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Simple question that doesn't appear to have a simple answer: is there a formula for the volume of regular hyperbolic polyhedra? I'm most interested in the regular dodecahedron, but others are interesting too.

Computing volume in hyperbolic space seems very hard; the general case for tetrahedra seems to have been cracked only recently-ish (or maybe rediscovered): https://arxiv.org/abs/1302.4919 I suppose technically the volume for the regular dodecahedron could be constructed by decomposing it into orthoschemes and using the Schläfli function, but that seems... messy.

I'm hoping someone has used the special structure of the regular polyhedra to come up with something nicer? For instance, you can construct honeycombs from most (all?) of them, so if you could come up with a recurrence relation expressing how many cells were at Manhattan distance x from the origin cell, you could relate that to the volume of the hyperbolic ball in the limit to get the volume.

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  • $\begingroup$ By looking at that article one sees that the volume formulas, far from being "rediscovered", are being reviewed with many citations to the literature. $\endgroup$
    – Lee Mosher
    Sep 17, 2020 at 21:19

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