The apeirogon is an extension of the definition of regular polygon to a figure with an infinite number of sides. Its Schläfli symbol is {infty}.

The apeirogon can produce a regular tiling on the hyperbolic plane. This is achieved by letting each edge of a regular polygon have length s, and each internal angle of the polygon be theta. Then construct triangle ABC, where C is the midpoint of an edge, B is an adjacent vertex, and A is the center of the polygon. This is a right triangle, with the right angle at C. But the length of side a is s/2, and the angle B is theta/2. The length of side c, 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,


The value of sec(theta/2) is never less than one, while the value of tanh(s/2) increases from zero to one as s increases. Only for small values of s is tanhc less than one, which it has to be for any real value of c. Thus, by making s large enough, the figure has an infinite number of sides and is an apeirogon. If tanh(s/2)sec(theta/2)=1, 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 {infty,p}, indicating that p apeirogons meet at each vertex. The interior angle theta is then equal to 2pi/p. To find the minimum edge length, solve the equation


For example, if p=3, then


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 {infty,2}. To construct it, divide a line into equal segments which are the edges of the apeirogons. The internal angles are pi, and the interiors of the two apeirogons that tile the plane are the two half planes on either side of the line.

See also

Circle, Regular Polygon

This entry contributed by Robert A. Russell

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Coxeter, H. S. M. "Regular Honeycombs in Hyperbolic Space." In Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Vol. 3. Groningen, Netherlands: Noordhoff, pp. 155-169, 1956. Reprinted as Ch. 10 in The Beauty of Geometry: Twelve Essays. New York: Dover, pp. 200-214, 1999.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 45, 1973.Coxeter, H. S. M. "Various Definitions for a Circle." §11.1 inNon-Euclidean Geometry, 6th ed. Washington, DC: Math. Assoc. Amer., p. 213, 1988.Leys, J. "Fractals by Jos Leys: Hyperbolic05.", S. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. Washington, DC: Math. Assoc. Amer., 1994.

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Cite this as:

Russell, Robert A. "Apeirogon." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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