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
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.
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.
-gon of unit area is
(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
is the largest root 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.
