Introduction to Ultra Fractal Instructor Guide

Introduction to Ultra Fractal Instructor Guide
Introduction
to
Ultra Fractal
Instructor Guide
Text and images © 2001 Kerry Mitchell
Table of Contents
Introduction ..................................................................................................................................... 2
Lesson 1: What Are Fractals? ........................................................................................................ 4
Lesson 2: Using Ultra Fractal ........................................................................................................ 5
Lesson 3: Other Fractal Types ....................................................................................................... 6
Lesson 4: Rendering Images .......................................................................................................... 7
Reproducible Masters ..................................................................................................................... 8
Creating Mandelbrot And Julia Fractals ............................................................................. 9
Experimenting With Formulas And Colorings ................................................................. 10
Quiz ................................................................................................................................... 11
Quiz Answer Key .............................................................................................................. 13
Instructor Background Information .............................................................................................. 15
What Are Fractals? ........................................................................................................... 16
The Mandelbrot And Julia Sets......................................................................................... 19
Ultra Fractal Information .................................................................................................. 20
Glossary ........................................................................................................................................ 22
1
Introduction
The area of fractal geometry is a relatively new one that has generated interest in many areas.
While being fundamentally mathematical, the aesthetic beauty of fractals appeals to artists and
people not traditionally drawn to math. Fractals typically are generated using a computer, so
more technically oriented people can find them intriguing, too.
This unit is an introduction to the fractal-generating software, “Ultra Fractal.” The intended
users are undergraduate students at the University of Advancing Computer Technology. The
instruction has been developed such that it could be delivered to any student at the University,
irrespective of their major, by any of the faculty in the mathematics area. The unit has been
written to be approximately 4 hours long, covering two 2-hour class periods.
Objectives
1. Students generate images of Mandelbrot and Julia fractals.
2. Students demonstrate the techniques for saving and exporting fractal image and parameter
files.
3. Students demonstrate the use of formulas and inside and outside colorings in creating fractal
images.
4. Students demonstrate the use of gradients and layering in manipulating fractal images.
Students will also learn:
• what are fractals
• what differentiates fractals and non-fractal shapes
• about Mandelbrot and Julia sets
• about other types of fractals
• how to use anti-aliasing to improve their fractal images
At the conclusion of the instruction, the students will have a basic introduction to “Ultra Fractal,”
a program that is currently available to all students at the University.
Organization
The material is organized into 4 lessons. Each lesson is designed to be delivered in one hour (50
clock minutes) of classroom time. The lessons consist of small lectures, demonstrations, and
opportunities for the students to try out the material for themselves.
2
Materials
The “Introduction to Ultra Fractal” unit consists of a complete set of instructor and student
materials, as well as addition material about Ultra Fractal:
• Instructor Guide: Contains instructional procedures to use with each lesson, reproducible
exercises and quiz, instructor background information about fractals, and a glossary. The
unit contains one color copy of the Instructor Guide.
• Student Guide: Contains 25 pages of instructional text and practice opportunities. The unit
contains 15 black and white copies.
• Supplemental material on CD-ROM:
ƒ Ultra Fractal program: The latest release, version 2.05. This is the same version that is
available from www.ultrafractal.com. See that site or the Ultra Fractal Information on
page 20 of this guide for more information on installing the program.
ƒ Sample images: Over 40 sample images created by artists who work with Ultra Fractal.
The images are saved in fractal file (ufr), parameter (upr), and image (jpg) formats. The
jpg files can be viewed with any standard image processing program or web browser.
ƒ Electronic copies of the Student and Instructor Guides, in Microsoft Word format.
Instructor Background Information
This unit was designed such that it can be taught without any special knowledge about fractals.
However, if you would like additional information before you teach the unit, or at times during
the unit, you can refer to the Instructor Background Information starting on page 15 of this guide.
General information about fractals is presented, along with specific information about the
Mandelbrot set and about the Ultra Fractal program.
3
Lesson 1: What Are Fractals?
Preparation
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Read “What Are Fractals?” in the Student Guide (pages 3 – 5).
Read “What Are Fractals?” (page 16 – 18) and, “The Mandelbrot and Julia Sets” (page
19) in the Instructor Guide.
