Apeirogon

In geometry, an apeirogon (from the Greek word ἄπειρος apeiros, "infinite, boundless" and γωνία gonia, "angle") or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.

The regular apeirogon
Edges and vertices	∞
Schläfli symbol	{∞}
Coxeter diagram
Internal angle (degrees)	180°
Dual polygon	Self-dual

An apeirogon can be defined as a partition of the Euclidean line into infinitely many equal-length segments.

In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.[1]

Definitions

Classical constructive definition

Given a point A₀ in a Euclidean space and a translation S, define the point A_i to be the point obtained from i applications of the translation S to A₀, so A_i = Sⁱ A₀. The set of vertices A_i with i any integer, together with edges connecting adjacent vertices, is a sequence of equal segments of a line. The constructed sequence A_i is called the regular apeirogon as defined by H. S. M. Coxeter.[1]

An example of apeirogonal tiling in the hyperbolic plane, visualised using the Poincaré disk model.

A regular apeirogon can be defined as a partition of the Euclidean line E¹ into infinitely many equal-length segments, generalizing the regular n-gon, which can be defined as a partition of the circle S¹ into finitely many equal-length segments.[2]

Modern abstract definition

An abstract polytope is a partially ordered set P (whose elements are called faces) with properties modeling those of the inclusions of faces of convex polytopes. The rank (or dimension) of an abstract polytope is determined by the length of the maximal ordered chains of its faces, and an abstract polytope of rank n is called an abstract n-polytope.[3]^:22–25

For abstract polytopes of rank 2, this means that the elements of the partially ordered set are sets of vertices with either zero vertices (the empty set), one vertex, two vertices (an edge), or the entire vertex set, ordered by inclusion of sets, that each vertex belongs to exactly two edges, and that the undirected graph formed by the vertices and edges is connected.[3]^:22–25[4]^:224 An abstract polytope is called an abstract apeirotope if it has infinitely many elements, and an abstract 2-apeirotope is called an abstract apeirogon.[3]^:25

In an abstract polytope, a flag is a collection of one face of each dimension, all incident to each other (that is, comparable in the partial order) and an abstract polytope is called regular if it has symmetries (structure-preserving permutations of its elements) that take every flag to every other flag. In the case of a two-dimensional abstract polytope, this is automatically true; the symmetries of the apeirogon form the infinite dihedral group.[3]^:31

Pseudogon

The regular pseudogon is a partition of the hyperbolic line H¹ (instead of the Euclidean line} into segments of length 2λ, as an analogue of the regular apeirogon.[2]

Realizations of abstract apeirogons

A realization of an abstract apeirogon is defined as a mapping from its vertices to a finite-dimensional geometric space (typically a Euclidean space) such that every symmetry of the abstract apeirogon corresponds to an isometry of the images of the mapping.[3]^:121[4]^:225 Two realizations are called congruent if the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces.[3]^:126[4]^:229 The classical definition of an apeirogon as an equally-spaced subdivision of the Euclidean line is a realization in this sense, as is the convex subset in the hyperbolic plane formed by the convex hull of equally-spaced points on a horocycle. Other realizations are possible in higher-dimensional spaces.

The infinite dihedral group G of symmetries of a realization V of an abstract apeirogon P is generated by two reflections, the product of which translates each vertex of P to the next.[3]^:140–141[4]^:231 The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and a possibly trivial reflection.[3]^:141[4]^:231

Generally, the moduli space of realizations of an abstract polytope is a convex cone of infinite dimension.[3]^:127[4]^:229–230 The realization cone of the abstract apeirogon has uncountably infinite algebraic dimension and cannot be closed in the Euclidean topology.[3]^:141[4]^:232

Classification of Euclidean apeirogons

The realizations of two-dimensional abstract polytopes (including both polygons and apeirogons), in Euclidean spaces of at most three dimensions can be classified into six types:

• convex polygons,
• star polygons,
• regular apeirogons in the Euclidean line,
• infinite skew polygons (infinite zig-zag polygons in the Euclidean plane),
• antiprisms (including star prisms and star antiprisms), and
• infinite helical polygons (evenly spaced points along a helix).[5]

Abstract apeirogons may be realized in all of these ways, in some cases mapping infinitely many different vertices of an abstract apeirogon onto finitely many points of the realization. An apeirogon also admits star polygon and antiprismatic realizations with a non-discrete set of infinitely many points.

Generalizations

Higher dimension

Main articles: Apeirotope and Apeirohedron

The 3-dimensional analogues of apeirogons are apeirohedra, which are themselves the infinite analogues of polyhedra.[6] More generally, an apeirotope, or infinite polytope, is the n-dimensional analogue of apeirogons and the infinite analogue of n-polytopes.[3]^:22–25

See also

• Apeirogonal tiling
• Apeirogonal prism
• Apeirogonal antiprism

References

1. Coxeter, H. S. M. (1948). Regular polytopes. London: Methuen & Co. Ltd. p. 45.
2. Johnson, Norman W. (2018). "11: Finite Symmetry Groups". Geometries and transformations. Cambridge University Press. p. 226.
3. McMullen, Peter; Schulte, Egon (December 2002). Abstract Regular Polytopes (1st ed.). Cambridge University Press. ISBN 0-521-81496-0.
4. McMullen, Peter (1994), "Realizations of regular apeirotopes", Aequationes Mathematicae, 47 (2–3): 223–239, doi:10.1007/BF01832961, MR 1268033
5. Grünbaum, B. (1977). "Regular polyhedra – old and new". Aequationes Mathematicae. 16 (1–2): 119. doi:10.1007/BF01836414.
6. Coxeter, H. S. M. (1937). "Regular Skew Polyhedra in Three and Four Dimensions". Proc. London Math. Soc. 43: 33–62.

External links

• Russell, Robert A.. "Apeirogon". MathWorld.
• Olshevsky, George. "Apeirogon". Glossary for Hyperspace. Archived from the original on 4 February 2007.