94 video: open avi stream 93 video: view avi 94 video: write frames 94

© 2003 - 2010 ... Mystic Fractal

Index

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