tag:blogger.com,1999:blog-1948420587779787298.post6020009257638254372..comments2010-10-03T19:46:33.962-07:00Comments on Elliptica: "But if you can't picture it..."Lynethttp://www.blogger.com/profile/06357023675142716573noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-1948420587779787298.post-54183498184685638732007-03-18T23:50:00.000-07:002007-03-18T23:50:00.000-07:00Oh, and see John Baez's diary for the full story, ...Oh, and see John Baez's diary for the full story, with diagrams, which starts:<BR/><BR/>"Richard Block... is an expert on Lie algebras, and there's a fun story about him and Murray Gell-Mann, the physicists who one the Nobel prize for inventing 'quarks'. Actually quarks were part of a mathematical scheme which Gell-Mann called the Eightfold Way, because it was based on the 8-dimensional Lie algebra su(3)..."<BR/><BR/>Worth the reminder that multidimensional geometry is beautiful for its own sake, but mysteriously tells things about the fundamental Physics of the universe we live in, and maybe all possible universes.<BR/><BR/>http://math.ucr.edu/home/baez/diary/index.html#now<BR/><BR/>Well, "now" as of mid March 2007. The hotlink will change later.<BR/><BR/>-- Jonathan Vos PostAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-1948420587779787298.post-16255529897891929002007-03-18T23:38:00.000-07:002007-03-18T23:38:00.000-07:00Very exciting and (as you'll read) BIG news on vis...Very exciting and (as you'll read) BIG news on visualizing a higher dimensional geometry, see:<BR/><BR/>http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html<BR/><BR/>News about E8<BR/>Posted by John Baez<BR/><BR/>The exceptional Lie group E8 is a marvelous 248-dimensional monster, with mysterious connections to the octonions and string theory. Here’s a nice webpage about a new calculation involving E 8:<BR/><BR/> * American Institute of Mathematics, Mathematicians map E8.<BR/><BR/>http://aimath.org/E8/<BR/><BR/>Wonderful stuff -- even if nobody alive can actually visualize one of the most complicated structures ever studied: the object known as the exceptional Lie group E8.<BR/><BR/>-- Prof. Jonathan Vos PostAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-1948420587779787298.post-32364542443287871802007-03-15T21:11:00.000-07:002007-03-15T21:11:00.000-07:00Actually, I've been struggling the past few days w...Actually, I've been struggling the past few days with a 4-dimensional geometry problem where my visualization is good enough to pose the problem, yet is almost but not quite good enough to clearlyt see a solution, which I'm therefore approaching by brute force.<BR/><BR/>The problem is find the hypervolume of the convex hull of a set of points of known coordinates in Euclidean 4-space.<BR/><BR/>Excerpt:<BR/><BR/>The messy way is to pentatopalize (4-D version of 2-D triangularize and 3-D tetrahedralize) the convex hull into 4-simplexes, and then calculate the hypervolume of each pentatope by the Cayley-Menger determinant (generalization of 2-D Heron's formula). This invokes the 4th coefficient in -1, 2, -16, 288, -9216, 460800, ... (Sloane's A055546).<BR/><BR/>If V_4 is the 4-D hypervolume of the general 4-simplex, then:<BR/><BR/>(V_4)^2 = ((-1)^4)/((2^4)(4!)^2) det B where B is the 6x6 matrix determined by d(ij) = the L2-norms of edges between vertex i and<BR/>vertex j.<BR/><BR/>[the spacing might not work well in your font]<BR/><BR/>B =<BR/><BR/>0..........1..........1..........1 ..........1..........1<BR/>1...0...(d_12)^2 (d_13)^2 (d_14)^2 (d_15)^2<BR/>1...(d_21)^2...0...(d_23)^2 (d_24)^2 (d_25)^2<BR/>1...(d_31)^2 (d_32)^2...0... (d_34)^2 (d_35)^2<BR/>1...(d_41)^2 (d_42)^2 (d_43)^2...0 ... (d_45)^2<BR/>1 ...(d_51)^2 (d_52)^2 (d_53)^2 (d_54)^2 ...0Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1948420587779787298.post-62552297107297363842007-03-10T11:40:00.000-08:002007-03-10T11:40:00.000-08:00I apologize for my tone, Lynet.I blogged on this a...I apologize for my tone, Lynet.<BR/><BR/>I blogged on this at such great length once that I annoyed the blogmistress, and I was inadvertantly rude because of my unhappy memories of that.<BR/><BR/>I used to be able to visualize 4-dimensional geometry as a precocious child Mathematician, but the ability faded to my now merely being able to sometime vaguely visualize what once was clear "in my mind's eye."<BR/><BR/>My son, about to graduate university at age eighteen with a double B.S. in Math and Computer Science considers it one of my few failures as a parent not to have been able to teach him how to visualize 4-D geometry.<BR/><BR/>It is fascinating to read postings between two people who CAN so visualize.<BR/><BR/>See my:<BR/><BR/>Table of Polytope Numbers, Sorted, Through 1,000,000<BR/>http://magicdragon.com/poly.html<BR/><BR/>-- Prof. Jonathan Vos PostAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-1948420587779787298.post-18164317159692367632007-03-10T05:00:00.000-08:002007-03-10T05:00:00.000-08:00Although I left space for the possibility that the...Although I left space for the possibility that there might be people out there who can visualise four dimensions that I haven't heard of, I admit it never occured to me to look for them.<BR/><BR/>Thanks for the information. With all due respect, however, I feel unduly patronized by your tone.Lynethttps://www.blogger.com/profile/06357023675142716573noreply@blogger.comtag:blogger.com,1999:blog-1948420587779787298.post-10171840767750217512007-03-09T23:20:00.000-08:002007-03-09T23:20:00.000-08:00With all due respect, I strongly disgaree. There ...With all due respect, I strongly disgaree. There are plenty of people who can visualize 4-dimensional geometry. I've blogged about this before, and don't feel like doing it again. But google these names to start:<BR/><BR/>Alicia Boole Stott<BR/><BR/>Ludwig Schläfli.<BR/><BR/>"H. S. M. Coxeter, Regular Polytopes"<BR/><BR/>This will be a better thread once you've done this basic search, and you will find some cool things about interesting people.<BR/><BR/>-- Prof. Jonathan Vos PostAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-1948420587779787298.post-4648034495573641672007-03-01T13:12:00.000-08:002007-03-01T13:12:00.000-08:00Well, either this one or one of the other couple o...Well, either this one or one of the other couple of maths posts I've got in mind, yeah :)Lynethttps://www.blogger.com/profile/06357023675142716573noreply@blogger.comtag:blogger.com,1999:blog-1948420587779787298.post-15008749257732129402007-03-01T04:49:00.000-08:002007-03-01T04:49:00.000-08:00Actually, now that I think of it, the explanation ...Actually, now that I think of it, the explanation of why a sphere is two dimensional is even better than I thought it was.<BR/><BR/>Just to be sure, you're sending this to the Carnival of Mathematics, right?Alon Levyhttps://www.blogger.com/profile/10512585759732763307noreply@blogger.com