# Regular convex polytope

### Points and edges[edit | edit source]

There is exactly one regular 0D polytope: the point. There is also exactly one regular 1D polytope: the line segment.

### Polygons[edit | edit source]

For every integer n > 2, there is exactly one regular convex polygon with n sides and vertices. This polygon is called the n -gon, and is given the Schläfli symbol {n}.

### Polyhedra[edit | edit source]

Regular polyhedra have Schläfli symbols of the form {p,q}, with p -gonal faces with a q -gonal vertex figure. There are five convex regular polyhedra, known as the **Platonic solids**:

- {3,3} - Tetrahedron
- {4,3} - Cube
- {3,4} - Octahedron
- {5,3} - Dodecahedron
- {3,5} - Icosahedron

### Polychora[edit | edit source]

Regular polychora have Schläfli symbols of the form {p,q,r}, where the cells are regular convex polyhedra {p,q} and there is an r -gonal edge figure. Their vertex figure is then a regular polyhedron {q,r}. There are 6 convex regular polychora:

- {3,3,3} - Pentachoron
- {4,3,3} - Tesseract
- {3,3,4} - Hexadecachoron
- {3,4,3} - Icositetrachoron
- {5,3,3} - Hecatonicosachoron
- {3,3,5} - Hexacosichoron

### 5-polytopes and higher[edit | edit source]

In all higher dimensions, there are only 3 infinite families of regular polytopes:

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