An apeirotope or infinite polytope is a polytope which has infinitely many facets. There are two main geometric classes of apeirotope:[1]


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

Skew apeirogons

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

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.[2] 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)


  1. Grünbaum, B.; "Regular Polyhedra—Old and New", Aeqationes mathematicae, Vol. 16 (1977), pp 1–20.
  2. 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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