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An apeirogon is a degenerate polygon with an infinite number of sides. It is the limit of a sequence of polygons with more and more sides.

Like any polygon, it is a sequence of line segments (edges) and angles (corners). But whereas an ordinary polygon is bounded because it is closed, an apeirogon may be unbounded. On the other hand, it is easy to show that bounded apeirogons exist. This can happen when the lengths of the sides form sequences (one in each direction, starting from any point) whose sums converge.

A regular apeirogon has equal-length sides and equal corner angles, just like any regular polygon.

If the corner angles are 180 degrees, the overall form of the apeirogon resembles a circle of infinite radius or a straight line:

  . . . --o-----o-----o-----o-----o-----o-----o-----o-----o-- . . .

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:

              o       o       o       o       o
             / \     / \     / \     / \     / \
  . . . \   /   \   /   \   /   \   /   \   /   \ . . .
         \ /     \ /     \ /     \ /     \ /
          o       o       o       o       o

If each corner angle is displaced out of the plane of the previous angle, the apeirogon resembles a helix (which defeats ASCII art). A polygon which, like this family, does not lie in a plane, is said to be skew.

• Coxeter, H. S. M. (1973). Regular Polytopes, 3rd ed., New York: Dover Publications, 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

• Russell, Robert, Apeirogon at MathWorld.
• Olshevsky, George, Apeirogon at Glossary for Hyperspace.