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regular polyhedron
A regular polyhedron is a polyhedron such that

Every face is a regular polygon.

The dihedral angle between any two faces is always the same.
These polyhedra are also known as Platonic solids, since Plato described them in his work. There are only 5 regular polyhedra, as was first shown by Theaetetus, one of Plato’s students. Some sources ascribe to Theaetetus also the discovery of the dodecahedron.
The five solids are:
 Regular Tetrahedron

It has 6 edges and 4 vertices and 4 faces, each one being an equilateral triangle. Its symmetry group is $S_{4}$.
 Regular Hexahedron
 Regular Octahedron

It has 6 vertices, 12 edges and 8 faces, each one being an equilateral triangle Its symmetry group is $S_{4}\times C_{2}$.
 Regular Dodecahedron
 Regular Icosahedron

It has 12 vertices, 30 edges and 20 faces, each one being an equilateral triangle. Its symmetry group is $A_{5}\times C_{2}$.
Note: $A_{n}$ is the alternating group of order $n$, $S_{n}$ is the symmetric group of order $n$ and $C_{n}$ is the cyclic group with order $n$.
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