A074455 - OEIS (original) (raw)

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.

8

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

COMMENTS

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.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 67.

LINKS

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)

/* David W. Cantrell */

(PARI) 2 * (solve(x=3, 4, psi(x) - log(Pi)) - 1) \\ Jianing Song, May 12 2025