hyperbolic

hyperbolic

Hyperbolic flowers.

I managed to fix the calculations around the edges. Now there are now missing parts like in my previous drawing.

Hyperbolic

Hyperbolic tiling.

I could fix a lot of things faster than expected. As you can see this version is much better in quality and in details around the edge. Now comes the harder part, I will make it move!

i did eventually figure out how to build a hyperbolic tiling like this with a compass and a straightedge (but i was too lazy to do it on paper so i did it in geogebra)

ok so I want to point out a neat geometrical fact and I don’t feel like drawing these things so I’ll just use some Wikimedia images

- first of all, if you take
**hexagons**, attach squares to each side, and fill in the gaps with triangles, you get a very nice (“semi-regular” or “Archimedean”) tiling of**Euclidean**2-space, like this:

- now we can replace the hexagons with
**pentagons**(leaving the squares and triangles alone), but obviously there’s too much room in the Euclidean plane for that pattern to work, so we go to a place where there’s less room for stuff:**spherical**2-space (aka the sphere), which we can see is completely tiled by this pattern, like this:

- we can
*also*try to replace the hexagons with**heptagons**, and clearly the Euclidean plane doesn’t have enough room for that, so it turns out that this variant tiles**hyperbolic**2-space (where there’s more room), here presented in the form of the Poincaré disk, like this: