Transcript Fractals Jennifer Trinh Benoît Mandelbrot, “father of fractal geometry” They’re SO BADASS! I’m badass too!

Fractals
Jennifer Trinh
Benoît Mandelbrot, “father of fractal
geometry”
They’re SO BADASS!
I’m badass
too!
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Fractals are
•Self-similar (will go into details in
a moment)
•Cannot be described accurately
with Euclidean geometry (they’re
complex)
•Have a higher HausdorffBesicovitch dimension than
topological dimension (will go into
details in a moment)
•Have infinite length or detail
Romanesco
Broccoli
With Euclidean geometry…
Exact Self-Similarity: Koch
Snowflake
Can be formed with Lsystems
Approximate Self-Similarity:
Mandelbrot Set
Statistical Self-Similarity
Hausdorff-Besicovitch Dimension:
Fractal Dimension?
• relationship between the measured length and
the ruler length is not linear, i.e.: 1 dimensional
• The fractal/Hausdorff-Besicovitch dimension
is d in the equation N = M^d, where N is the
number of pieces left after an object is
divided M times. E.g., we divide the sides of
a square into thirds, we have 9 total pieces
left. 9 = 3^2, so the fractal dimension is 2.
• More formally seen as log(N(l)) = log(c) - D log(l)
• Doesn’t have to be an integer
Sierpinski Triangle
Generating Fractals
•Escape-time
fractals: give each point a value and
•“Escape-time fractals:
plug into a recursive function (Mandelbrot set consists
of complex numbers such that x(n+1)=x(n)^2 + c does
not go to infinity, like i; they remain bounded).
Depending on what a value does, that point gets a
certain color, causes fractal picture
•Iterated function systems: fixed geometric
replacement
•Random fractals: determined by stochastic
processes (place a seed somewhere. Allow a particle
to randomly travel until it hits the seed, then start a
new randomly placed particle; see here)
“Measuring” Fractals
•Smaller and smaller rulers
•Box methods: counting the number of
non-overlapping boxes or cubes (went
over in Kenkel)
•See Kenkel
•Lacunarity: measuring how much
space a fractal takes up (kind of like
density). Another way to classify
Sources
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http://tiger.towson.edu/~gstiff1/fractalpage.htm
http://www.fractal-animation.net/ufvp.html
http://local.wasp.uwa.edu.au/~pbourke/fractals/
http://www.fractalus.com/info/layman.htm
http://en.wikipedia.org/wiki/Fractal
http://mathworld.wolfram.com/KochSnowflake.ht
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