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The regular polyhedra are those that have congruent and regular vertices, faces and edges. They are listed here, along with their Schläfli symbols.

## Platonic Edit

The five Platonic solids are the convex regular polyhedra.

• Tetrahedron $\{3, 3\}$
• Cube $\{4, 3\}$
• Dodecahedron $\{5, 3\}$
• Octahedron $\{3, 4\}$
• Icosahedron $\{3, 5\}$

## Kepler-Poinsot Edit

The four Kepler-Poinsot polyhedra are concave, intersect themselves, and some have star polygons as faces rather than convex ones.

## Abstract Edit

Abstract polyhedra are topologically equivalent to tilings of hyperbolic space. Therefore, they are not considered part of the nine main regular polyhedra, but are listed nonetheless.

## Spherical Edit

Spherical polyhedra can only exist on the surface of a sphere, and are degenerate in normal Euclidean space. Therefore, they are not considered part of the nine main regular polyhedra.