Given a point A0 in a Euclidean space and a translationS, define the point Ai to be the point obtained from i applications of the translation S to A0, so Ai = Si A0. The set of vertices Ai with i any integer, together with edges connecting adjacent vertices, is a sequence of equal segments of a line. The constructed sequence Ai is called the regular apeirogon as defined by H. S. M. Coxeter. in the hyperbolic plane, visualised using the Poincaré disk model. A regular apeirogon can be defined as a partition of the Euclidean lineE1 into infinitely many equal-length segments, generalizing the regular n-gon, which can be defined as a partition of the circleS1 into finitely many equal-length segments.
Modern abstract definition
An abstract polytope is a partially ordered setP with properties modeling those of the inclusions of faces of convex polytopes. The rank 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. For abstract polytopes of rank 2, this means that the elements of the partially ordered set are sets of vertices with either zero vertices, one vertex, two vertices, 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. An abstract polytope is called an abstract apeirotope if it has infinitely many elements, and an abstract 2-apeirotope is called an abstract apeirogon. In an abstract polytope, a flag is a collection of one face of each dimension, all incident to each other and an abstract polytope is called regular if it has symmetries 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.
Pseudogon
The regular pseudogon is a partition of the hyperbolic lineH1 (instead of the Euclidean line
Realizations of abstract apeirogons
A realization of an abstract apeirogon is defined as a mapping from its vertices to a finite-dimensional geometric space such that every symmetry of the abstract apeirogon corresponds to an isometry of the images of the mapping. Two realizations are called congruent if the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces. 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. 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. Generally, the moduli space of realizations of an abstract polytope is a convex cone of infinite dimension. The realization cone of the abstract apeirogon has uncountably infinitealgebraic dimension and cannot be closed in the Euclidean topology.
Classification of Euclidean apeirogons
The realizations of two-dimensional abstract polytopes, 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,
antiprisms, and
infinite helical polygons.
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
The 3-dimensional analogues of apeirogons are apeirohedra, which are themselves the infinite analogues of polyhedra. More generally, an apeirotope, or infinite polytope, is the n-dimensional analogue of apeirogons and the infinite analogue of n-polytopes.