Open Ultra Fractal.
Prepare the “lesson 1” folder on the CD-ROM for access.
Materials
•
Student Guide
Procedures
A. Introduction
• Distribute the student materials.
• Open Ultra Fractal.
• Show some or all of the images in the “Sample Images” folder on the CD-ROM.
• Explain that the students will learn how to use Ultra Fractal to create similar images.
B. Demonstrate the smooth nature of the circle.
• Open the “circle zoom” fractal.
• Zoom in by pressing the F9 key repeatedly. Zoom out by pressing the F10 key.
C. Demonstrate the fractal nature of the Mandelbrot set.
• Open the “mandelbrot zoom” fractal in the “lesson 1” folder.
• Zoom in by pressing the F9 key repeatedly. Zoom out by pressing the F10 key.
• Alternatively, use the zoom box (see page 9 of the Student Guide) to zoom into any
point of your choosing, or simply move the cursor to an interesting spot in the image
and double-click.
D. On the white board, demonstrate iterating and how a sequence can be bounded or
divergent.
• Use f(z) = z2 + c with the initial z = 0.
• For divergent examples, use c = 1, 2, or larger integer.
• For bounded examples, use c = 0 or c = -1.
E. Demonstrate the effect of increasing the number of iterations.
• Open the “mandelbrot iteration” fractal.
• On the “Formula” tab, repeatedly double the “Maximum Iterations” up to 128 or so,
for the students to see the effect.
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Lesson 2: Using Ultra Fractal
Preparation
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Read “Using Ultra Fractal” in the Student Guide (pages 6 – 14).
Read “Ultra Fractal Information” in the Instructor Guide (pages 20 – 21).
Visit the Ultra Fractal website (www.ultrafractal.com) for additional information.
Prepare the “lesson 2” folder on the CD-ROM for access.
Open Ultra Fractal.
Procedures
A. Introduction
• Show some or all of the images of Mandelbrot and Julia fractals in the “Lesson 2”
folder on the CD-ROM.
• Explain that, in this lesson, the students will learn how to use Ultra Fractal to create
similar images.
B. Administer tutorial on creating a Mandelbrot set fractal.
• Demonstrate creating a new fractal and loading the “Mandelbrot” formula.
• Show how to size the image.
• Save the fractal file, allowing the students time to do so. Check to see if everyone’s
fractal opened properly.
• Demonstrate the use of the zoom box.
• Save the fractal parameters, allowing the students time to do so. Check to see if
everyone’s parameters loaded properly.
• Demonstrate entering parameters into the Location tab.
• Export the image to a JPEG file, allowing the students time to do so. Check to see if
everyone’s image exported properly.
C. Discuss Julia sets and switching from Mandelbrot to Julia sets.
• Briefly discuss the relationship between the Mandelbrot set and Julia sets.
• Show how to use the Switch feature to generate a Julia set from a Mandelbrot set.
D. Conduct exercise, “Creating Mandelbrot and Julia Fractals.”
• Distribute handout.
• Circulate among the students to check progress and answer questions.
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Lesson 3: Other Fractal Types
Preparation
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•
•
Read “Other Fractal Types” in the Student Guide (pages 15 – 17).
Open Ultra Fractal. Select “Options...” from the “Options” menu. On the “Previews”
tab, check the boxes, “Use size from original window,” “Use gradient from original
window,” and, “Use color settings from original window.”
Prepare the “lesson 3” folder on the CD-ROM for access.
Procedures
A. Introduction
• Show some or all of the images in the “Sample Images” folder on the CD-ROM.
• Explain that, in this lesson, the students will learn how to use Ultra Fractal to create
images other than the Mandelbrot and Julia set fractals.
B. Demonstrate the effect of changing powers on the Mandelbrot fractal.
• Open the “mandelbrot” fractal from the “lesson 3” folder.
• On the Formula tab, change the real part of the “power” parameter to 3, 4, 5, etc.
Discuss the symmetry of the Mandelbrot set for various powers.
• Use various non-zero values of the imaginary part of the “power” parameter. Discuss
the branch cuts in the fractal.
