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In geometry, an apeirogon () or infinite polygon is a polygon with an Infinity number of sides. Apeirogons are the rank 2 case of Apeirotope. In some literature, the term "apeirogon" may refer only to the Regular polygon apeirogon, with an infinite dihedral group of symmetry.
A regular apeirogon can be defined as a partition of the Euclidean line E1 into infinitely many equal-length segments. It generalizes the regular polygon, which may be defined as a partition of the circle S1 into finitely many equal-length segments.
For 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.
An abstract polytope is called an abstract apeirotope if it has infinitely many elements; an abstract 2-apeirotope is called an abstract apeirogon.
A realization of an abstract polytope is a mapping of its vertices to points a geometric space (typically a Euclidean space). A faithful realization is a realization such that the vertex mapping is injective. provide a stricter definition, requiring that the induced maps on higher rank faces be injective as well. However a regular polytope is either degenerate in which case it has no faithful realizations, or every vertex-faithful realization is faithful. The apeirogon is not degenerate and thus this condition is sufficient to show its realizations are faithful. Every geometric apeirogon is a realization of the abstract apeirogon.
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); an abstract polytope is called regular if it has symmetries (structure-preserving permutations of its elements) that take any flag to any 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.
A symmetric 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.
In two dimensions the discrete regular apeirogons are the Skew apeirogon, resulting from the blend of the 1-dimensional apeirogon with the digon, represented with the Schläfli symbol , , or .
In three dimensions the discrete regular apeirogons are the infinite helical polygons, with vertices spaced evenly along a helix. These are the result of blending the 1-dimensional apeirogon with a 2-dimensional polygon, or .
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