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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

The five Platonic solids are the convex regular polyhedra.

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

## Kepler-Poinsot

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

• Great Stellated Dodecahedron ${\displaystyle \{5/2, 3\}}$
• Great Icosahedron ${\displaystyle \{3, 5/2\}}$
• Great Dodecahedron ${\displaystyle \{5, 5/2\}}$
• Small Stellated Dodecahedron ${\displaystyle \{5/2, 5\}}$

## Abstract

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.

• Dodecadodecahedron ${\displaystyle \{5,4\}}$
• Medial Rhombic Triacontahedron ${\displaystyle \{4,5\}}$
• Medial Triambic Icosahedron ${\displaystyle \{6,5\}}$
• Ditrigonal Dodecadodecahedron ${\displaystyle \{5,6\}}$
• Excavated Dodecahedron ${\displaystyle \{6,6\}}$

## Spherical

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.

• Henagonal Henahedron ${\displaystyle \{1, 1\} }$
• Henagonal Hosohedron ${\displaystyle \{2,1\}}$
• Henagonal Dihedron ${\displaystyle \{1,2\}}$
• Digonal Dihedron / Digonal Hosohedron ${\displaystyle \{2,2\}}$
• Trigonal Hosohedron ${\displaystyle \{2,3\}}$
• Trigonal Dihedron ${\displaystyle \{3, 3\}}$