Apeirotope
In geometry, an apeirotope or infinite polytope is a generalized polytope which has infinitely many facets.
Definition
[edit]Abstract apeirotope
[edit]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
[edit]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
[edit]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
[edit]Skew apeirogons
[edit]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
[edit]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)
References
[edit]- ^ McMullen & Schulte (2002), pp. 22–25.
- ^ McMullen (1994), p. 224.
- ^ McMullen & Schulte (2002), p. 25.
- ^ McMullen & Schulte (2002), p. 31.
- ^ Grünbaum (1977).
- ^ McMullen & Schulte (2002, Section 7E)
Bibliography
[edit]- Grünbaum, B. (1977). "Regular Polyhedra—Old and New". Aeqationes mathematicae. 16: 1–20.
- McMullen, Peter (1994), "Realizations of regular apeirotopes", Aequationes Mathematicae, 47 (2–3): 223–239, doi:10.1007/BF01832961, MR 1268033
- McMullen, Peter; Schulte, Egon (2002), Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511546686, ISBN 0-521-81496-0, MR 1965665