: 22–25 : 224Īn abstract polytope is called an abstract apeirotope if it has infinitely many elements an abstract 2-apeirotope is called an abstract apeirogon. : 22–25įor abstract polytopes of rank 2, this means that: A) the elements of the partially ordered set are sets of vertices with either zero vertex (the empty set), one vertex, two vertices (an edge), or the entire vertex set (a two-dimensional face), ordered by inclusion of sets B) each vertex belongs to exactly two edges C) the undirected graph formed by the vertices and edges is connected. 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.

Modern abstract definition Īn 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. It generalizes the regular n-gon, which may be defined as a partition of the circle S 1 into finitely many equal-length segments. Ī regular apeirogon can be defined as a partition of the Euclidean line E 1 into infinitely many equal-length segments. The set of vertices A i with i any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the regular apeirogon as defined by H. Given a point A 0 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 A 0, so A i = S i(A 0). Definitions Classical constructive definition In some literature, the term 'apeirogon' may refer only to the regular apeirogon, with an infinite dihedral group of symmetries. Apeirogons are the two-dimensional case of infinite polytopes. In geometry, an apeirogon (from Ancient Greek ἄπειρος apeiros 'infinite, boundless', and γωνία gonia 'angle') or infinite polygon is a polygon with an infinite number of sides. An apeirogon can be defined as a partition of the Euclidean line into infinitely many equal-length segments.