Volumes of n-balls: what is so special about n=5? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:19:46Z http://mathoverflow.net/feeds/question/53119 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53119/volumes-of-n-balls-what-is-so-special-about-n5 Volumes of n-balls: what is so special about n=5? Andrey Rekalo 2011-01-24T20:27:02Z 2011-10-22T17:22:40Z <p>I am reposting <a href="http://math.stackexchange.com/questions/15656/volumes-of-n-balls-what-is-so-special-about-n-5" rel="nofollow">this question</a> from math.stackexchange where it has not yet generated an answer I had been looking for. </p> <ul> <li><p>The volume of an $n$-dimensional ball of radius $R$ is given by the classical formula $$V_n(R)=\frac{\pi^{n/2}R^n}{\Gamma(n/2+1)}.$$ It is not difficult to see that the "dimensionless" ratio $V_n(R)/R^n$ attains its maximal value when $n=5$.</p></li> <li><p>The "dimensionless" ratio $S_n(R)/R^n$ where $S_n(R)$ is the $n$-dimensional volume of an $n$-sphere attains its maximum when $n=7$.</p></li> </ul> <p><strong>Question.</strong> Is there a purely geometric explanation of why the maximal values in each case are attained at these particular values of the dimension?</p> <p>[EDIT. Thanks to all for the answers and comments.]</p> http://mathoverflow.net/questions/53119/volumes-of-n-balls-what-is-so-special-about-n5/53128#53128 Answer by Marty for Volumes of n-balls: what is so special about n=5? Marty 2011-01-24T21:18:49Z 2011-01-24T21:18:49Z <p>In my opinion, nothing is special about $n = 5$.</p> <p>The "dimensionless ratio" $V_n(R) / R^n$ is the ratio of the volume of the $n$-ball of radius $1$ to the volume of the $n$-cube of side-length $1$. So this is maximized at $n =5$, but bluntly, so what?</p> <p>More interesting geometrically might be the equally dimensionless ratio $V_n(R) / (2R)^n$, which is the ratio of the volume of the $n$-ball to the volume of the smallest $n$-cube containing it. This is monotonic decreasing (for $n \geq 1$), showing that balls decrease in volume relative to their smallest cubical container, as the dimension increases. This has more geometric content, since there is a simple geometric relationship between the sphere and cube here. </p> <p>One could consider many similar problems, involving inscribing a cube inside a sphere (instead of the other way around), or using an orthoplex or polycylinder or other figure instead of a cube. All of these have some geometric content, and are expressed as a sequence of dimensionless ratios.</p> http://mathoverflow.net/questions/53119/volumes-of-n-balls-what-is-so-special-about-n5/53129#53129 Answer by Igor Rivin for Volumes of n-balls: what is so special about n=5? Igor Rivin 2011-01-24T21:25:17Z 2011-01-24T21:35:41Z <p>A very celebrated result on a related subject is the following, which was a very major advance in the Busemann-Petty problem (don't worry, the math review has all you need to know). <strong>EDIT</strong> by popular demand, the review can be seen here: <a href="http://dl.dropbox.com/u/5188175/BallReview.pdf" rel="nofollow">http://dl.dropbox.com/u/5188175/BallReview.pdf</a></p> <p>@incollection {MR950983,</p> <pre><code>AUTHOR = {Ball, Keith}, TITLE = {Some remarks on the geometry of convex sets}, </code></pre> <p>BOOKTITLE = {Geometric aspects of functional analysis (1986/87)},</p> <pre><code>SERIES = {Lecture Notes in Math.}, VOLUME = {1317}, PAGES = {224--231}, </code></pre> <p>PUBLISHER = {Springer},</p> <p>ADDRESS = {Berlin},</p> <pre><code> YEAR = {1988}, </code></pre> <p>MRCLASS = {52A40},</p> <p>MRNUMBER = {950983 (89h:52009)},</p> <p>MRREVIEWER = {G. D. Chakerian},</p> <pre><code> DOI = {10.1007/BFb0081743}, URL = {http://dx.doi.org/10.1007/BFb0081743}, </code></pre> <p>}</p> http://mathoverflow.net/questions/53119/volumes-of-n-balls-what-is-so-special-about-n5/53130#53130 Answer by Bill Thurston for Volumes of n-balls: what is so special about n=5? Bill Thurston 2011-01-24T21:27:24Z 2011-01-24T21:27:24Z <p>There are many "dimensionless" ratios: choosing $R$ as the linear measurement is arbitrary. For instance, the ratio of the volume of the sphere to the volume of the circumscribed cube has a maximum at $n=1$. The ratio to the volume of the inscribed cube never attains a maximum. There are intermediate geometrically-related "midscribed" cubes, where all faces of some dimension are tangent to the unit sphere. Here is the graph for the ratio of volumes when the codimension 2 faces of a cube are tangent. It attains the maximum for $n = 12$ (just barely more than for $n=11$). There are many other reasonable dimensionless comparisons, for instance comparing to a simplex, etc. etc. Since the Gamma function grows super-exponentially, these simple geometric variations tend to shift the maximum --- there's nothing special about 5 or 7. </p> <p><img src="http://dl.dropbox.com/u/5390048/MidScribed.jpg" alt="alt text"></p> http://mathoverflow.net/questions/53119/volumes-of-n-balls-what-is-so-special-about-n5/78836#78836 Answer by I. J. Kennedy for Volumes of n-balls: what is so special about n=5? I. J. Kennedy 2011-10-22T17:22:40Z 2011-10-22T17:22:40Z <p>Brian Hayes has <a href="http://www.americanscientist.org/issues/pub/an-adventure-in-the-nth-dimension" rel="nofollow">very nice article</a> about the volume of the n-ball in the current issue of <em>American Scientist</em> (Nov 2011). In particular, he discusses the surprising fact that the maximum volume occurs at $n=5$.</p>