• Use the “Switch” mode to generate sample Julia sets, as desired.
C Show the “Newton’s Method” fractal.
• Briefly discuss Newton’s Method.
• Open the “newtons method” fractal. The colors relate to the solution found. Discuss
how the fractal structure relates to the method finding different solutions.
• On the Formula tab, change the “Degree” parameter from 3 to 4, 5, 6, etc. Discuss
the number of arms present for each degree.
D Show the “Phoenix” fractal.
• Open the “phoenix mandelbrot” fractal. Show how the Phoenix is a variation of the
Mandelbrot.
• On the Formula tab, roll over the formula-specific parameter boxes with the cursor.
Show how hints are displayed.
• Change various parameters and see the effect on the image. Zoom in if desired.
E Show the “General Tent” fractal.
• Open the “general tent mandelbrot” fractal.
• On the Formula tab, emphasize how parameters are entered and how to use the dropdown menus.
6
Lesson 4: Rendering Images
Preparation
•
•
Read “Rendering Images” in the Student Guide (pages 18 – 25).
Open Ultra Fractal. Select “Options...” from the “Options” menu. On the “Previews”
tab, check the boxes, “Use size from original window,” “Use gradient from original
window,” and, “Use color settings from original window.”
Procedures
A. Administer tutorial on loading a coloring formula.
• Create the “basic julia” fractal by following the directions on page 18 of the Student
Guide.
• Show how to load the “Basic” coloring from the “lkm” folder on the Outside tab.
• Change the “color by” parameter to “polar angle.”
• Check to see if everyone’s fractal is colored properly.
B. Administer tutorial on using the gradient editor.
• Open the gradient editor, using the “basic julia” image with the “polar angle”
coloring.
• Show how to cycle the colors through the “Position” parameter.
• Show how to change the colors and locations of the control points.
• Demonstrate loading pre-existing and random gradients.
• Check to see if everyone is following along.
C. Administer tutorial on layering, found on pages 22 – 23 of the Student Guide.
• Create the bottom layer.
• Create the top layer.
• Select the merge mode.
• Check to see if everyone’s fractal is colored properly.
D. Administer the quiz. Depending on the time, you may have the students perform the quiz
right then in class, or at their leisure within the next day.
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Reproducible Masters
ƒ
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“Creating Mandelbrot and Julia Fractals” exercise
“Experimenting with Formulas and Colorings” exercise
Quiz
Quiz answer key
8
Creating Mandelbrot and Julia Fractals
Here’s your chance to experiment with Mandelbrot and Julia fractals.
1.
Create a zoom into the Mandelbrot set. On the Location tab, set the Center to 0/1
(Re/Im) and the Magnification to 1. Zoom in repeatedly by pressing F9. How far in can
you go? (When the image stays blocky and doesn’t refine, you’ve run out of computer
precision.) Export this image as a JPEG file. View this image in a web browser or an
image-processing program to verify that you exported it properly.
2.
There’s a small copy of the Mandelbrot set centered around -1.9990958/0 (Re/Im). Can
you find it? You may need to increase the Maximum Iterations on the Formula tab as
you zoom in. Save this as a fractal file (.ufr). Open this file to verify that you saved it
correctly.
3.
Create a Mandelbrot set image and zoom in to an interesting spot of your own choosing.
Zoom in to a magnification of at least 1 million (1000000 or 1e6), increasing the
maximum iterations if necessary. Then, choose a spot in that image and generate the
corresponding Julia set. Is there any resemblance between the zoom of the Mandelbrot
set and the Julia set? Save the parameters for this Julia set as a .upr file. Reload these
parameters to verify that you saved them correctly.
Experimenting with Formulas and Colorings
Now you can try out what you’ve learned about formulas and coloring methods. Be sure to save
any images that you like to your own disk, either as parameters (upr) or as fractal files (ufr).
1.
Create an image using the “General Tent Mandelbrot” formula (in the “lkm” folder) and
the “Statistics” coloring (in the “lkm” folder) for the Inside pixels. Be sure to try various
settings for the “r type” parameter in the formula and “statistic” parameter in the
coloring.
2.
Take the “layering.ufr” image from the layering example and play with different
gradients for both layers. Vary the merge mode and opacity for the top layer until you
find a combination that you like.
3.
Trade images with someone else. See if you can improve their image and if they can
improve yours.
Introduction to Ultra Fractal
Quiz
1.
Create a Mandelbrot fractal with the following characteristics. Any item that is not
specified is to be left at the default value. Save the fractal file as “Mandelbrot.ufr” and
email it to the instructor. Export the image as “Mandelbrot.jpg” with the default quality
setting and email the image to the instructor.
Formula
• “Mandelbrot” from “standard” folder
• Periodicity Checking: off
• Maximum Iterations: 2500
• Power (Re/Im): 2.01/0
• Bailout value: 1000
Location
• Start with these parameters:
ƒ Center (Re/Im): -1.7578/0
ƒ Magnification: 300
ƒ Rotation angle: 90
• Then, zoom in to create an image where a small Mandelbrot fills the window. The
magnification must be at least 1 million (1000000).
Gradient
• “Royal” from “standard” folder
• Invert it
• Rotate it to a position of 50.
Image
• Width: 640 pixels
• Height: 480 pixels
2.
Create a Julia fractal with the following characteristics. Any item that is not specified is
to be left at the default value. Save the parameters as “Julia” in “julia.upr” and email that
file to the instructor.
Formula
• “Julia” from “standard” folder
• Drawing Method: Multi-pass Linear
• Periodicity Checking: off
• Maximum Iterations: 243
• Julia seed (Re/Im): -0.044662513/0.643849337
• Power (Re/Im): 2/0
Location
• Center (Re/Im): 0.1560925/ 0.7658098
• Magnification: 110
• Rotation angle: 286
Inside
• “Range Lite” from “lkm” folder
• range center: 0.56
• range width: 0.01
• color by: last mag
Outside
• same coloring as Inside
Gradient
• “45/225” from “lkm” folder
Image
• Width: 500 pixels
• Height: 400 pixels
Introduction to Ultra Fractal
Quiz Answer Key
1.
Create a Mandelbrot fractal with the following characteristics. Any item that is not
specified is to be left at the default value. Save the fractal file as “Mandelbrot.ufr” and
email it to the instructor. Export the image as “Mandelbrot.jpg” with the default quality
setting and email the image to the instructor.
Formula (5 points)
• “Mandelbrot” from “standard” folder
• Periodicity Checking: off
• Maximum Iterations: 2500
• Power (Re/Im): 2.01/0
• Bailout value: 1000
Location (2 points)
• Start with these parameters:
ƒ Center (Re/Im): -1.7578/0
ƒ Magnification: 300
ƒ Rotation angle: 90
• Then, zoom in to create an image where a
small Mandelbrot fills the window. The
magnification must be at least 1 million
(1000000).
Award 1 point each for:
• fractal file named properly
• fractal file emailed (or otherwise handed in)
properly
• image file (.jpg) named properly
• image file emailed (or otherwise handed in)
properly
To check the ufr file, save it to disk from email.
Within Ultra Fractal, open it by choosing “File
| Open,” and select the appropriate folder and
file. Check the “Formula,” “Location,” and
“Image” tabs to verify that the specifications
were met.
The image should resemble this:
Image (2 points)
• Width: 640 pixels
• Height: 480 pixels
Gradient (2 points)
• “Royal” from “standard” folder
• Invert it
• Rotate it to a position of 50.
The gradient should look like this:
13
2.
Create a Julia fractal with the following characteristics. Any item that is not specified is
to be left at the default value. Save the parameters as “Julia” in “julia.upr” and email that
file to the instructor.
Formula (7 points)
• “Julia” from “standard” folder
• Drawing Method: Multi-pass Linear
• Periodicity Checking: off
• Maximum Iterations: 243
• Julia seed (Re/Im): 0.044662513/0.643849337
• Power (Re/Im): 2/0
Location (4 points)
• Center (Re/Im): 0.1560925/0.7658098
• Magnification: 110
• Rotation angle: 286
Inside (4 points)
• “Range Lite” from “lkm” folder
• range center: 0.56
• range width: 0.01
• color by: last mag
Award 1 point each for:
• parameter file named properly
• parameter file emailed (or otherwise handed
in) properly
To check the upr file, save it to disk from email.
