(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

[helenangel.livejournal.com]

Very, very cool. And great shots! =)

[mrlogic.livejournal.com]

Actually it puts me in mind of the triangles generated by certain 1-dimensional cellular automata, as in examples on this page. (http://psoup.math.wisc.edu/mcell/ca_gallery.html)

Note that some rules generate straight-up Sierpinski triangles, while others introduce more apparent chaos, much like these hills.

Neat!

[anansi133.livejournal.com]

There's a model of Mount St Helen's that is made up of just the ridge lines, everything else is hollow. It's very cool. I wish I had the software to convert DEMs into such models, it would make mountanous areas much more comprehensible.

[gfish.livejournal.com]

Yeah, that's exactly the kind of thing I was thinking of. Sloppy terminology.

Somewhere I have a shell like this one (http://users.frii.com/davejen/shell.jpg), but with a denser pattern more like this (http://psoup.math.wisc.edu/mcell/rules/1dto_marvel.gif). Pretty neat to see nature's laziness at work.

[grinninfoole.livejournal.com]

Who was Sierpinski and what, exactly, defines his or her triangles?

[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.

[mrlogic.livejournal.com]

In case you haven't seen this (http://www.stephenwolfram.com/publications/talks/jmm2004/index.html) -- I think you'd enjoy it.

I can see my house from here!

[gement.livejournal.com]

Nothing to do with math, and I know you're headed for Points North right now, but this was a little creepy:

You took that photo from the parking lot of my old workplace.

Past the initial shiver of scrolling down past a picture of familiar hills next to the icon of a Seattle identity (which is pretty big cognitive dissonance in and of itself), the utter certainty of location was disconcerting.

Re: I can see my house from here!

[gfish.livejournal.com]

Weird. I just rolled down the window and took it while driving.