hyperbolic tiling

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

  • 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:

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)