From: "Glyn Holton" To: Subject: Points on a sphere and Financial Applications Date: Fri, 4 Sep 1998 11:47:15 -0400 Hi Dave: My financial research raised the issue of how to evenly space points on a sphere, and that lead to your FAQ on the subject. Wow, I knew the problem was challenging. I didn't know how challenging it was until I read the FAQ. It is a great piece. You mentioned that you were thinking of splitting the FAQ into a 3-dimensional FAQ and one for spheres in higher dimensions. Actually, my problem is very high dimensional ... fitting (n^2 + n)/2 points on a sphere in n dimensions where n is anywhere from 3 to several thousand, but typically about 150. I am also interested in filling a sphere with evenly spaced points. I am writing a book (Academic Press ... financial stuff) that touches on the topic briefly. I wanted to cite applicable references, but found few books or articles mentioned in the FAQ or related links. I hesitate to simply list URLs that might be outdated in a year or two. Can you point me toward appropriate references, particularly for the higher dimensional problems I am working on? If you are interested in my field of application, check out my website (below). Thanks, Glyn Holton ______________________________ In case the music stops ... http://www.contingencyanalysis.com ============================================================================== From: Dave Rusin Date: Fri, 4 Sep 1998 11:05:58 -0500 (CDT) To: glyn@contingencyanalysis.com Subject: Re: Points on a sphere and Financial Applications >I am writing a book (Academic Press ... financial stuff) that touches on >the topic briefly. I wanted to cite applicable references, but found few >books or articles mentioned in the FAQ or related links. I hesitate to >simply list URLs that might be outdated in a year or two. Can you point >me toward appropriate references, particularly for the higher >dimensional problems I am working on? Good thinking. John Conway's "Sphere packings, lattices, and groups" contains just oodles of information about this and many other topics. (In this context, a lattice is a repeating arrangement of points along lines -- technically, a finitely-generated subgroup of R^n with finite covolume -- such as the square or honeycomb lattice in the plane, or the grocer's-cart display of oranges in R^3. You get a high-density arrangement of point around a sphere by placing a sphere around one of these points, then drawing lines from that one point to its nearest neighbors, looking to see where those lines pierce the sphere; you stop checking more distant neighbors when you have enough points on your sphere.) Interesting answers are known about spheres in very small dimensions, in dimensions 24 or so (e.g. the Leech lattice), and a few other special cases (240, I think); a fair amount of data has been collected in many dimensions in a range up to about 1000 or so -- Noam Elkies (at harvard) has found an interesting connection between moderate-dimensional lattices and elliptic curves. Neil Sloane (Bell labs) has worked on lower-dimensional sphere packings for quite some time; a reference to his web site is probably safe for quite a long time. (I think his interest has been on techniques only useful in dimensions about up to 100). Fitting O(n^2) points on a sphere in R^n sounds like it ought to be comparatively easy: that sounds like a "small" number of points, and so it should be straightforward to describe arrangements with a reasonably large minimal separate. I have no idea whether there is an optimal arrangement or not. dave ============================================================================== From: "Glyn Holton" To: "Dave Rusin" Subject: Re: Points on a sphere and Financial Applications Date: Tue, 8 Sep 1998 16:26:27 -0400 Hi Dave: Thanks for your insights. I got a copy of Saaf and Kuijlaars paper and ordered a copy of Conway's book - A new edition comes out this month. I found a typo in your FAQ cover page. You list the Saaf and Kuijlaars paper as being in Vol. 29. As I found out in the stacks of the MIT library today, it is actually Vol. 19. The rest of your citation is fine. The reason I am interested in the problem is that I have to fit a quadratic function to another function in a neighborhood of the origin. I thought that having one interpolation point at the origin and other points uniformly dispersed on a unit sphere would provide good results. As it turns out, Saaf and Kuijlaars suggest that this is a typical application of this stuff. If you have any citations relating to interpolation, please pass them on. Best Regards, Glyn Holton [inclusion of previous message deleted -- djr]