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The hyperbolic tiling order-6 pentagonal tiling is an apeirohedron.

An apeirotope is a polytope with infinitely many facets.[1] The most common examples of these are tilings or honeycombs. This term is almost always used exclusively for polytopes with countably many facets, as any (strongly connected) polytope must have countably many elements.

The apeirogon is the unique connected apeirotopic polygon.

A polytope with finitely many elements may be called finite. Conversely, a polytope that isn't finite may be called infinite. Every apeirotope must be infinite, but not every infinite polytope is an apeirotope – a counterexample is given by the apeirogonal dihedron, and a nondegenerate example is the mucubic honeycomb.

References[edit | edit source]

  1. McMullen, Peter (1994). "Realizations of regular apeirotopes". Aequationes Mathematicae. 47 (2–3): 223–239. doi:10.1007/BF01832961. MR 1268033.