The set of vertices Ai 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. S. M.

[1] A regular apeirogon can be defined as a partition of the Euclidean line E1 into infinitely many equal-length segments.

It generalizes the regular n-gon, which may be defined as a partition of the circle S1 into finitely many equal-length segments.

The infinite dihedral group G of symmetries of a regular geometric apeirogon is generated by two reflections, the product of which translates each vertex of P to the next.

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 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.

[3]: 121 [4]: 225 Generally, the moduli space of a faithful realization 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] In three dimensions the discrete regular apeirogons are the infinite helical polygons,[5] with vertices spaced evenly along a helix.