Volume of an n-ball

Function to calculate volume of n-balls

# Input 
#     n - a vector of dimensions
#     r - radius of the n-ball
# Output
#     v - a vector of volumes

v.n.ball <- function(n, r){
  v <- sapply(n, function(n) {(pi ^ (n / 2)) * (r ^ n) / gamma(n / 2 +1)})
  return(v)
}

Set dimension as 1 to 20, Set radius as 1

dim <- seq(1, 20, by = 0.1)

volume.n.ball <- v.n.ball(dim, 1)

volume.n.square <- 2 ^ dim

ratio.ball.vs.square <- volume.n.ball / volume.n.square

Plot volume of an n-ball with increase of dimension

plot(x = dim, y = volume.n.ball, type = 'l', col = 'skyblue', lwd = 5)
lines(x = dim, y = ratio.ball.vs.square, col = 'tomato', lwd = 5)
legend(12, 4, c('Vol: n-ball', 'Ratio: n-ball/n-square'),  lty = c(1, 1), col = c('skyblue', 'tomato'), lwd = c(5,5))

Reference

http://www.americanscientist.org/libraries/documents/201110101628308738-2011-11CompSciHayes.pdf

Appendix

dim

##   [1]  1.0  1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9  2.0  2.1  2.2  2.3
##  [15]  2.4  2.5  2.6  2.7  2.8  2.9  3.0  3.1  3.2  3.3  3.4  3.5  3.6  3.7
##  [29]  3.8  3.9  4.0  4.1  4.2  4.3  4.4  4.5  4.6  4.7  4.8  4.9  5.0  5.1
##  [43]  5.2  5.3  5.4  5.5  5.6  5.7  5.8  5.9  6.0  6.1  6.2  6.3  6.4  6.5
##  [57]  6.6  6.7  6.8  6.9  7.0  7.1  7.2  7.3  7.4  7.5  7.6  7.7  7.8  7.9
##  [71]  8.0  8.1  8.2  8.3  8.4  8.5  8.6  8.7  8.8  8.9  9.0  9.1  9.2  9.3
##  [85]  9.4  9.5  9.6  9.7  9.8  9.9 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7
##  [99] 10.8 10.9 11.0 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 12.0 12.1
## [113] 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13.0 13.1 13.2 13.3 13.4 13.5
## [127] 13.6 13.7 13.8 13.9 14.0 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9
## [141] 15.0 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 16.0 16.1 16.2 16.3
## [155] 16.4 16.5 16.6 16.7 16.8 16.9 17.0 17.1 17.2 17.3 17.4 17.5 17.6 17.7
## [169] 17.8 17.9 18.0 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 19.0 19.1
## [183] 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 20.0

volume.n.ball

##   [1] 2.00000000 2.11151898 2.22427206 2.33802172 2.45252625 2.56754075
##   [7] 2.68281813 2.79811007 2.91316807 3.02774431 3.14159265 3.25446952
##  [13] 3.36613478 3.47635258 3.58489217 3.69152866 3.79604371 3.89822628
##  [19] 3.99787319 4.09478973 4.18879020 4.27969839 4.36734797 4.45158295
##  [25] 4.53225792 4.60923838 4.68240095 4.75163354 4.81683548 4.87791758
##  [31] 4.93480220 4.98742319 5.03572585 5.07966685 5.11921405 5.15434636
##  [37] 5.18505350 5.21133577 5.23320377 5.25067810 5.26378901 5.27257608
##  [43] 5.27708781 5.27738124 5.27352155 5.26558161 5.25364158 5.23778843
##  [49] 5.21811550 5.19472205 5.16771278 5.13719739 5.10329011 5.06610922
##  [55] 5.02577664 4.98241743 4.93615939 4.88713259 4.83546898 4.78130195
##  [61] 4.72476597 4.66599614 4.60512786 4.54229648 4.47763691 4.41128334
##  [67] 4.34336890 4.27402537 4.20338290 4.13156978 4.05871213 3.98493373
##  [73] 3.91035579 3.83509675 3.75927214 3.68299435 3.60637257 3.52951261
##  [79] 3.45251678 3.37548384 3.29850890 3.22168334 3.14509475 3.06882693
##  [85] 2.99295983 2.91756954 2.84272829 2.76850447 2.69496262 2.62216349
##  [91] 2.55016404 2.47901753 2.40877353 2.33947802 2.27117341 2.20389867
##  [97] 2.13768936 2.07257773 2.00859284 1.94576059 1.88410388 1.82364265
## [103] 1.76439403 1.70637242 1.64958958 1.59405479 1.53977488 1.48675441
## [109] 1.43499572 1.38449908 1.33526277 1.28728318 1.24055496 1.19507105
## [115] 1.15082285 1.10780030 1.06599193 1.02538504 0.98596571 0.94771894
## [121] 0.91062875 0.87467822 0.83984959 0.80612437 0.77348336 0.74190679
## [127] 0.71137433 0.68186522 0.65335826 0.62583196 0.59926453 0.57363396
## [133] 0.54891808 0.52509461 0.50214120 0.48003549 0.45875513 0.43827784
## [139] 0.41858143 0.39964388 0.38144328 0.36395797 0.34716649 0.33104763
## [145] 0.31558047 0.30074438 0.28651902 0.27288443 0.25982095 0.24730932
## [151] 0.23533063 0.22386636 0.21289839 0.20240900 0.19238087 0.18279709
## [157] 0.17364117 0.16489706 0.15654909 0.14858204 0.14098111 0.13373189
## [163] 0.12682043 0.12023316 0.11395693 0.10797901 0.10228705 0.09686912
## [169] 0.09171366 0.08680951 0.08214589 0.07771237 0.07349890 0.06949581
## [175] 0.06569373 0.06208367 0.05865697 0.05540528 0.05232058 0.04939516
## [181] 0.04662160 0.04399279 0.04150189 0.03914234 0.03690786 0.03479242
## [187] 0.03279023 0.03089577 0.02910373 0.02740906 0.02580689

