Fractals Project
Natalie Rowe
Types of Fractals
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Mandelbrot Set
Pythagoras Trees
Sierpinski’s Triangle
Plasma Fractals
Star Fractals
Circle Fractals
Lorenz Fractals
Levy Curve
The Golden Ratio
Snowflakes
Crystals
Dragon Fractal
Julia Sets
Newton Fractals
Pascal’s Triangle
Peano Curves
Formula Fractals
Ginger Fractals
Orbital Fractals
Quaternion
Mandelbrot Set
• Mandelbrot sets are mathematical sets of points
closely related to a general class of fractal. This
fractal is closely related to the Julia set.
• The Mandelbrot set was named after
mathematician Benoit Mandelbrot in 1978.
• http://www.youtube.com/watch?v=G_GBw
uYuOOs
Julia Sets
• The Julia Set was discovered by
Gaston Julia early in the century.
• The Julia set is mostly found in a
modified Mandelbrot set.
• http://www.youtube.com/watch?v=NGwim
5kxRUM&feature=related
Pythagoras Tree
• Pythagoras Tree was invented by mathematician
Albert E. Bosman in 1942.
• It is constructed by squares, each triple square
touching on another encloses right triangles.
• By using 30-60-90 degree triangles it can make the
tree bend one side called Lopsided Pythagoras tree
which also creates a Levy Curve.
http://www.youtube.com/watch?v=B0YM_j
kmv8A
Levy Curve
• The Levy Curve was analyzed by Ernesto Cesàro in
1906 and G. Farber in 1910, but now bears the name
of French mathematician Paul Pierre Levy, who was
the first to describe its self-similarity properties.
• The Levy Curve also known as the C-curve. The C
curve is built using 45* isosceles triangles. It can be
used at any degrees other then 45* but below 60*.
Angles less then 45* make a fractal that is tightly
“curled.”
• http://scratch.mit.edu/projects/ngmr/34534
Sierpinski’s Triangle
• Sierpinski’s Triangle was discovered by
mathematician Waclaw Sierpinski in 1916.
• It’s a process of subdivision; with screen revolution
it can be seen to continue indefinitely also known as
the “chaos game.”
• The Sierpinski’s triangle starts with a normal
equilateral triangle. Then connect the midpoints of
each side to form four separate triangles and cut out
the triangle in the center. For each of the remaining
triangles perform this act indefinitely.
Plasma Fractals
• Due to their randomness, plasma fractals closely
resemble nature.
• Plasma Fractals are really useful to use in
making landscapes and cloud like features.
• By using the atlas colors we can tell the height of
the colors to make a 3D landscape.
• http://www.youtube.com/watch?v=5fFHAmcQdvI
Star Fractals
• Star fractals are formed by taking a 5-corner star
and connecting the stars that are 3 times smaller
to every corner. Which makes the process an act
of Iteration.
• A repeated process with the aim of approaching
a target.
• There are different types of star fractals such as:
the Pentagon fractal that's made up of many
pentagons, another of triangles and the other
one as squares.
Circle Fractals
• Circle fractals was introduced by
Apollonius of Perga.
• Circle fractals are based on repeated
placement of two equal tangent circles
within each circle of the figure.
• Circle fractals are closely resembled to the
Cantor set.
Cantor Set
• The Cantor set was introduced by German
mathematician Georg Cantor in 1833.
• It was originally discovered by Henry John
Stephen Smith in 1875.
Van Koch Line Fractal
• Van Koch line fractal involves a line that is
breaking into segments to produce a
pyramid like segment.
• First you start with a line, break it into 3
segments (AC,CD,DF), bring the middle
segments (CD) up, then connect B to AC
segments and E to DF segment. Repeat
the process and the Van Koch line fractal
is started.
Snowflakes
• The Koch snowflake is based on the Koch
Curve which was described by Helge Von
Koch in 1904.
• It is built by starting with an equilateral
triangle and removing the inner third of
each side and building another equilateral
triangle at the location where it was
removed, the process in repeated
indefinitely.
Crystals
• Fractal crystals is most constructed by
starting with a first generation cube and
placing a half-scale cube on the center of
each face. The second-generation cubes
have the same orientation as the firstgeneration cube and so on indefinitely.
• http://www.youtube.com/watch?v=uGhtt4OB
yoU
• Here’s also a cool link to play with fractal
cubes.
http://www.coolmath.com/fractals/tmapcube/i
ndex.htm
Dragon Fractals
• A dragon curve is a repeated nonintersecting
curve whose name comes from its resemblance
to a certain mythical creature the Dragon.
• Imagine taking a strip of paper in half, and then
unfolding it to that angle formed at 90 degrees.
Then consider folding it twice and three times
and then infinitely.
• http://www.youtube.com/watch?v=ZBOE4Tt4OQ
o&feature=related
Pascal’s Triangle
• Pascal’s Triangle was named after French
mathematician Blaise Pascal.
• The Pascal’s triangle starts with number 1 at the
top, then continue placing the number in a
triangular pattern.
• Each number is just two numbers above it added
together (except the edges which are all number
1).
Peano Curve
• In 1880 Italian mathematician Giuseppe
Peano constructed the Peano curve.
• The Peano curve is mostly known as the
space-filling curve.
• It’s a curve that has a range that contains
an entire 2-dimensional.
• http://www.youtube.com/watch?v=4RQmL
Na5ZNo
Quaternion
• Quaternion are number of systems that extends
the complex numbers.
• It was first described by Irish mathematician Sir
William Rowan Hamilton in 1843.
• Its applied to mechanics in 3-dimensional space.
• Unit quaternion provide a convenient
mathematical notation for representing
orientations and rotations of objects in 3dimension.
Software for Fractals
• Mandelbulber- is an easy to use software that helps
you design 3D Mandelbrot sets. Free to download.
• XenoDream- has unique 3D graphics which allows
you to explore or create your own fractal. It allows
you to create different lighting and textures. You can
also make animations.
• These are only a couple of software’s that you can
use. There are a lot more you can use, here’s a link I
found with more software programs.
• http://fractalfoundation.org/resources/fractal-software/
Resources
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http://www.olympus.net/personal/dewey/mandelbrot.html
http://mathworld.wolfram.com/MandelbrotSet.html
http://library.thinkquest.org/26242/full/types/ch5.html
http://www.mcgoodwin.net/julia/juliajewels.html
http://library.thinkquest.org/26242/full/types/ch11.html
http://www.phidelity.com/blog/blog/nerdy-stuff/fractal/pythagorastree/
http://library.thinkquest.org/26242/full/fm/fm18.html
http://scratch.mit.edu/projects/ngmr/34534
http://serendip.brynmawr.edu/playground/sierpinski.html
http://www.zeuscat.com/andrew/chaos/sierpinski.html
http://library.thinkquest.org/26242/full/types/ch10.html
http://bocoup.com/processingjs/docs/index.php?page=Plasma%20Fractals
More Sources
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http://library.thinkquest.org/26242/full/types/ch13.html
http://www.tnlc.com/eep/circles/
http://mathworld.wolfram.com/CantorSet.html
http://www.tgmdev.be/curvevonkoch.php
http://library.thinkquest.org/26242/full/fm/fm16.html
http://www.coolmath.com/fractals/tmapcube/index.htm
http://cloudscape.blogspirit.com/archive/2009/12/20/crystalfractals.html
http://library.thinkquest.org/26242/full/fm/fm8.html
http://ptri1.tripod.com/
http://planetmath.org/encyclopedia/PeanoCurve.html
http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/
quaternions/index.htm
http://fractalfoundation.org/resources/fractal-software/