tag:blogger.com,1999:blog-264226589944705290.post4138154343406835759..comments2021-12-14T05:53:12.175-08:00Comments on God Plays Dice: Probabilistic fun with the n-sphereMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-264226589944705290.post-4047343403646984752009-02-02T13:41:00.000-08:002009-02-02T13:41:00.000-08:00Yemon, you weren't supposed to give it away! ;-) Y...Yemon, you weren't supposed to give it away! ;-) <BR/><BR/>Yes, it came from those guys. (I was at the conference where they were first discussing this problem.)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-46714030750008796852009-01-30T15:40:00.000-08:002009-01-30T15:40:00.000-08:00Todd, the problem/puzzle you mention rang a bell s...Todd, the problem/puzzle you mention rang a bell somewhere. I can't remember when or how I came across the following, but does the construction of Blass and Schanuel<BR/><BR/>http://www.math.lsa.umich.edu/~ablass/geom.html<BR/><BR/>do the trick?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-50686602832017082392009-01-27T14:42:00.000-08:002009-01-27T14:42:00.000-08:00There are plenty of other surprises in high-dimens...There are plenty of other surprises in high-dimensional spaces. <BR/><BR/>I wrote a <A HREF="http://mark.reid.name/iem/warning-high-dimensions.html" REL="nofollow">post</A> the other day that looks at what happens when you arrange n-spheres in n-dimensional space. I use the ratio of volumes argument you have here to try to understand what is going on.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-24662683905001734512009-01-27T04:11:00.000-08:002009-01-27T04:11:00.000-08:00Have you ever looked at the area and volume of neg...Have you ever looked at the area and volume of negative dimensional spheres?Dhttps://www.blogger.com/profile/15141040566131538098noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-85355162511375444552009-01-26T23:46:00.000-08:002009-01-26T23:46:00.000-08:00I recently mentioned the pi^{n/2}/(n/2)! result ov...I recently mentioned the pi^{n/2}/(n/2)! result over at John Armstrong's blog, but I didn't mention the curious puzzle this gives rise to. <BR/><BR/>Take n = 2m, and think of the n-dimensional ball B_n as sitting in C^m, m-dimensional complex space. On the other hand, we have the polydisk <BR/><BR/>(B_2)^m \subseteq C^m<BR/><BR/>and the symmetric group on m letters acts on (B_2)^m by permuting coordinates. The orbit space (B_2)^m/S_m has the same volume as B_n. <BR/><BR/>Is there a continuous volume-preserving map from one of these spaces to the other? (Add on other conditions as you see fit: symplectic, diffeomorphism, etc. -- the question is a little open-ended.)Anonymousnoreply@blogger.com