volume.n.square

##   [1] 2.000000e+00 2.143547e+00 2.297397e+00 2.462289e+00 2.639016e+00
##   [6] 2.828427e+00 3.031433e+00 3.249010e+00 3.482202e+00 3.732132e+00
##  [11] 4.000000e+00 4.287094e+00 4.594793e+00 4.924578e+00 5.278032e+00
##  [16] 5.656854e+00 6.062866e+00 6.498019e+00 6.964405e+00 7.464264e+00
##  [21] 8.000000e+00 8.574188e+00 9.189587e+00 9.849155e+00 1.055606e+01
##  [26] 1.131371e+01 1.212573e+01 1.299604e+01 1.392881e+01 1.492853e+01
##  [31] 1.600000e+01 1.714838e+01 1.837917e+01 1.969831e+01 2.111213e+01
##  [36] 2.262742e+01 2.425147e+01 2.599208e+01 2.785762e+01 2.985706e+01
##  [41] 3.200000e+01 3.429675e+01 3.675835e+01 3.939662e+01 4.222425e+01
##  [46] 4.525483e+01 4.850293e+01 5.198415e+01 5.571524e+01 5.971411e+01
##  [51] 6.400000e+01 6.859350e+01 7.351669e+01 7.879324e+01 8.444851e+01
##  [56] 9.050967e+01 9.700586e+01 1.039683e+02 1.114305e+02 1.194282e+02
##  [61] 1.280000e+02 1.371870e+02 1.470334e+02 1.575865e+02 1.688970e+02
##  [66] 1.810193e+02 1.940117e+02 2.079366e+02 2.228609e+02 2.388564e+02
##  [71] 2.560000e+02 2.743740e+02 2.940668e+02 3.151730e+02 3.377940e+02
##  [76] 3.620387e+02 3.880234e+02 4.158732e+02 4.457219e+02 4.777129e+02
##  [81] 5.120000e+02 5.487480e+02 5.881336e+02 6.303459e+02 6.755881e+02
##  [86] 7.240773e+02 7.760469e+02 8.317465e+02 8.914438e+02 9.554258e+02
##  [91] 1.024000e+03 1.097496e+03 1.176267e+03 1.260692e+03 1.351176e+03
##  [96] 1.448155e+03 1.552094e+03 1.663493e+03 1.782888e+03 1.910852e+03
## [101] 2.048000e+03 2.194992e+03 2.352534e+03 2.521384e+03 2.702352e+03
## [106] 2.896309e+03 3.104188e+03 3.326986e+03 3.565775e+03 3.821703e+03
## [111] 4.096000e+03 4.389984e+03 4.705068e+03 5.042768e+03 5.404704e+03
## [116] 5.792619e+03 6.208375e+03 6.653972e+03 7.131550e+03 7.643406e+03
## [121] 8.192000e+03 8.779968e+03 9.410137e+03 1.008554e+04 1.080941e+04
## [126] 1.158524e+04 1.241675e+04 1.330794e+04 1.426310e+04 1.528681e+04
## [131] 1.638400e+04 1.755994e+04 1.882027e+04 2.017107e+04 2.161882e+04
## [136] 2.317048e+04 2.483350e+04 2.661589e+04 2.852620e+04 3.057363e+04
## [141] 3.276800e+04 3.511987e+04 3.764055e+04 4.034214e+04 4.323764e+04
## [146] 4.634095e+04 4.966700e+04 5.323177e+04 5.705240e+04 6.114725e+04
## [151] 6.553600e+04 7.023975e+04 7.528110e+04 8.068428e+04 8.647527e+04
## [156] 9.268190e+04 9.933400e+04 1.064635e+05 1.141048e+05 1.222945e+05
## [161] 1.310720e+05 1.404795e+05 1.505622e+05 1.613686e+05 1.729505e+05
## [166] 1.853638e+05 1.986680e+05 2.129271e+05 2.282096e+05 2.445890e+05
## [171] 2.621440e+05 2.809590e+05 3.011244e+05 3.227371e+05 3.459011e+05
## [176] 3.707276e+05 3.973360e+05 4.258542e+05 4.564192e+05 4.891780e+05
## [181] 5.242880e+05 5.619180e+05 6.022488e+05 6.454742e+05 6.918022e+05
## [186] 7.414552e+05 7.946720e+05 8.517084e+05 9.128384e+05 9.783560e+05
## [191] 1.048576e+06

