A349605 - OEIS

A349605

Decimal expansion of the probability that the intersection of a cube with random plane that passes through its center is a hexagon.

3, 5, 0, 9, 5, 9, 3, 1, 2, 1, 8, 3, 6, 4, 3, 6, 2, 1, 0, 2, 5, 1, 3, 3, 3, 5, 5, 3, 3, 4, 5, 8, 5, 4, 6, 7, 8, 9, 9, 7, 7, 1, 8, 9, 6, 6, 3, 6, 4, 0, 1, 7, 2, 3, 7, 2, 7, 6, 2, 9, 7, 8, 8, 1, 3, 2, 0, 0, 8, 4, 5, 2, 0, 6, 4, 6, 5, 3, 4, 8, 1, 2, 2, 1, 2, 7, 0, 9, 5, 7, 0, 5, 4, 6, 4, 7, 0, 7, 8, 4, 7, 7, 1, 6, 0

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OFFSET

0,1

COMMENTS

The normal to the random plane is in the direction from the center of the cube to a point uniformly chosen at random on the surface of a sphere whose center coincides with the center of the cube.

LINKS

Table of n, a(n) for n=0..104.

Su Pernu Mero, Problem 2065, Mathematics Magazine, Vol. 92, No. 1, (2019), p. 73; Solution, by Bill Cowieson, ibid., Vol. 93, No. 1 (2020), pp. 76-77.

Yawen Zhang, A purely 3-D geometrical solution to Mathematics Magazine Problem 2065, arXiv:2010.12349 [math.HO], 2020.

FORMULA

Equals 1 - 6*arcsin(1/3)/Pi.

Equals 6*arccos(1/3)/Pi - 2.

EXAMPLE

0.35095931218364362102513335533458546789977189663640...

MATHEMATICA

RealDigits[1 - 6 * ArcSin[1/3] / Pi, 10, 100][[1]]

PROG

(PARI) 1 - 6*asin(1/3)/Pi \\ Michel Marcus, Nov 23 2021

CROSSREFS

Cf. A137914, A188615.