cp's OEIS Frontend

A074455 Consider the volume of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the best d.

%I A074455 #52 May 12 2025 10:13:43
%S A074455 5,2,5,6,9,4,6,4,0,4,8,6,0,5,7,6,7,8,0,1,3,2,8,3,8,3,8,8,6,9,0,7,6,9,
%T A074455 2,3,6,6,1,9,0,1,7,2,3,7,1,8,3,2,1,4,8,5,7,5,0,9,8,7,9,6,7,8,7,7,7,1,
%U A074455 0,9,3,4,6,7,3,6,8,2,0,2,7,2,8,1,7,7,2,0,2,3,8,4,8,9,7,9,2,4,6,9,2,6
%N A074455 Consider the volume of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the best d.
%C A074455 From _David W. Wilson_, Jul 12 2007: (Start)
%C A074455 For an integer d, the volume of a d-dimensional unit ball is v(d) = Pi^(d/2)/(d/2)! and its surface area is area(d) = d*Pi^(d/2)/(d/2)! = d*v(d). If we interpolate n! = gamma(n+1) we can define v(d) and area(d) as continuous functions for (at least) d >= 0.
%C A074455 A074457 purports to minimize area(d). Since area(d+2) = 2*Pi*v(d), area() is minimized at y = x+2; therefore A074457 coincides with the current sequence except at the first term. (End)
%D A074455 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9.
%D A074455 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.
%D A074455 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 67.
%H A074455 Brian Hayes, <a href="https://doi.org/10.1515/9781400844678-006">An Adventure in the Nth Dimension</a>, pp. 30-42 of M. Pitici, editor, The Best Writing on Mathematics 2012, Princeton Univ. Press, 2012. See p. 42. - From _N. J. A. Sloane_, Jan 13 2013; <a href="http://www.peterbeerli.com/classes/images/2/20/AmSci2011Hayes2.pdf">alternative link</a>.
%H A074455 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Ball.html">Ball</a>.
%F A074455 d = root of Psi((1/2)*d + 1) = log(Pi).
%F A074455 d is 2 less than the number with decimal digits A074457 (the hypersphere dimension that maximizes hypersurface area). - _Eric W. Weisstein_, Dec 02 2014
%e A074455 5.256946404860576780132838388690769236619017237183214857509879678777109...
%t A074455 x /. FindRoot[ PolyGamma[1 + x/2] == Log[Pi], {x, 5}, WorkingPrecision -> 105] // RealDigits // First (* _Jean-François Alcover_, Mar 28 2013 *)
%o A074455 (PARI)
%o A074455 hyperspheresurface(d)=2*Pi^(d/2)/gamma(d/2)
%o A074455 hyperspherevolume(d)=hyperspheresurface(d)/d
%o A074455 FindMax(fn_x,lo,hi)=
%o A074455 {
%o A074455 local(oldprecision, x, y, z);
%o A074455 oldprecision = default(realprecision);
%o A074455 default(realprecision, oldprecision+10);
%o A074455 while (hi-lo > 10^-oldprecision,
%o A074455 while (1,
%o A074455 z = vector(2, i, lo*(3-i)/3 + hi*i/3);
%o A074455 y = vector(2, i, eval(Str("x = z[" i "]; " fn_x)));
%o A074455 if (abs(y[1]-y[2]) > 10^(5-default(realprecision)), break);
%o A074455 default(realprecision, default(realprecision)+10);
%o A074455 );
%o A074455 if (y[1] < y[2], lo = z[1], hi = z[2]);
%o A074455 );
%o A074455 default(realprecision, oldprecision);
%o A074455 (lo + hi) / 2.
%o A074455 }
%o A074455 default(realprecision, 105);
%o A074455 A074455=FindMax("hyperspherevolume(x)", 1, 9)
%o A074455 A074457=FindMax("hyperspheresurface(x)", 1, 9)
%o A074455 A074454=hyperspherevolume(A074455)
%o A074455 A074456=hyperspheresurface(A074457)
%o A074455 /* David W. Cantrell */
%o A074455 (PARI) 2 * (solve(x=3, 4, psi(x) - log(Pi)) - 1) \\ _Jianing Song_, May 12 2025
%Y A074455 Cf. A074457.
%Y A074455 The volume is given by A074454. Cf. A072345 & A072346.
%K A074455 cons,nonn
%O A074455 1,1
%A A074455 _Robert G. Wilson v_, Aug 22 2002
%E A074455 Corrected by _Eric W. Weisstein_, Aug 31 2003
%E A074455 Corrected by _Martin Fuller_, Jul 12 2007