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A361620
Decimal expansion of the median of the distribution of the least of the nine acute angles between pairs of edges of two randomly disoriented cubes (in radians).
3
4, 0, 4, 6, 2, 6, 8, 0, 0, 8, 3, 8, 5, 0, 1, 3, 8, 4, 7, 5, 1, 4, 4, 5, 0, 0, 3, 5, 7, 4, 1, 4, 1, 8, 3, 6, 4, 7, 2, 6, 7, 2, 3, 3, 6, 3, 2, 8, 7, 8, 7, 1, 8, 8, 0, 0, 2, 1, 0, 5, 9, 0, 6, 4, 9, 0, 1, 2, 9, 7, 2, 4, 8, 6, 7, 3, 5, 2, 3, 2, 0, 8, 3, 1, 7, 2, 2, 6, 8, 7, 8, 9, 1, 7, 1, 8, 2, 7, 8, 7, 9, 1, 0, 9, 7
OFFSET
0,1
COMMENTS
The corresponding value in degrees is 23.1834079659...
LINKS
J. K. Mackenzie, Second Paper on Statistics Associated with the Random Disorientation of Cubes, Biometrika, Vol. 45, No. 1-2 (1958), pp. 229-240.
J. K. Mackenzie and M. J. Thomson, Some Statistics Associated with the Random Disorientation of Cubes, Biometrika, Vol. 44, No. 1-2 (1957), pp. 205-210.
Eric Weisstein's World of Mathematics, Median.
Wikipedia, Misorientation.
FORMULA
Equals c such that Integral_{t=0..c} P(t) dt = 1/2, where P(t) is given in the Formula section of A361618.
EXAMPLE
0.40462680083850138475144500357414183647267233632878...
MATHEMATICA
wp = 110; p[a_?NumericQ] := If[a <= 0 || a >= Pi/4, 0, (288/Pi^2) * Sin[a]*(Pi^2/32 - NIntegrate[ArcSin[Tan[a/2]*Cos[x]], {x, 0, Pi/4}, WorkingPrecision -> wp])]; f[y_?NumericQ] := NIntegrate[p[a], {a, 0, y}, WorkingPrecision -> wp]; RealDigits[y /. FindRoot[f[y] == 1/2, {y, 0.5}, WorkingPrecision -> wp], 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Mar 18 2023
STATUS
approved