Within Ultra Fractal, choose “File | Browse |
Parameters...” Click the “Add File” icon and
select the correct folder and file. Verify that the
parameter set is named “Julia”, and click on it
in the browser to load it. Check the “Formula,”
“Location,” “Image,” “Inside” and “Outside”
tabs to verify that the specifications were met.
The gradient should resemble this:
Outside (4 points)
• same coloring as Inside
Gradient (1 point)
• “45/225” from “lkm” folder
And the image should resemble this:
Image (2 points)
• Width: 500 pixels
• Height: 400 pixels
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Instructor Background Information
ƒ
ƒ
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What Are Fractals?
The Mandelbrot and Julia Sets
Ultra Fractal Information
15
What Are Fractals?
(This material was taken from the Fractalus website, www.fractalus.com. It is © Damien M.
Jones.)
This is a simple question with a very complicated (and very long) answer. A technical answer,
while accurate, doesn’t help much because it uses other fractalspeak jargon that few people
understand, so I won’t even give that definition here.
The simple answer is that a fractal is a shape that, when you look at a small part of it, has a
similar (but not necessarily identical) appearance to the full shape. Take, for example, a rocky
mountain. From a distance, you can see how rocky it is; up close, the surface is very similar.
Little rocks have a similar bumpy surface to big rocks and to the overall mountain.
A Simple Example
This concept of self-similarity can be a little hard to come to
grips with, but it’s a fundamental part of fractals. So take a look
at this first image—from a Julia set, a very simple fractal type.
I’ve highlighted a small box near the left side (it’s a little faint).
The portion of the image in that box is shown in the second
image, below. That image also has a small box highlighted, and
that area is shown in full in the third image.
You can see in these images that smaller areas of the fractal
shape look very much like the larger, full-size image. With Julia
fractals, you can continue this enlarging (“zooming”) process as
often as you like, and you will still see the same sort of details
and shapes at very tiny sizes that you see on the full-size image.
This is what is meant by self-similarity.
Now of course, with something so rigidly self-similar, there’s not
really much point in zooming in. After all, everything is the
same; small detail looks like large detail. So while it’s interesting
that fractals are self-similar, if this is all there is to it, there isn’t
much point.
Not Quite So Similar
Fortunately for fractal enthusiasts, that isn’t all there is to it. Many fractal types get wildly
different as you zoom in. They’re still self-similar, but they’re not rigidly self-similar.
This is what makes fractal exploration so intriguing. The features you see as you zoom are
always changing—teasing you with a little bit of familiarity, and tantalizing you with new and
unexpected twists. With just a single fractal shape, you can explore forever and never see
everything it has to offer. The further you zoom, the more likely you are seeing something that
16
nobody has ever seen before. And with modern computers, it’s very easy to zoom and zoom and
zoom. With just a few clicks you can have zoomed so far that the original fractal image is larger
than the sun.
Take, for example, the zoom sequence on the left.
Starting with the upper left image, the center of each
image was magnified tenfold. Whereas the Julia image
looked virtually identical as it was zoomed, this
Mandelbrot image shows new variations as the
magnification increases. (Because of the small size of
these images, it may not be obvious that each image is a
magnification of the previous one, but they are.)
How It All Works
The basic technique of these fractals can actually be
explained without resorting to confusing mathematical
equations and jargon. It’s rather simple, really.
First, give every point on the screen a unique number.
Now take that number and stick it into a formula; you’ll
get a result from the formula. Take that result and stick
it back into the formula. Keep doing this and watch
what happens to the numbers you get. Color each point
based on what happens.
That’s it. Really—that’s it. Now, with most formulas it
probably won’t do much of interest, but with the
formulas used in fractal creation, some interesting
things happen. Sometimes the numbers you get by
feeding the results of a formula back into the formula
(iterating) explode into enormous numbers, that just
keep getting bigger and bigger. Those points get
colored one way. Other times, the numbers “home in”
on a number, getting closer and closer to it. They get
colored a different way.
