Hypersphere

Author

Eric W. Weisstein
May 2, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/MultidimensionalGeometry/Hypersphere.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Hypersphere.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

Surface Area

HypersphereSurfaceArea[n_]:=RegionMeasure[Sphere[n],n-1]

HypersphereSurfaceArea[n]

bug(391474): RegionMeasure[Ball[n]] and RegionMeasure[Sphere[n], n - 1] could evaluate for symbolic n

HypersphereSurfaceArea1[n_]:=((2(π^(n/2)))/(Gamma[(n/2)]))

HypersphereSurfaceArea2[n_]:=Piecewise[{{(((2^((n+1)/2)) (π^((n-1)/2)))/((n-2)!!)),Mod[n,2]==1},{((2 (π^(n/2)))/(((n/2)-1)!)),Mod[n,2]==0}}]

HypersphereSurfaceArea[n]

RegionMeasure[Sphere[n],n-1]

Table[RegionMeasure[Sphere[n],n-1]-((2(π^(n/2)))/(Gamma[(n/2)])),{n,20}]//Union

FullSimplify[Subtract@@Through[{HypersphereSurfaceArea,HypersphereSurfaceArea2}[n]],n∈Integers]

TextGrid[Table[{n,Through[{HypersphereSurfaceArea,HypersphereSurfaceArea1,HypersphereSurfaceArea2}[n]]},{n,20}],Dividers->All]

A072478(n)/A072479(n)

Transpose[NumeratorDenominator[Table[HypersphereSurfaceArea[n],{n,20}]]]/.Pi->1

Maximum surface area

Plot[HypersphereSurfaceArea[n],{n,0,15},Epilog->{Red,Dashing[{.02,.02}],Line[{{#,0},{#,35}}&[7.256946404684528]]},AxesLabel->{n,Subscript[S,n]},LabelStyle->Directive[Black,FontFamily->"Times",12]]

eqn=FullSimplify[D[HypersphereSurfaceArea[n],n],n∈Integers&&n>0]

FullSimplify[n/.Solve[eqn==0,n,PositiveReals][[1]]]

N[%,110]

Maximize[((2(π^(n/2)))/(Gamma[(n/2)])),n,PositiveReals]//FullSimplify//Timing

N[%,110]//Column

Volume

bug(391474): RegionMeasure[Ball[n]] and RegionMeasure[Sphere[n], n - 1] could evaluate for symbolic n

RegionMeasure[Ball[n]]

RegionMeasure[Ball[n],n]

Table[RegionMeasure[Ball[n],n],{n,20}]

Table[RegionMeasure[Ball[n]],{n,20}]

Table[((π^(n/2))/(((n/2))!)),{n,20}]

A072345/A072346

Transpose[NumeratorDenominator[Table[RegionMeasure[Ball[n],n],{n,20}]]]/.Pi->1

Maximum

Plot[((π^(n/2))/(((n/2))!)),{n,0,10}]

eqn=FullSimplify[D[((π^(n/2))/(((n/2))!)),n],n∈Integers&&n>0]

First[n/.Solve[eqn==0,n,PositiveReals]]

n/.ToRules[Reduce[eqn==0&&n>0,Reals]]

A074455

N[%,20]

Maximize[((π^(n/2))/(((n/2))!)),n,PositiveReals]//FullSimplify//Timing

Maximize[((π^(n/2))/(Gamma[(n/2)+1])),n,PositiveReals]//FullSimplify//Timing