What are Fractals?
A fractal is a mathematical object, a geometric pattern that is repeated (iterated) at ever smaller (or larger) scales to produce (self similar) irregular shapes and surfaces that cannot be represented by classical (Euclidian) geometry. Fractals are used especially in computer modeling of irregular patterns and structures found in nature.
Benoit Mendelbrot
Benoit Mandelbrot is a Polish-born French mathematician best known as the father of fractal geometry.
Mandelbrot showed how Fractals can occur in many different places in both Mathematics and elsewhere in Nature.
'Mandelbrot was educated at the École Polytechnique (1945-47) in Paris and at the California Institute of Technology (1947-49). He studied for his doctorate in Paris between 1949 and 1952 and then did research for a year under John von Neumann at the Institute for Advanced Study in Princeton, New Jersey. From 1958 to 1993 he worked for IBM at its Thomas J. Watson Research Center in New York, becoming a research fellow there in 1974. From 1987 he taught at Yale University, becoming the Sterling Professor of Mathematical Sciences in 1999.' www.britannica.com
'Mandelbrot was educated at the École Polytechnique (1945-47) in Paris and at the California Institute of Technology (1947-49). He studied for his doctorate in Paris between 1949 and 1952 and then did research for a year under John von Neumann at the Institute for Advanced Study in Princeton, New Jersey. From 1958 to 1993 he worked for IBM at its Thomas J. Watson Research Center in New York, becoming a research fellow there in 1974. From 1987 he taught at Yale University, becoming the Sterling Professor of Mathematical Sciences in 1999.' www.britannica.com
Fractals Lessons
Try some of these interesting lessons.
- The Sierpinski Triangle
- Waclaw Sierpinski (1882-1969) was a Polish mathematician. His work predated Mandelbrot's discovery of fractals. He is best known for the 'Sierpinski triangle', but there are many other Sierpinski-style fractals.
Sierpinski Triangle
The Sierpinski Triangle is the classic example of an Orbital fractal
Resources for Sierpinski Triangle
- JavaTM Version
- The Sierpinski Triangle is an interesting geometric pattern formed by connecting the midpoints of the sides of an equilateral triangle. This creates 4 other equilateral triangles. You could repeat this infinitely many times. Try the Java demonstration here.
- Sierpinski meets Pascal
- Pascal's Triangle was originally developed by the ancient Chinese, but Blaise Pascal was the first person to discover the importance of all the patterns it contained. Click here
here to learn more about the Pascal's triangle.
Sierpinski Arrowhead Curve
The Sierpinski Curve is a base-motif fractal formed using half a hexagon as a motif.
Resources for Sierpinski Arrowhead Curve
- Arrowhead Curve - Wikimedia Commons
- This short, interesting demonstration clearly shows the power of fractal mathematics to create complex, everchanging shapes.
- Robert Dickau and the Sierpinski Arrowhead Curve
- Figures created with Mathematica 6 by Robert Dickau.
- Two-dimensional L-systems
- Designed and rendered using Mathematica versions 2.2 and 3.0 for the Apple Macintosh.
- Dancing Sierpinski
- This is a very creative form of Sierpinski's triangle. Constructed using the Fractal Movie Creator in IFS Construction Kit.
Von Koch's Snowflake
The Koch Snowflake is generated by a simple recursive geometric procedure:
- draw an equilateral triangle
- divide each side into three equal parts
- remove the middle segment (= 1/3 of the original line segment)
replace the middle segment with two segments of the same length (= 1/3 the original line segment) such that they all connect (i.e. 3 connecting segments of length 1/3 become 4 connecting segments of length 1/3.)
To complete the shape, the above procedure is repeated indefinitely on each line segment on the side of a triangle.
Images showing the procedure and complete Koch Snowflake are shown here.
Resources for Koch's Snowflake
- Infinite Perimeter
- If the perimeter of the equilateral triangle that you start with is 9 units, what is the perimeter of the other figures?
Answer: click here. - Finate Area
- 'Area contained inside the Koch Snowflake:
To compute the area of the snowflake we will sum up the areas contained in each of the little triangles that we have added. Suppose that the initial area of the triangle is . We first observe that the area of each of the pieces added at the next stage is 1/9 A.' This can be seen from this diagram.
Fractals Books
The Fractal Geometry of Nature
Amazon Price: $33.65 (as of 12/20/2009) 
Fractal Geometry: Mathematical Foundations and Applications
Amazon Price: $56.45 (as of 12/20/2009)
NOVA: Fractals - Hunting the Hidden Dimension
Amazon Price: $22.49 (as of 12/20/2009)
Fractals: The Patterns of Chaos: Discovering a New Aesthetic of Art, Science, and Nature (A Touchstone Book)
Amazon Price: (as of 12/20/2009)
Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise
Amazon Price: $16.47 (as of 12/20/2009)
Fantastic Fractals
Roller Poster
Muppet Owl
Bearly There Bear
BirdHouse
curly Qs
Clothes Pin Bug
Setting Sun
Elevation
point
Pregnant
automatically generated by Flickr
Fractals Bookmarks
- Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal
- Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal
- The Unravelling of the Real 3D Mandelbrot Fractal
- Geeky Math Equation Creates Beautiful 3-D World
- Hypercomplex Fractals
- Apophysis.org
- Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal
- Illuminations: Fractal Tool
- Welcome - Welcome to Fractal Forums - Index
- rfractals.net
What is your favourite book on fractals?
Fractal Time: The Secret of 2012 and a New World Age by Gregg Braden
 In this fascinating book, Gregg Braden merges t more...0 points
Fractals, Googols, and Other Mathematical Tales by Theoni Pappas
A new treasure trove of stories that make mathemat more...0 points
The Fractal Geometry of Nature by Benoit B. Mandelbrot
Clouds are not spheres, mountains are not cones, a more...0 points
Chaos and Fractals: New Frontiers of Science by Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe
For almost 15 years chaos and fractals have been r more...0 points
Fractals and Scaling In Finance by Benoit B. Mandelbrot
BLECK: PHYSICS TODAY "At once a compendium of more...0 points
Fractals Blogs
- Apple Bloog » Blog Archive » Genuine Fractals 6.0.4 - Photoshop ...
- Genuine Fractals is a revolutionary step forward for image enlargements. We've taken the industry standard and given it a complete overhaul. New scaling technology, faster performance, new features, new user interface, greater ease of ...
- For Your Inspiration: Magnificent Fractals | Bit Rebels
- Since I love both math and art, I've always been fascinated with fractals. The Mandelbrot set is one of the most popular ways to play with fractals, and if.
- Ephemeral 1.0.1 – Screensaver that draws animated IFS fractals ...
- Ephemeral is a screen saver that draws animated IFS fractals, giving an ever-changing display of complex, natural shapes. It uses OpenCL (a new feature of OS X introduced with Snow Leopard) to harness the power of your GPU to ...
- Entertainment Web: Amazing flame fractals take your breath away
- Witness the beauty and elegance of mathematics with 50 breathtaking fractals. Water Lilies Vulture over Prometheus Cats in a Bag Martian Missile Defense Exotic flame. Passion Flame Watercolor flame. Hovering Coccoon Sillouette ...
Fractals on CafePress
Fractals ... what marvelous mathematical forms!
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- James-Falconer James-Falconer Sep 10, 2009 @ 9:24 am
- Nice lens . . . great to teach people about fractals. I am so glad I learned about them . . . I have written a few bits of fractal software for my own use, and love the results.
See my Fractal Lens, NEW today!!. Beautiful Fractals
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