An abstract n-polytope is a partially ordered setP (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]
Classificationedit
There are two main geometric classes of apeirotope:[5]
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.
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)
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, ISBN0-521-81496-0, MR 1965665
November 27, 2023
apeirotope, geometry, apeirotope, infinite, polytope, generalized, polytope, which, infinitely, many, facets, contents, definition, abstract, apeirotope, classification, honeycombs, skew, apeirotopes, skew, apeirogons, infinite, skew, polyhedra, references, bi. In geometry an apeirotope or infinite polytope is a generalized polytope which has infinitely many facets Contents 1 Definition 1 1 Abstract apeirotope 2 Classification 2 1 Honeycombs 2 2 Skew apeirotopes 2 2 1 Skew apeirogons 2 2 2 Infinite skew polyhedra 3 References 3 1 BibliographyDefinition editAbstract 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 G P acts transitively on all of the flags of P 4 Classification editThere 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 spaceHoneycombs edit 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 edit Skew apeirogons edit 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 edit 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 References edit McMullen amp Schulte 2002 pp 22 25 McMullen 1994 p 224 McMullen amp Schulte 2002 p 25 McMullen amp Schulte 2002 p 31 Grunbaum 1977 McMullen amp Schulte 2002 Section 7E Bibliography edit Grunbaum 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 Retrieved from https en wikipedia org w index php title Apeirotope amp oldid 1090213418, wikipedia, wiki, book, books, library,
