Platonic Solids

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