(no subject)

[gfish]

I was in Clarkston this weekend, doing this thing, when I noticed something really cool about the huuj, ginormous bluffs that tower over the city.



Sierpinski triangles! Roughly. It doesn't come through on the image as well as I'd like. (It's a link to a much larger version, but even that one doesn't really capture it. And half of you won't be able to connect to cyphertext.net anyway.) But yay, found math!

Edit:
I was bored, so I highlighted the ridge lines. (No, I don't claim I was particularly impartial in the process. The point here is to share an experience, not to prove a theory.)

Comments

[gfish.livejournal.com]

Waclaw Sierpinski, Polish mathematician.

It's a very basic fractal. (Also known as a Sierpinski gasket.) Take an equilateral triangle. Divide into 4 smaller equilateral triangles. Remove the center one. Recursively apply to the remaining 3. It quickly approaches something with an infinite border but zero surface area. A Menger sponge (http://mathworld.wolfram.com/mimg1726.gif) is a related, three-dimensional concept.

The same shape is created if you take Pascal's triangle (http://www.alunw.freeuk.com/pascal.html) and separate it into even and odd components.

They can also be generated through cellular automata, which is what I was thinking of here. Like Conway's Life, but one-dimensional. If a cell has one or two neighbors, turn it on, else turn it off. If you start with a single cell and plot the result into the second dimension, you get a Sierpinski triangle. If you start with random cells turned on, you end up with something like the seashell I linked to above.