Apeirotope

In geometry, an apeirotope or infinite polytope is a generalized polytope which has infinitely many facets.

The regular hexagonal tiling is an example of a 3-dimensional apeirotope

Definition

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][2] An abstract polytope is called an abstract apeirotope if it has infinitely many faces.^[3]

An abstract polytope is called regular if its automorphism group Γ(P) acts transitively on all of the flags of P.^[4]

Classification

There are two main geometric classes of apeirotope:^[5]

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

Honeycombs

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

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.^[6] 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)