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In $\mathbb{R}^3$, there are five regular polyhedrons (up to similarity), and can be parametrized by number of vertices, edges and faces. What is the number of regular polyhedrons in $\mathbb{R}^n$, and their parametrization? Please suggest the reference(s) also. (Thanks in advance.)

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There is a pretty decent article here which gives the basic facts and pointers to references. –  Old John Aug 6 '12 at 8:31

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up vote 9 down vote accepted

In short, what happens is the following. The $n$-dimensional analogue of a Platonic solid is called a regular polytope. In any dimension you are guaranteed three "boring" regular polytopes: the $n$-dimensional version of the tetrahedron (the $n$-simplex), the $n$-dimensional hypercube, and its dual, the $n$-dimensional version of the octahedron. In three dimensions, as you know, there are two others. In four dimensions there are others as well, called the 24-cell, 120-cell and 600-cell. In dimensions five and above the boring regular polytopes are the only ones that exist.

The wiki pages http://en.wikipedia.org/wiki/Regular_polytopes and http://en.wikipedia.org/wiki/Convex_regular_4-polytope are good places to start. Coxeter's book Regular Polytopes is very comprehensive. Another approach is to look at these things through their reflection symmetry groups: Coxeter's book is a good source for this too, see also http://en.wikipedia.org/wiki/Coxeter_group

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Just a comment, but "comments" don't support gif's: The Tesseract

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... don't get hypnotized! –  draks ... Aug 6 '12 at 8:47

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