Fractals For the Classroom
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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. The first fractals were described at the end of the XIXth Century and Cantor Set and Cantor Dust are probably the first fractals described by teh German mathematician Georg Cantor (1845-1918) in 1883. To produce a Cantor Set, also known as a Cantor Comb, start with a line segment of any length, draw another line segment equal in length with the first one and divide it into three equal segments. Remove the middle third, and repeat this process indefinitely.
The first six steps of the iteration are represented in the image below:
Cantor Set
Image from Wikipedia
Cantor Dust
The Cantor Dust uses a square segment as a base. The first step in the construct ion of a Cantor Dust is to make a square. The next step is to construct an identical square as the first one and divide it into 9 equal squares. Remove all the inside squares and retain only the corner squares. Repeat this iteration to infinity. The fractal is a base motif fractal. In this fractal the base is the square and the motif is the removal of the inner squares.
Georg Cantor on Amazon Search
Benoit Mandelbrot
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.'
From ENCYCLOPEDIA Britannica
Green Mandelbrot Photographic Print
A Clockwork Mandelbrot
Benoit Mandelbrot on Amazon Search
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. The Sierpinski Triangle is the classic example of an Orbital fractal.
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.
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.
Fractals for the Classroom on Amazon Search
Fractal and Pi Products on Zazzle
http://www.zazzle.com/mazcloset/fractal gifts
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Kerri's House of Fractals
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Original Mandelbrot and Julia fractal art, Fractal more...2 points
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Fractal Art - A Fish, A Nothing Is Impossible Shell, A Christmas Tree And More
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animotaxis: Optical Illusions: Zazzle.ca Store
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Abandon Reason - Digital Art by Susan Wallace
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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.
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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
What is your favourite book on fractals?
Fractal Time: The Secret of 2012 and a New World Age by Gregg Braden
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Fractals, Googols, and Other Mathematical Tales by Theoni Pappas
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The Fractal Geometry of Nature by Benoit B. Mandelbrot
Clouds are not spheres, mountains are not cones, and 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
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Fractals Blogs
- Ritchey exhibits high qi fractal art at Mr. Helsinki
- Mr. Helsinki Restaurant and Wine Bar in downtown Fish Creek is showing the high qi holo movement fractal artwork of higher psychic healer artist Gale Ritchey through June 30. A reception with light hors d'oeuvres will take place from 5 to 7 pm June 7.
- Science for Designers: Scaling and Fractals
- Most designers know something about ?fractals,? those beautiful patterns that mathematicians like Benoît Mandelbrot have described in precise structural detail. In essence, fractals are patterns of elements that are ?self-similar? at different scales.
- Technical Analysis: Understanding Elliott Wave Theory
- Elliott also discovered that these market fractals seem to follow many natural mathematical laws, such as the golden ratio (1:1.618) found throughout the natural world. Certain personalities find this outrageous: How could a stock market have anything ...
- Let them eat fractal pancakes
- What else can explain his creation of edible fractals and mathematical constants? Oh, you prefer biology? Then chow down on some marine invertebrates, or perhaps a few human organs. An archaeologist at heart? Try his dinosaurs.
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blackspanielgallery
Feb 2, 2012 @ 8:42 pm | delete
- Nice lens.
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Poetryman Jan 25, 2012 @ 7:30 am | delete
- very cool. a nice way to get kids interested in math.
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COUNTRYLUTHIER
Jan 8, 2012 @ 8:27 pm | delete
- This is so interesting. THanks for sharing something nerdy yet refreshingly interesting!
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Comfortdoc
May 28, 2011 @ 11:46 pm | delete
- Math would have been a lot more interesting if we'd been able to study fractals.
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Heard_Zazzle May 8, 2011 @ 5:37 am | delete
- Fractals are so beautiful. There was a PBS show featuring fractals. Really interesting.
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by sorana
5 January 2012 more »
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