The interesting thing—and the reason fractals work at
all—is that sometimes, just a tiny little change in the
number you start with can completely change what
happens as you keep iterating the number. And the
boundary between numbers that explode and numbers
that home in is complicated and twisted—it’s the shape
of the fractal.
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The Enormous Task at Hand
Calculating fractals this way involves a lot of work. A small fractal image—perhaps only
640x480—contains over 300,000 points. Each of those points may require running a number
through the fractal formula more than 1,000 times. This means the formula has to be computed
more than three hundred million times. And that’s a mild example. Extreme images (such as
poster-size fractals) can involve more than one trillion calculations.
Fortunately for the impatient among us, modern computers are fast enough to do the job in a few
minutes. Large fractals might take hours or days, but exploring fractals has never been easier.
Where To Go From Here
As I stated at the outset, fractals are a huge topic. All I’ve even talked about here are one
particular type of fractals (escape-time fractals), but there are many other types as well.
Unfortunately, the further you look into fractals, the more math you will need to know. There are
very few fractal-related books or web pages that don’t get into heavy mathematics. I’ve
attempted to assemble some pages here that will get you started on the mathematics behind
fractals in an accessible fashion, but there is no hiding the fact that it is math.
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The Mandelbrot and Julia Sets
The Mandelbrot Set
The Mandelbrot Set is a set of points in the complex plane. For each complex number c, a
sequence of iteratates (the orbit of c) is formed by iterating the function f(z) = z2 + c:
z0 = 0
zn = zn-12 + c, for n = 1, 2, 3, ...
If |zn| remains bounded as n goes to infinity, then c is a member of the Mandelbrot set (also
termed an “inside” point). Otherwise, |zn| goes to infinity, c is outside of the Mandelbrot set, and
c’s orbit diverges.
The Mandelbrot set is named after Benoit Mandelbrot, who discovered it while working at IBM
in the 1970s. Mandelbrot was studying the slightly different, but dynamically similar function z2
- c, which he called his “μ map.” In practice, images of the Mandelbrot set are generated in cspace, showing the c values that are in the set. The initial value of z is set to 0, as 0 is the critical
point for this function.
Julia Sets
Mandelbrot’s work was inspired by the work of Gaston Julia. Julia studied the dynamics of
iterated rational functions in the complex plane in the early 1900s. He won the French
equivalent of the Nobel Prize for his work, but his work was soon forgotten and languished until
Mandelbrot’s rediscovery of it.
The (standard) Julia set uses the same function as the Mandelbrot set, but differs in the
initialization. The orbit is formed based on the initial value of z, z0; c is a parameter. In
practice, the horizontal and vertical coordinates of the pixel are scaled into the real and
imaginary parts of z0. The parameter c is a user input. The function f(z) = z2 + c is iterated, as
with the Mandelbrot set. If the orbit does not diverge, then z0 is part of the Julia set for that
particular c. Each value of c has its own Julia set.
Part of Mandelbrot’s contribution was in discovering how his set “cataloged” all the Julia sets for
the same function. If c for a particular Julia set is outside of the Mandelbrot set (for example, c =
1), then the Julia set is a “dust,” a collection of isolated points. If c is taken from the interior of
the Mandelbrot set (for example, c = -1), then the Julia set has a definite interior. Points on the
edge of the Mandelbrot set (for example, c = i) have dendritic Julia sets, infinitely thin connected
lines with no interior.
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Ultra Fractal Information
(This material was taken from the Ultra Fractal website, www.ultrafractal.com. It is © Frederik
Slijkerman.)
Ultra Fractal is a versatile fractal plotter for Windows 95/98/NT/2000. Ultra Fractal allows you
to open any number of fractals at the same time. A fractal consists of one or more layers, which
are merged in Photoshop-like fashion. Each layer has its own fractal formula, location, and
coloring algorithms. You can write your own fractal formulas, coloring algorithms and
transformations, and Ultra Fractal will compile them on-the-fly. Besides being powerful, Ultra
Fractal also offers a friendly user interface and extensive online help to help you to get started
quickly.
