A074456 -id:A074456 - 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.

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

Robert G. Wilson v, Aug 22 2002

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)

			5.256946404860576780132838388690769236619017237183214857509879678777109...

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.

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.

Cf. A074457.

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

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

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 */

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

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

Corrected by Eric W. Weisstein, Aug 31 2003

Corrected by Martin Fuller, Jul 12 2007

A074457 Consider surface area 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.

Original entry on oeis.org

7, 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

Robert G. Wilson v, Aug 22 2002

			7.256946404860576780132838388690769236619017237183214857509879678777...

Nenad Cakic, Dusko Letic, and Branko Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

Dusko Letic, Nenad Cakic, Branko Davidovic, and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - From _N. J. A. Sloane_, Sep 04 2012
Eric Weisstein's World of Mathematics, Hypersphere.

Surface area is A074456. Cf. A072478, A072479, A074455.

RealDigits[ FindMinimum[ -n*Pi^(n/2)/(n/2)!, {n, 7}, WorkingPrecision -> 125] [[2, 1, 2]]] [[1]]
x /. FindRoot[ PolyGamma[x/2] == Log[Pi], {x, 7}, WorkingPrecision -> 105] // RealDigits // First (* Jean-François Alcover, Mar 28 2013 *)

Equals 2 + A074455.

Corrected by Eric W. Weisstein, Aug 31 2003

Corrected by Martin Fuller, Jul 12 2007

A244619 Decimal expansion of 'theta', the unique positive root of the equation polygamma(x) = log(Pi), where polygamma(x) gives gamma'(x)/gamma(x), that is the logarithmic derivative of the gamma function.

Original entry on oeis.org

3, 6, 2, 8, 4, 7, 3, 2, 0, 2, 4, 3, 0, 2, 8, 8, 3, 9, 0, 0, 6, 6, 4, 1, 9, 1, 9, 4, 3, 4, 5, 3, 8, 4, 6, 1, 8, 3, 0, 9, 5, 0, 8, 6, 1, 8, 5, 9, 1, 6, 0, 7, 4, 2, 8, 7, 5, 4, 9, 3, 9, 8, 3, 9, 3, 8, 8, 5, 5, 4, 6, 7, 3, 3, 6, 8, 4, 1, 0, 1, 3, 6, 4, 0, 8, 8, 6, 0, 1, 1, 9, 2, 4, 4, 8, 9, 6, 2, 3, 4, 6, 3, 4, 7, 8

Offset: 1

Jean-François Alcover, Jul 02 2014

This constant appears in d_a = 2*theta = 7.2569464... and d_v = 2*(theta-1) = 5.2569464..., the fractional dimensions at which d-dimensional spherical surface area and volume, respectively, are maximized. [after Steven Finch]

			3.6284732024302883900664191943453846183...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5.4 Gamma Function, p. 34.

Cf. A074454, A074455, A074456, A074457.

theta = x /. FindRoot[PolyGamma[x] == Log[Pi], {x, 4}, WorkingPrecision -> 105]; RealDigits[theta] // First

polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
solve(x=3.5, 3.7, polygamma(0, x) - log(Pi)) \\ Gheorghe Coserea, Sep 30 2018

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.

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A074457 Consider surface area 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.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A244619 Decimal expansion of 'theta', the unique positive root of the equation polygamma(x) = log(Pi), where polygamma(x) gives gamma'(x)/gamma(x), that is the logarithmic derivative of the gamma function.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

PARI