Convex regular 4-polytope

In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional (4D) polytope which is both regular and convex. These are the four-dimensional analogs of the Platonic solids (in three dimensions) and the regular polygons (in two dimensions).
These polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. Schläfli discovered that there are precisely six such figures. Five of these may be thought of as higher dimensional analogs of the Platonic solids. There is one additional figure (the 24-cell) which has no three-dimensional equivalent.
Each convex regular 4-polytope is bounded by a set of 3-dimensional "cells" which are all Platonic solids of the same type and size. These are fitted together along their respective faces in a regular fashion.
Properties.
The following tables lists some properties of the six convex regular polychora. The symmetry groups of these polychora are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group.
Since the boundaries of each of these figures is topologically equivalent to a 3-sphere, whose Euler characteristic is zero, we have the 4-dimensional analog of Euler's polyhedral formula:
where "N""k" denotes the number of "k"-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).
Visualizations.
The following table shows some 2 dimensional projections of these polytopes. Various other visualizations can be found in the other websites below. The Coxeter-Dynkin diagram graphs are also given below the Schläfli symbol.