An apeirotope or infinite polytope is a generalized polytope which has infinitely many facets.

Abstract apeirotope

An abstract n-polytope is a partially ordered set P (whose elements are called faces) such that P contains a least face and a greatest face, each maximal totally ordered subset (called a flag) contains exactly n + 2 faces, P is strongly connected, and there are exactly two faces that lie strictly between a and b are two faces whose ranks differ by two.[1]:22–25[2]:224 An abstract polytope is called an abstract apeirotope if it has infinitely many faces.[1]:25

An abstract polytope is called regular if its automorphism group Γ(P) acts transitively on all of the flags of P.[1]:31

There are two main geometric classes of apeirotope:[3]

honeycombs in n dimensions, which completely fill an n-dimensional space.
skew apeirotopes, comprising an n-dimensional manifold in a higher space

Main article: Honeycomb (geometry)

In general, a honeycomb in n dimensions is an infinite example of a polytope in n + 1 dimensions.

Tilings of the plane and close-packed space-fillings of polyhedra are examples of honeycombs in two and three dimensions respectively.

A line divided into infinitely many finite segments is an example of an apeirogon.
Skew apeirotopes
Main article: Skew apeirotope
Skew apeirogons
Main article: Skew apeirogon

A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular.

Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left- or right-handed.
Infinite skew polyhedra
Main article: Regular skew apeirohedron

There are three regular skew apeirohedra, which look rather like polyhedral sponges:

6 squares around each vertex, Coxeter symbol {4,6|4}
4 hexagons around each vertex, Coxeter symbol {6,4|4}
6 hexagons around each vertex, Coxeter symbol {6,6|3}

There are thirty regular apeirohedra in Euclidean space.[4] These include those listed above, as well as (in the plane) polytopes of type: {∞,3}, {∞,4}, {∞,6} and in 3-dimensional space, blends of these with either an apeirogon or a line segment, and the "pure" 3-dimensional apeirohedra (12 in number)

McMullen, Peter; Schulte, Egon (December 2002). Abstract Regular Polytopes (1st ed.). Cambridge University Press. ISBN 0-521-81496-0.
McMullen, Peter (1994), "Realizations of regular apeirotopes", Aequationes Mathematicae, 47 (2–3): 223–239, doi:10.1007/BF01832961, MR 1268033
Grünbaum, B.; "Regular Polyhedra—Old and New", Aeqationes mathematicae, Vol. 16 (1977), pp 1–20.

McMullen & Schulte (2002, Section 7E)

McMullen, Peter; Schulte, Egon (2002), Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, 92, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511546686, ISBN 0-521-81496-0, MR 1965665

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