1. HONR 300/CMSC 491 Fractals (Flake, Ch. 5)
Prof. Marie desJardins, February 11, 2016
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2. Happy Valentine’s Day!
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3. Key Ideas Self-similarity Fractal constructions Fractal widths/lengths
Cantor set
Koch curve
Peano curve
Fractal widths/lengths
Recurrence relations
Closed-form solutions
Fractal dimensions
Fractals in nature
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4. Cantor Sets Construction and properties (activity!)
Description of points in Cantor set
Standard Cantor set: “middle third” removal
Variation: “middle half”
Distance between pairs of end points at iteration i = ?
Width of set at iteration i = ?
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5. Fractional dimensions
D = log N / log(1/a)
N is the length of the curve in units of size a
Cantor set: D = ?
Koch curve: D = ?
Peano curve: D = ?
Standard Cantor: D = ?
Middle-half Cantor: D = ?
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6. Hilbert Curve Another space-filling curve
Images: mathworld.com(T,L), donrelyea.com(R)
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7. Koch Snowflake
Same as the Koch curve but starts with an equilateral triangle
Images: ccs.neu.edu(L), commons.wikimedia.org(R)
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8. Sierpinski Triangle Generate by subdividing an equilateral triangle
Amazingly, you can also construct the Sierpinski triangle with the Chaos Game:
Mark the three vertices of an equilateral triangle
Mark a random point inside the triangle (p)
Pick one of the three vertices at random (v)
Mark the point halfway between p and v
Repeat until bored
This process can be used with any polygon to generate a similar fractal
Images: curvebank.calstatela.edu(L), egge.net(R)
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9. Mandelbrot and Julia Sets
...about which, more soon!!
Images: salvolavis.com(L), geometrian.com, nedprod.com, commons.wikimedia.org
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10. Fractals in Nature
Coming up soon!!
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