The apeirogon is an extension of the definition of regular polygon to a figure with an infinite number of sides. Its Schläfli
symbol is .
The apeirogon can produce a regular tiling on the hyperbolic plane. This is achieved by letting each edge of a regular polygon have length
, and each internal angle of the polygon
be . Then construct triangle , where is the midpoint of an edge, is an adjacent vertex, and is the center of the polygon. This is a right
triangle, with the right angle at . But the length of side is ,
and the angle
is . The length of side , the radius of the circumscribed circle, can the be determined
using the standard formula for a right triangle on a surface that has a constant
curvature of ,
(1)
(2)
The value of
is never less than one, while the value of increases from zero to one as increases. Only for small values of is
less than one, which it has to be for any real value of . Thus, by making large enough, the figure has an infinite number of sides and
is an apeirogon. If ,
the figure is inscribed in a horocycle. If that value is greater than one, the figure
is inscribed in a hypercycle or equidistant curve.
To tile the hyperbolic plane with apeirogons, select a Schläfli symbol , indicating that apeirogons meet at each vertex. The interior angle is then equal to . To find the minimum edge length, solve the equation
(3)
For example, if ,
then
(4)
Of course, a longer edge length could also be used for this tiling.
There are no apeirogons on the sphere, but there is a degenerate regular tiling of the Euclidean plane with apeirogons with Schläfli symbol . To construct it, divide a line into equal segments
which are the edges of the apeirogons. The internal angles are , and the interiors of the two apeirogons that tile the plane
are the two half planes on either side of the line.