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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
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 and 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

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

$\hskip2.1in$enter image description here

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

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