## Monday, December 7, 2015

### Fractals

Communicating Math
 From IBM's tumblr
Do you know about Benoit B. Mandelbrot?  If you don't, you should!  Dr. Mandelbrot was born in Poland, but had citizenship in France and the US.  He worked for IBM for 35 years and dabbled in such vast areas as finance, aeronautics, physics, and of course math.  Oh yeah, and he invented a thing called fractal geometry.

I may have undersold it.  Fractal geometry is a big deal.  It can be applied to coastlines, bronchi in lungs, clusters of galaxies, and financial price increments, to name a few.  The basic idea of fractal geometry is that as you zoom in on a picture, it looks pretty much the same as when you were zoomed out.  Also, while zooming in, the curve doesn't get simpler.  For example, zooming in on a section of a parabola makes it eventually look like a line segment (Jordan Ellenberg actually discusses this in his book "How Not to be Wrong: The Power of Mathematical Thinking" when introducing the logic behind calculus).  But when you zoom in on a section of a fractal it continues to be as detailed as it was before, if not more so.  Benoit referred to this detail as "roughness" which he would calculate.  He discusses an instant in this TED talk of a colleague giving him a picture of a curve and asking him to estimate the roughness.  His estimate was off by .02.  This idea of roughness is interesting though because it can describe so many things from mountains to cauliflower.

Anyway, fractals have a little wiggle room in their definition.  How else could these both be fractals?
 Sierpinski Carpet
 Mandelbrot Set

Both of these have some sort of self-similarity--that idea of the picture being about the same at different "zooms."  Plus, fractals have "fine or detailed structure at arbitrarily small scales. irregularity locally and globally that is not easily described in
traditional Euclidean geometric language, and simple and 'perhaps recursive' definitions" (https://en.wikipedia.org/wiki/Fractal).

The Mandelbrot set is a bit "complex" ;) but can be written as a simple recursive rule.  The Mandelbrot set is the set of complex numbers c for which the sequence (c, c² + c, (c²+c)² + c, ((c²+c)²+c)² +c, (((c²+c)²+c)²+c)² + c, …) does not approach infinity.  It's defined recursively as $z_{n+1}=z_n^2+c$.  Some numbers like c=1 do not belong to the Mandebrot set because the rule would go to infinity.  However, c= -1 is in the set because that sequence oscillates between 0 and -1 so it does not approach infinity. (https://en.wikipedia.org/wiki/Mandelbrot_set)

Anyway, fractals are fun to look at and play with (check out John Golden's Geogebra file) and Mandelbrot was a very interesting man.  He was highly revered for this discovery that is so applicable to nature.  Plus he has some great quotes:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
-The Fractal Geometry of Nature

Bottomless wonders spring from simple rules, which are repeated without end.
-TED talk "Fractals and the art of roughness"