Assignment • Hat Curve Fractal Handout

Assignment • Hat Curve Fractal Handout
Assignment
• Hat Curve Fractal
Handout
• Fractal Project
Extra Credit
Example 1
Find the function that generates the following
sequence:
2 4 8 16
, , , , ...
3 9 27 81
Example 2
Find the function that generates the following
sequence:
2 4 8 16
1, , ,
,
, ...
3 9 27 81
Example 3
Draw the next picture in the following
sequence then describe the pattern.
Creating a Hat Curve Fractal
Objectives:
1. To create a Hat Curve
fractal on Geometer’s
Sketchpad using
iteration.
2. To find the length of the
Hat Curve fractal at the
nth stage of iteration.
Iteration
Iteration is the act of
performing a
mathematical
operation or
transformation on an
initial value or state
and then repeating
that transformation
on the result.
Mathematical Fractals
The process of iteration can be used to
construct a figure called a fractal.
©Paul Carson
©J.C. Sprott
Mathematical Fractals
"I find the ideas in the fractals, both as a body of knowledge
and as a metaphor, an incredibly important way of
looking at the world.“
--Al Gore
©J.C. Sprott
©Paul Carson
Natural Fractals
"Clouds are not spheres, mountains are not cones,
coastlines are not circles, and bark is not smooth, nor
does lightning travel in a straight line."
--Benoit Mandelbrot
©Frame & Mandelbrot
©R. Kraft
Natural Fractals
"Clouds are not spheres, mountains are not cones,
coastlines are not circles, and bark is not smooth, nor
does lightning travel in a straight line."
--Benoit Mandelbrot
©Frame & Mandelbrot
©J.C. Sprott
Example 4
Fractals are said to
be self-similar.
Use the illustration
of the Fern fractal to
define selfsimilarity.
Is the human body
self-similar?
©J.C. Sprott
Fractals
A fractal is a
shape that has
self-similarity;
that is, it looks
approximately
the same at
any level of
magnification.
Fractals
The word
“fractal” was
coined by
Mandelbrot in
1975, in part
referring to
their fractional
dimension.
Gosper Island
Koch Snowflake
Anticross-Stitch Curve
Sierpinski Sieve
Fractal Gallery
Cantor Square
Sierpinski Curve
H-Fractal
Cesaro Torn Square
Fractal Gallery
Star Fractal
Levy Fractal
Dragon Curve
Fractal Gallery
Menger Sponge
Sierpinski Carpet
Tetrix
Pentaflake
Fractal Gallery
Peano Curve
Peano-Gosper
Curve
Fractal Gallery
Mandelbrot Tree
Barnsley’s Fern
Pythagoras Tree
Coastline Paradox
A coastline is a fractal. As
such, if you tried to
measure it, its length would
depend on the size of your
ruler: the smaller the ruler,
the longer the coastline.
This is known as the
Coastline Paradox or
Richardson Effect.
200 km
Coastline Paradox
A coastline is a fractal. As
such, if you tried to
measure it, its length would
depend on the size of your
ruler: the smaller the ruler,
the longer the coastline.
This is known as the
Coastline Paradox or
Richardson Effect.
100 km
Coastline Paradox
A coastline is a fractal. As
such, if you tried to
measure it, its length would
depend on the size of your
ruler: the smaller the ruler,
the longer the coastline.
This is known as the
Coastline Paradox or
Richardson Effect.
50 km
Investigation: Hat Curve Fractal
Use the GSP Activity to
create a Hat curve fractal.
Then complete Q1
through Q3 on a separate
sheet of paper.
Keep in mind that we are trying to find a function
that will help us predict how long the Hat Curve
will become as we increase the number of
iterations.
Assignment
• Hat Curve Fractal
Handout
• Fractal Project
Extra Credit
Assignment
Fractal Project Extra
Credit (+ 5 Points)
• Pick a fractal to create
using GSP
• Type up a set of
instructions detailing
how to construct your
fractal
• No two students can
do the same fractal
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