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There are thirty regular apeirohedra in
Euclidean space. 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)
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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.
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Tilings of the plane and close-packed space-fillings of polyhedra are examples of honeycombs in two and three dimensions respectively.
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Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular
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contains a least face and a greatest face, each maximal totally ordered subset (called a
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There are three regular skew apeirohedra, which look rather like polyhedral sponges:
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is strongly connected, and there are exactly two faces that lie strictly between
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are two faces whose ranks differ by two. An abstract polytope is called an
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A line divided into infinitely many finite segments is an example of an
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The regular hexagonal tiling is an example of a 3-dimensional apeirotope
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traces out a helical spiral and may be either left- or right-handed.
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Grünbaum, B. (1977). "Regular
Polyhedra—Old and New".
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6 hexagons around each vertex, Coxeter symbol {6,6|3}
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4 hexagons around each vertex, Coxeter symbol {6,4|4}
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6 squares around each vertex, Coxeter symbol {4,6|4}
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There are two main geometric classes of apeirotope:
177:dimensions is an infinite example of a polytope in
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158:-dimensional manifold in a higher space
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107:if it has infinitely many faces.
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75:(whose elements are called
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445:Multi-dimensional geometry
395:Abstract Regular Polytopes
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43:which has infinitely many
227:Regular skew apeirohedron
405:10.1017/CBO9780511546686
358:Aequationes Mathematicae
342:Aeqationes mathematicae
221:Infinite skew polyhedra
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70:partially ordered set
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169:Honeycomb (geometry)
105:abstract apeirotope
87:) contains exactly
56:Abstract apeirotope
370:10.1007/BF01832961
146:-dimensional space
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39:is a generalized
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434:Categories
250:References
163:Honeycombs
135:honeycombs
51:Definition
33:apeirotope
440:Polytopes
189:apeirogon
66:-polytope
62:abstract
391:(2002),
41:polytope
29:geometry
423:1965665
378:1268033
348:: 1–20.
112:regular
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45:facets
77:faces
68:is a
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