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).
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
Examples
0.40462680083850138475144500357414183647267233632878...
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.
Programs
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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]]
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.
Comments