In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, an apeirotope or infinite polytope is a generalized
polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
which has infinitely many
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
.
Definition
Abstract apeirotope
An
abstract ''n''-polytope is a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
''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. An abstract polytope is called an abstract apeirotope if it has infinitely many faces.
An abstract polytope is called ''regular'' if its automorphism group Γ(''P'') acts transitively on all of the flags of ''P''.
Classification
There are two main geometric classes of apeirotope:
*
honeycombs in ''n'' dimensions, which completely fill an
''n''-dimensional space.
*
skew apeirotope
In geometry, an apeirotope or infinite polytope is a generalized polytope which has infinitely many facets.
Definition Abstract apeirotope
An abstract ''n''-polytope is a partially ordered set ''P'' (whose elements are called ''faces'') such tha ...
s, comprising an ''n''-dimensional manifold in a higher space
Honeycombs
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
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to the ...
.
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
In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lin ...
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 hexagons around each vertex, Coxeter symbol
* 6 hexagons around each vertex, Coxeter symbol
There are thirty regular apeirohedra in Euclidean space.
These include those listed above, as well as (in the plane) polytopes of type: , , 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
Bibliography
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Multi-dimensional geometry