{"id":737,"date":"2014-01-25T21:06:41","date_gmt":"2014-01-26T02:06:41","guid":{"rendered":"http:\/\/www.curtisbright.com\/bln\/?p=737"},"modified":"2014-01-25T21:06:41","modified_gmt":"2014-01-26T02:06:41","slug":"a-curious-hypersphere-property","status":"publish","type":"post","link":"http:\/\/localhost\/blog\/index.php\/2014\/01\/25\/a-curious-hypersphere-property\/","title":{"rendered":"A curious hypersphere property"},"content":{"rendered":"

Last time<\/a>\u00a0when I derived the formula for the volume of a hypersphere in $n$ dimensions I forgot to point out a curious consequence of the formula, namely that the volume tends to zero as $n$ tends to infinity.<\/p>\n

When I was an undergraduate I remember a professor of mine pointing this out and then declaring “That doesn’t make sense!”. At the time it didn’t seem too surprising to me, since I could see that the unit circle in $\\newcommand{\\R}{\\mathbb{R}}\\R^2$ took up more of the surrounding square $[-1,1]^2$ than the unit sphere in $\\R^3$ took up of $[-1,1]^3$. Consquently, I thought it likely that the ratio of the volume of the unit sphere in $\\R^n$ to the volume of $[-1,1]^n$ should go to zero as $n\\to\\infty$.<\/p>\n

However, I misunderstood the claim being made: not only does the above ratio of hypersphere-to-hypercube volume go to zero, the volume of the hypersphere itself<\/em> goes to zero. This was something I hadn’t even considered: since as $n\\to\\infty$ the hypersphere is “growing”, I presumably took for granted that its volume should go to infinity, not zero!<\/p>\n

Of course, one can consider the unit sphere in $\\R^{n-1}$ as a subset of the unit sphere in $\\R^n$, since for example the unit sphere in $\\R^3$ contains the unit circle as a “slice”. In this way as $n\\to\\infty$ the hypersphere is<\/em> growing. However, though the “slice” has volume in $\\R^{n-1}$, it has no volume in $\\R^n$; as the dimension increases it becomes “harder” to make volume in a sense. This allows the hypersphere to “grow” as $n\\to\\infty$ while still shrink in volume.<\/p>\n

Algebraically, as we’ve seen, the volume of the unit sphere in $\\R^n$ is given by<\/p>\n

\\[ V_n = \\frac{\\pi^{n\/2}}{(n\/2)!} . \\]<\/p>\n

If one knows Stirling’s approximation<\/a><\/p>\n

\\[ n! \\sim \\sqrt{2\\pi n}\\Bigl(\\frac{n}{e}\\Bigr)^n \\]<\/p>\n

then it isn’t too hard to see that the denominator of $V_n$ grows asymptotically faster than the numerator, and therefore $V_n$ tends to $0$. Explicitly, we have<\/p>\n

\\[ \\lim_{n\\to\\infty} V_n = \\lim_{n\\to\\infty}\\frac{\\pi^{n\/2}}{\\sqrt{\\pi n}(\\frac{n}{2e})^{n\/2}} = \\lim_{n\\to\\infty}\\frac{1}{\\sqrt{\\pi n}}\\cdot\\lim_{n\\to\\infty}\\Bigl(\\frac{2\\pi e}{n}\\Bigr)^{n\/2} = 0 \\]<\/p>\n

since $\\lim_{n\\to\\infty}1\/\\sqrt{\\pi n}=0$ and<\/p>\n

\\[ \\lim_{n\\to\\infty}\\Bigl(\\frac{2\\pi e}{n}\\Bigr)^{n\/2} = \\lim_{n\\to\\infty}\\exp\\Bigl(\\frac{n}{2}\\ln\\Bigl(\\frac{2\\pi e}{n}\\Bigr)\\Bigr) = \\lim_{m\\to-\\infty}\\exp(m) = 0 .\u00a0\\]<\/p>\n","protected":false},"excerpt":{"rendered":"

Last time\u00a0when I derived the formula for the volume of a hypersphere in $n$ dimensions I forgot to point out a curious consequence of the formula, namely that the volume tends to zero as $n$ tends to infinity. When I … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"_links":{"self":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/737"}],"collection":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=737"}],"version-history":[{"count":0,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/737\/revisions"}],"wp:attachment":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=737"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=737"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}