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Transcript of Fractals
By Stephen Jevons, Vahid Kheirollah, and Nick Paras
Burger, E. B., Starbird, M. 2013. The Heart of Mathematics: An Invitation to Effective Thinking (4th ed). Hoboken: John Wiley & Sons, Inc.
Fractal Foundation. What are Fractals? 13 April 2014. <http://fractalfoundation.org/>
Fractals in nature and applications.
13 April 2014. <http://kluge.in-chemnitz.de/documents/fractal/node2.html>
What are Fractals?
Fractals are infinitely repeating patterns. Each smaller section is a copy of the entire shape.
Having the same pattern under varying scales.
Begin with a line. Then split it into 4 equal sections. Continue splitting the segments.
Infinitely continued additions of line segments.
Each line of the triangle
is split into 4 same size pieces.
Begin with a triangle.
Replace the middle triangle with an empty triangle. Notice the self-similarity with each sub-triangle.
Barnsley Fern Zoom:
Take an image and make various copies of different sizes. Use that collage and add it to a new collage of the same image at even smaller sizes. Continue creating collagception.
Collage of Triangles
Using the collage rules, we can see
the Sierpinski triangle emerging from
other images. If we start with a square and continue to split it into 3 separate squares we see the Sierpinski triangle emerge.
The Chaos Game
Uses in Real Life
Fractals show us how small repetitions can lead to very large outcomes. Fractals can be used to model complex patterns in astronomy, nature, and computer science.
Take an equilateral triangle and label the three vertices. Start at one and then choose another at random. Place a dot halfway between the two vertices. Choose a new vertex and repeat the process. A pattern will begin to emerge.
Synthetic and Imaginary Fractals
Start with a cube. "Punch out" all the cores of the cube. This leaves 20 sub-cubes. Continue this to get 400 and then 160,000 sub-cubes.