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I'm currently working on a specialized game board, which will be a hyperbolic plane tiled with octagons, as a personal that I'm getting someone to crochet for me. Right now, the plan is to crochet a bunch of octagons and stitch them together. My (very rough and possibly inaccurate) models indicate that I will need four colours to tile the plane. At the moment, I'm wondering if the four colour theorem extends to hyperbolic space.

If it does not, then how many colours will I need?

1337w0n
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    One very quick way of seeing this: the hyperbolic plane has many different models in the Euclidean plane - they don't necessarily preserve lines as lines (or angles as angles), but since they're continuous mappings they preserve all of the connectivity information, and the 4CT doesn't require any particular shape of its regions so the underlying geometry of the space doesn't matter. – Steven Stadnicki Aug 10 '17 at 04:11
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    Well, the four-colour theorem is valid for all tilings of the plane, even ones that don't look regular when viewed as euclidean. If there were a tiling of the hyperbolic plane that needed more than four colours (at least if it had a finite portion which required five colours), that would violate the four-colour theorem, even if it happens inside a circle with a tiling of octagons with curved edges. That being said, depending on what exactly your tiling is, you might get away with $2$ or $3$ colours. – Arthur Aug 10 '17 at 07:53
  • Why octagons? You will be only able to make a very small board out of octagons. For my own game project (HyperRogue) even heptagons were too large, so I have used a tesselation of heptagons and hexagons. Also, how did the game turn out? :) – Zeno Rogue Feb 23 '18 at 21:04

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