A074455 Consider 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.

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, 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, 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

Offset

1,1

Comments

From David W. Wilson, Jul 12 2007: (Start)

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.

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)

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9.

Links

Table of n, a(n) for n=1..102.

Brian Hayes, An Adventure in the Nth Dimension, 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; alternative link.

Eric Weisstein's World of Mathematics, Ball.

Formula

d = root of Psi((1/2)*d + 1) = log(Pi).

d is 2 less than the number with decimal digits A074457 (the hypersphere dimension that maximizes hypersurface area). - Eric W. Weisstein, Dec 02 2014

Example

5.256946404860576780132838388690769236619017237183214857509879678777109...

Mathematica

x /. FindRoot[ PolyGamma[1 + x/2] == Log[Pi], {x, 5}, WorkingPrecision -> 105] // RealDigits // First (* Jean-François Alcover, Mar 28 2013 *)

Prog

(PARI)
hyperspheresurface(d)=2*Pi^(d/2)/gamma(d/2)
hyperspherevolume(d)=hyperspheresurface(d)/d
FindMax(fn_x, lo, hi)=
{
local(oldprecision, x, y, z);
oldprecision = default(realprecision);
default(realprecision, oldprecision+10);
while (hi-lo > 10^-oldprecision,
while (1,
z = vector(2, i, lo*(3-i)/3 + hi*i/3);
y = vector(2, i, eval(Str("x = z[" i "]; " fn_x)));
if (abs(y[1]-y[2]) > 10^(5-default(realprecision)), break);
default(realprecision, default(realprecision)+10);
);
if (y[1] < y[2], lo = z[1], hi = z[2]);
);
default(realprecision, oldprecision);
(lo + hi) / 2.
}
default(realprecision, 105);
A074455=FindMax("hyperspherevolume(x)", 1, 9)
A074457=FindMax("hyperspheresurface(x)", 1, 9)
A074454=hyperspherevolume(A074455)
A074456=hyperspheresurface(A074457)
/* David W. Cantrell */

Crossrefs

Cf. A074457.

The volume is given by A074454. Cf. A072345 & A072346.

Sequence in context: A082571 A354195 A087300 * A339161 A142702 A236184

Adjacent sequences: A074452 A074453 A074454 * A074456 A074457 A074458

Keyword

cons,nonn

Author

Robert G. Wilson v, Aug 22 2002

Extensions

Corrected by Eric W. Weisstein, Aug 31 2003

Corrected by Martin Fuller, Jul 12 2007

Status

approved