Features
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MDI design, allowing you to open multiple documents (fractals, gradients, and formula files)
at the same time
Multi-threading, which makes the application more responsive while calculating fractals, and
takes full advantage of multi-processor systems (Windows NT only)
True-color (24-bit) fractals, with dithering in 256-color and high-color modes
Multiple layers per fractal, with all Photoshop merge modes
Formula compiler, compatible with Fractint 19.6, 2-6 times faster, with new advanced
features
Custom fractal formulas
Custom coloring algorithms
Custom geometric transformations to create your own coordinate mappings
Easy-to-use gradient editor for color designing
Built-in formula editor
Fractint PAR and MAP import
Export to BMP, JPEG, PNG, PSD (with layers) and TGA images
Render to disk feature to create poster-size images (up to 100,000x100,000 pixels) with antialiasing
Batch render feature to render an entire parameter set to disk
Browsers to manage your parameter/formula/coloring/transformation collection
System requirements
Basically, Ultra Fractal can run on any computer with Windows 95, 98 or NT installed. For good
performance, it is recommended to use at least an Intel Pentium processor (Ultra Fractal will be
much slower on AMD K5/K6 and Cyrix processors). Pentium II or III processors are even better,
of course. The new AMD K7 Athlon processor seems to be the best choice for fractal
calculations now.
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An accelerated PCI video card is strongly recommended. To work comfortably, your display
should be set to at least 800x600 resolution in high-color (65536 colors), but 1024x768 with
true-color (16777216 colors) is much better.
The minimum amount of memory (RAM) you need is 16 MB, but if you want to create large
images or use a lot of layers, you need more, such as 32 MB or 64 MB.
Installing Ultra Fractal on your hard drive takes only a few megabytes, but some extra space for
storing fractals and rendered images won’t hurt. Make sure there is still enough free disk space
(about 50 MB) for Windows to use as virtual memory.
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Glossary
anti-aliasing: Refining the image by adding sub-pixels during the rendering, and averaging their
colors to achieve the final pixel color.
aspect ratio: Ratio of an image’s width to its height.
bailout value: How large a number must become before the iteration ceases and the pixel is
declared to be “outside.”
complex number: A number with two parts, the real and imaginary parts, used in the calculation
of fractal images.
control point: A point used to specify the colors that compose the gradient.
export: To save the fractal image only, without saving the underlying information.
formula, calculation: A small program used to compute the fractal shape, chosen from the
“Formula” tab.
formula, coloring: A small program used to compute the distribution of colors in an image,
chosen from the “Inside” and “Outside” tabs.
fractal: A shape characterized by infinite detail, typically made of small copies of the original
shape, and generated by iteration.
gradient: The set of colors used to render a fractal image.
inside: A point or pixel whose sequence of iterated numbers does not grow to be larger than the
bailout value. Also, the tab used for setting the coloring parameters for inside pixels.
iteration: A step in creating the orbit for a pixel. The previous step’s formula output is used as
the input for the formula in then next step.
Julia set: A fractal created from the same formula as the Mandelbrot set, but with different
initializations. Each point in a Mandelbrot set fractal has its own Julia set fractal.
layer: A single, independent fractal, which is combined with other layers to create a finished
image.
magnification: How big the entire set would be, compared to the view in the current window.
Mandelbrot set: One of the most famous fractals, discovered by Benoit Mandelbrot.
merge mode: How the current layer is combined with the stack of layers below it.
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opacity: The degree to which the current layer interacts with the stack of layers below it.
orbit: The sequence of complex numbers that is generated for each pixel during the iteration.
orbit trap: Monitoring the orbit and coloring the pixel according to some characteristic of the
orbit.
outside: A point or pixel whose sequence of iterated numbers grows to be larger than the bailout
value. Also, the tab used for setting the coloring parameters for outside pixels.
parameters: The “recipe” for generating a fractal, stored in a upr file.
render to disk: To create a fractal image in “batch” mode, without direct interaction.
switch: To generate a Julia set based on the cursor location in a corresponding Mandelbrot set
image.
ufr: A binary file containing a fractal’s parameters and image data.
upr: A text file used for storing a fractal’s parameters.
zoom box: A rectangular outline used to control the location of a new fractal view.
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