|Edges and vertices||∞|
Like any polygon, it is a sequence of line segments (edges) and angles (corners). But whereas an ordinary polygon has no ends because it is a closed circuit, an apeirogon can also have no ends because you can never make the infinite number of steps needed to get to the end in either direction. Closed apeirogons also exist. They occur when the corners form sequences (one in each direction, starting from any point) whose limits converge on the same point. Such a point is called an accumulation point, and any closed apeirogon must have at least one of them.
If the corner angles are 180°, the overall form of the apeirogon resembles a straight line:
This line may be considered as a circle of infinite radius, by analogy with regular polygons with great number of edges, which resemble a circle.
For some time, people thought this was the only regular example. Then Branko Grünbaum discovered two more.
If the corner angles alternate either side of the figure, the apeirogon resembles a zig-zag, and has 2*∞ Frieze group symmetry. However, this form is only regular if one does not label one side of the plane the body (or interior) of the apeirogon, and instead treats the apeirogon as a bodyless figure.
If each corner angle is displaced out of the plane of the previous angle, the apeirogon resembles a three-dimensional helix. A polygon such as this which does not lie in a plane, is said to be skew. The sketch on the right is a 3D perspective view of such a regular skew apeirogon.
This polygon can be constructed from a sequential subset of edges within an infinite stack of uniform n-gonal antiprisms, although unlike the antiprisms, the twist angle is not limited to an integer divisor of 180°. This polygon has screw axis. A sequence of edges of a Boerdijk–Coxeter helix can represent a regular skew apeirogon.
- Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed. ed.). New York: Dover Publications. pp. 121–122. ISBN 0-486-61480-8. p.296, Table II: Regular honeycombs
- Grünbaum, B. Regular polyhedra - old and new, Aequationes Math. 16 (1977) p. 1-20
- Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 25)