ratio.ball.vs.square

##   [1] 1.000000e+00 9.850584e-01 9.681706e-01 9.495319e-01 9.293337e-01
##   [6] 9.077627e-01 8.849999e-01 8.612194e-01 8.365878e-01 8.112640e-01
##  [11] 7.853982e-01 7.591319e-01 7.325976e-01 7.059189e-01 6.792101e-01
##  [16] 6.525762e-01 6.261137e-01 5.999099e-01 5.740438e-01 5.485859e-01
##  [21] 5.235988e-01 4.991375e-01 4.752497e-01 4.519761e-01 4.293512e-01
##  [26] 4.074030e-01 3.861541e-01 3.656217e-01 3.458182e-01 3.267514e-01
##  [31] 3.084251e-01 2.908394e-01 2.739909e-01 2.578732e-01 2.424774e-01
##  [36] 2.277921e-01 2.138037e-01 2.004971e-01 1.878554e-01 1.758605e-01
##  [41] 1.644934e-01 1.537340e-01 1.435616e-01 1.339552e-01 1.248932e-01
##  [46] 1.163540e-01 1.083160e-01 1.007574e-01 9.365689e-02 8.699321e-02
##  [51] 8.074551e-02 7.489335e-02 6.941675e-02 6.429624e-02 5.951291e-02
##  [56] 5.504846e-02 5.088517e-02 4.700598e-02 4.339449e-02 4.003494e-02
##  [61] 3.691223e-02 3.401194e-02 3.132029e-02 2.882415e-02 2.651105e-02
##  [66] 2.436913e-02 2.238715e-02 2.055446e-02 1.886101e-02 1.729729e-02
##  [71] 1.585434e-02 1.452373e-02 1.329751e-02 1.216823e-02 1.112889e-02
##  [76] 1.017293e-02 9.294213e-03 8.486992e-03 7.745899e-03 7.065926e-03
##  [81] 6.442400e-03 5.870970e-03 5.347586e-03 4.868481e-03 4.430155e-03
##  [86] 4.029362e-03 3.663088e-03 3.328544e-03 3.023144e-03 2.744497e-03
##  [91] 2.490395e-03 2.258794e-03 2.047812e-03 1.855710e-03 1.680886e-03
##  [96] 1.521867e-03 1.377294e-03 1.245919e-03 1.126595e-03 1.018269e-03
## [101] 9.199726e-04 8.308197e-04 7.499972e-04 6.767603e-04 6.104273e-04
## [106] 5.503745e-04 4.960315e-04 4.468773e-04 4.024358e-04 3.622728e-04
## [111] 3.259919e-04 2.932319e-04 2.636635e-04 2.369871e-04 2.129298e-04
## [116] 1.912434e-04 1.717022e-04 1.541012e-04 1.382541e-04 1.239917e-04
## [121] 1.111607e-04 9.962203e-05 8.924946e-05 7.992877e-05 7.155649e-05
## [126] 6.403898e-05 5.729151e-05 5.123746e-05 4.580759e-05 4.093934e-05
## [131] 3.657620e-05 3.266720e-05 2.916632e-05 2.603206e-05 2.322704e-05
## [136] 2.071755e-05 1.847324e-05 1.646678e-05 1.467358e-05 1.307152e-05
## [141] 1.164073e-05 1.036331e-05 9.223205e-06 8.206001e-06 7.298745e-06
## [146] 6.489819e-06 5.768801e-06 5.126345e-06 4.554076e-06 4.044488e-06
## [151] 3.590860e-06 3.187175e-06 2.828046e-06 2.508655e-06 2.224692e-06
## [156] 1.972306e-06 1.748054e-06 1.548859e-06 1.371976e-06 1.214953e-06
## [161] 1.075600e-06 9.519674e-07 8.423126e-07 7.450842e-07 6.588990e-07
## [166] 5.825248e-07 5.148643e-07 4.549403e-07 4.018834e-07 3.549199e-07
## [171] 3.133617e-07 2.765968e-07 2.440815e-07 2.153325e-07 1.899206e-07
## [176] 1.674644e-07 1.476256e-07 1.301039e-07 1.146327e-07 1.009758e-07
## [181] 8.892365e-08 7.829041e-08 6.891154e-08 6.064122e-08 5.335031e-08
## [186] 4.692450e-08 4.126260e-08 3.627506e-08 3.188268e-08 2.801542e-08
## [191] 2.461137e-08