Fractal calculations, Kanagawa vibes
November 4, 2022 12:44 PM Subscribe
Koch snowflake a little too symmetrical for you? Consider the Kochawave Curve [pdf] (arxiv.org page), a variant that leans hard to the side and has a number of interesting properties and tilings of its own.
What the heck?!?! https://en.wikipedia.org/wiki/The_Great_Wave_off_Kanagawa
posted by metametamind at 3:39 PM on November 4, 2022 [2 favorites]
posted by metametamind at 3:39 PM on November 4, 2022 [2 favorites]
Ahhhhhhh this is so beautiful! I hope that Etsy artist who makes fractal jewelry does something with this. Thank you, cortex.
posted by eirias at 4:48 AM on November 5, 2022
posted by eirias at 4:48 AM on November 5, 2022
Very cool. I didn't even bother looking at the math. The pictures were very pretty though.
posted by kathrynm at 10:05 AM on November 5, 2022
posted by kathrynm at 10:05 AM on November 5, 2022
Weird. I'd seen the construction of the basic Cantor set like this, where you remove the (open) middle 1/3 of the unit interval [0,1] and repeat with the segments.
It was my favorite example of a set of uncountably infinite numbers that still have Lebesgue measure of zero, the same as a single point. Poking around at that turns out you can get similar uncountably infinite sets of numbers, none of them "next" to each other, but still with a Lebesgue measure > zero. They're called "fat Cantor sets".
posted by aleph at 12:08 PM on November 5, 2022
It was my favorite example of a set of uncountably infinite numbers that still have Lebesgue measure of zero, the same as a single point. Poking around at that turns out you can get similar uncountably infinite sets of numbers, none of them "next" to each other, but still with a Lebesgue measure > zero. They're called "fat Cantor sets".
posted by aleph at 12:08 PM on November 5, 2022
I like how the first drawing is...okay that's an interesting thing to draw. Then you see it iterated and it's a very cool wave fractal. Then they show you how to take the center out of a triangle and rotate it and...whoa that's the same thing, a triangle of the same pattern.
And then they keep showing cooler versions of it.
I really like how it looks like it just barely fits inside itself as it curves around. And then when they shade it on Page 9, you can see, no there isn't any space at all between the big wave and the smaller one it curls around. Which should be obvious at the second iteration when you see that the center part touches the one on the right. (Which is, I think, what they're talking about at 3.11)
posted by straight at 12:34 PM on November 5, 2022
And then they keep showing cooler versions of it.
I really like how it looks like it just barely fits inside itself as it curves around. And then when they shade it on Page 9, you can see, no there isn't any space at all between the big wave and the smaller one it curls around. Which should be obvious at the second iteration when you see that the center part touches the one on the right. (Which is, I think, what they're talking about at 3.11)
posted by straight at 12:34 PM on November 5, 2022
The thing I like about fractals is how they show up everywhere.
posted by aniola at 8:35 PM on November 6, 2022
posted by aniola at 8:35 PM on November 6, 2022
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[I need to clean my grout]
posted by clew at 12:57 PM on November 4, 2022