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### Platonic Solids

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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
```

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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

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

```
Associated Topics:
College Polyhedra

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