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Regular apeirogon
Regular apeirogon.png
Edges and vertices
Schläfli symbol {∞}
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.png
Dual polygon Self-dual

In Euclidean geometry, an apeirogon is a degenerate polygon with a countably infinite number of sides.

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.

Two apeirogons can tessellate the plane, and the Schläfli symbol for this tessellation is {∞, 2}.

Regular apeirogons[edit]

A regular apeirogon has equal-length sides and equal corner angles, just like any regular polygon. Its Schläfli symbol is {∞}.

If the corner angles are 180°, the overall form of the apeirogon resembles a straight line:

...Regular apeirogon.png...

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.

Skew forms[edit]

A regular apeirogon as an equilateral triangular helix, drawn in perspective.

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.

...Regular apeirogon zig-zag.png...

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.

Hyperbolic geometry[edit]

In hyperbolic geometry, apeirogons are no longer degenerate, and appear in tilings such as {∞, 3}, with the vertices along horocycles.

H2 tiling 23i-1.png

See also[edit]


  • 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)

External links[edit]