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Polygon Triangle Picking


PolygonTrianglePicking

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

 A^__n=(9cos^2omega+52cosomega+44)/(36n^2sin^2omega),
(1)

where omega=2pi/n (Alikoski 1939; Solomon 1978, p. 109; Croft et al. 1991, p. 54). Prior to Alikoski's work, only the special cases n=3, 4, 6, 8, and infty had been determined. The first few cases are summarized in the following table, where A^__7 is the largest root of

 784147392x^3-84015792x^2+2125620x-15289=0,
(2)

and A^__9 is the largest root of

 24794911296x^3-2525407632x^2+55366092x-312427=0.
(3)

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


See also

Hexagon Triangle Picking, Square Triangle Picking, Pentagon Triangle Picking, Sylvester's Four-Point Problem, Triangle Triangle Picking

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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.

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Polygon Triangle Picking

Cite this as:

Weisstein, Eric W. "Polygon Triangle Picking." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolygonTrianglePicking.html

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