Date: 1/31/96 at 7:14:1 From: Martin Cizek Subject: platonic solids Hello Dr. Math! What are the Platonic solids in 4 and more dimensions? I have been curious about the solution of this problem for a long time, but a first attempt to find the solution I have done only recently. Now my colleague and I lead a correspondence seminar for high school students. We have put the above question to our correspondents and they found only trivial cases - hypercubes and simplexes. The only nontrivial case was succeeding terms in the series: interval <-1,1>, square, octahedron, ... (solids with vertices (1,0,0,...), (-1,0,0,...),(0,1,0,0,..),(0,-1,0,0,...),... in Cartesian coordinates). I tried to find other solids and I think I have found one in 4D-case. Its boundary consists of 120 dodecahedrons. I would like to know what are the other platonic solids in 4 and more dimensions, their properties and if it is possible to find somewhere literature or people who would be interested in the subject (maybe internet conference?). Martin Cizek - student of Charles University Prague
Date: 3/23/96 at 0:53:29 From: Doctor Jodi Subject: Re: platonic solids Hello! I found some information by searching the Math Forum Internet Mathematics Library http://mathforum.org/library/ for polytope: 1. The 4-simplex, constructed from five tetrahedra, three tetrahedra meeting at an edge. 2. The hypercube, constructed from eight cubes meeting three per edge. 3. The 16-cell, constructed from sixteen tetrahedra, with four tetrahedra meeting at an edge. 4. The 120-cell, constructed from 120 dodecahedra, with three dodecahedra meeting per edge. 4. The monstrous 600-cell, constructed from 600 tetrahedra, with five tetrahedra meeting at an edge. (from http://mathforum.org/news.archives/geometry.college/article165.html ) For pictures and further information, look at: http://www.uccs.edu/~eswab/hyprspac.htm The geometry newsgroups do have some information: --Rigidity of Convex Polytopes (for higher dimensions) at http://mathforum.org/~sarah/HTMLthreads/articletocs/rigid.convex.polytopes.html You might want to mail your questions to geometry.research or geometry.college if you have newsgroup access. Hope this helps! -Doctor Jodi, The Math Forum
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