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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. 6
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 (list; constant; graph; refs; listen; history; text; internal format)
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.

Brain Hayes, An Adenture 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

LINKS

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

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

5.2569464048605767801328383886907692366190172371832148575098796787771093\

4673682027281772023848979246926957...

MATHEMATICA

x /. FindRoot[ PolyGamma[1 + x/2] == Log[Pi], {x, 5}, WorkingPrecision -> 105] // RealDigits // First (* Jean-Fran├žois 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: A196626 A082571 A087300 * A142702 A236184 A201530

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

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Last modified August 2 18:07 EDT 2015. Contains 260251 sequences.