Upload
misha
View
106
Download
2
Embed Size (px)
DESCRIPTION
Fractals Project. Natalie Rowe. Types of Fractals. Mandelbrot Set Pythagoras Trees Sierpinski’s Triangle Plasma Fractals Star Fractals Circle Fractals Lorenz Fractals Levy Curve The Golden Ratio Pascal’s Triangle SnowflakesPeano Curves CrystalsFormula Fractals - PowerPoint PPT Presentation
Citation preview
Fractals Project
Natalie Rowe
Types of Fractals• Mandelbrot Set• Pythagoras Trees• Sierpinski’s Triangle• Plasma Fractals • Star Fractals• Circle Fractals• Lorenz Fractals • Levy Curve• The Golden Ratio Pascal’s Triangle• Snowflakes Peano Curves• Crystals Formula Fractals• Dragon Fractal Ginger Fractals• Julia Sets Orbital Fractals• Newton 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_GBwuYuOOs
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=NGwim5kxRUM&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_jkmv8A
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 first-generation cube and so on indefinitely.
• http://www.youtube.com/watch?v=uGhtt4OByoU • Here’s also a cool link to play with fractal
cubes. http://www.coolmath.com/fractals/tmapcube/index.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=ZBOE4Tt4OQo&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=4RQmLNa5ZNo
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 3-dimension.
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• 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/pythagoras-tre
e/
• 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/processing-js/docs/index.php?page=Plasma%20
Fractals
More Sources• 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/crystal-fractals.
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/