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:
I am currently still working on hyperbolic tools. This is the first tiling I choose to drew. Here 7 equilateral triangles meet at each point. The quality is not too good yet and I want to make it into an animation. Also I have to fix some of the calculation.