The Sierpinski Gasket [1, Weisstein] is a charming little image. It's actually got some pretty impressive properties. On an (x,y) grid, the easiest way to determine the next gasket goes like this: the color of pixel (a,b) is determined by the color of ((a-1,b-1) + (a+1,b-1)) mod 2.
Consider this: if we set a field of width 2n, and wrap the field (so that -n+1 is equivalent to n and n+1 is equivalent to -n), then we have interesting results with the gasket. If we assume that the only pixel with color value 1 in the first row of the gasket has x-cord 0, then if n is of the form 2^a for a any natural number, the gasket ceases to be complex after n iterations.
I find the generating equation for this gasket to be a very simple one: in fact, I stumbled upon it accidentally while I was playing around with cellular automatons. That such an obvious formula for automatons creates such an impressive object is intriguing. I've got a bit on cellular automatons and Stephen Wolfram in the post dated Jan 6, 2004.
Skiing moguls is a real challenge. I'm able to slam my way through them, and relatively rapidly, but it takes so much work. I'd like to know an easier way to do them; I've found a few sites [2] on the topic, but nothing extremely useful.
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