This page shows all the regular polyhedra that can be drawn on the genus-0 orientable manifold, the sphere. For the purpose of these pages, a "regular polyhedron" is regarded as a regular map on the sphere.

This definition gives five regular polyhedra, also known as the five platonic solids; and some other less interesting things. There are many better web pages about the five platonic solids, for instance on Wikipedia, on Wolfram, and a page which lets you rotate them.

name | Schläfli symbol | picture | V F E Eu | dual | rotational symmetry group | Antipodes, etc. | Comments | qy |
---|---|---|---|---|---|---|---|---|

tetrahedron | {3,3} | 4 4 6 2 | self-dual | A4 | Each face is antipodal to a vertex and vice versa; and each edge to an edge. Cantellation of the tetrahedron yields the octahedron. | These are the five "Platonic solids". | 2 | |

cube | {4,3} | 8 6 12 2 | octahedron | S4 | Each face is antipodal to one face, each vertex to one vertex, and each edge to one edge. | 3 | ||

octahedron | {3,4} | 6 8 12 2 | cube | S4 | Each face is antipodal to one face, each vertex to one vertex, and each edge to one edge. | 3 | ||

dodecahedron | {5,3} | 20 12 30 2 | icosahedron | A5 | Each face is antipodal to one face, each vertex to one vertex, and each edge to one edge. | 6 | ||

icosahedron | {3,5} | 12 20 30 2 | dodecahedron C | A5 | Each face is antipodal to one face, each vertex to one vertex, and each edge to one edge. | 6 | ||

The things below are not normally considered to be polyhedra. However they fit my definition. | ||||||||

n-hosohedron(heptahosohedron, etc.) | {2,n} | 2n n 2 |
di-n-hedron
| D2n |
n odd: Each face is antipodal to one edge and vice versa; each vertex is antipodal to the other vertex.n even: Each face is antipodal to one face, each edge to one edge, and each vertex to the other vertex. |
The image uses 7 as an example value for n. | 1 | |

di-n-hedron(diheptagon, etc.) | {n,2} | n2 n 2 | n-hosohedron
| D2n |
n odd: Each vertex is antipodal to one edge and vice versa; each face is antipodal to the other face.n even: Each vertex is antipodal to one vertex, each edge to one edge, and each face to the other face. |
The image uses 7 as an example value for n. | 1 | |

2-hosohedron, digonal dihedron | {2,2} | 2 2 2 2 | self-dual self-Petrie dual | C2×C2 | Each face is antipodal to the other face, each vertex to the other vertex, and each edge to the other edge. Cantellation of the 2-hosohedron yields the 4-hosohedron. |
1 | ||

monodigon | {2,1} | 2 1 1 2 | dimonogon self-Petrie dual | C2 | Each vertex is antipodal to the other vertex; the face is antipodal to the edge and vice versa. | ½ | ||

dimonogon | {1,2} | 1 2 1 2 | monodigon | C2 | Each face is antipodal to the other face; the vertex is antipodal to the edge and vice versa. | ½ | ||

edgeless map | {0,0} | 1 1 0 2 | self-dual self-Petrie dual | C1 | The face is antipodal to the vertex and vice versa. There is no edge. Our definition of a regular map requires that each face must have the topology of a disk. So this is a valid regular map; but in all other manifolds there must be at least one edge. |
0 |

Our definition of a regular map requires each face to have the topology of a disk. This cannot be achieved by embedding the empty set in any compact manifold, so every regular map must have at least one vertex.

Index to other pages on regular maps;

indexes to those on
S^{0}
C^{1}
S^{1}
S^{2}
C^{4}
C^{5}
S^{3}
C^{6}
S^{4}.

Some pages on groups

Copyright N.S.Wedd 2009,2010