Polygon Triangle Picking

The mean triangle area of a triangle picked inside a regular -gon of unit area is

(1)

where  (Alikoski 1939; Solomon 1978, p. 109; Croft et al. 1991, p. 54). Prior to Alikoski's work, only the special cases , 4, 6, 8, and  had been determined. The first few cases are summarized in the following table, where  is the largest root of

(2)

and  is the largest root of

(3)

		problem
3		triangle triangle picking
4		square triangle picking
5		pentagon triangle picking
6		hexagon triangle picking
7
8
9
10

Amazingly, the algebraic degree of  is equal to , where  is the totient function, giving the first few terms for , 4, ... as 1, 1, 2, 1, 3, 2, 3, 2, 5, 2, 6, 3, 4, 4, 8, ... (OEIS A023022). Therefore, the only values of  for which  is rational are , 4, and 6.

REFERENCES:

Alikoski, H. A. "Über das Sylvestersche Vierpunktproblem." Ann. Acad. Sci. Fenn. 51, No. 7, 1-10, 1939.

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1991.

Kendall, M. G. "Exact Distribution for the Shape of Random Triangles in Convex Sets." Adv. Appl. Prob. 17, 308-329, 1985.

Kendall, M. G. and Le, H.-L. "Exact Shape Densities for Random Triangles in Convex Polygons." Adv. Appl. Prob. 1986 Suppl., 59-72, 1986.

Sloane, N. J. A. Sequence A023022 in "The On-Line Encyclopedia of Integer Sequences."

Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, pp. 109-114, 1978.