This page shows all the regular maps that can be drawn in the projective plane. For the purpose of these pages, a "regular map" is defined here.

These include analogues of four of the five platonic solids, each with half as many vertices, faces and edges. The tetrahedron has no analogue in the projective plane.

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

hemi-cube | {4,3} | 4 3 6 1 | hemi-octahedron | S4 | 1½ | ||

hemi-octahedron | {3,4} | 3 4 6 1 | hemi-cube self-Petrie dual | S4 | 1½ | ||

hemi-dodecahedron | {5,3} | 10 6 15 1 | hemi-icosahedron self-Petrie dual | S5 | The Petersen graph. If you take hemi-dodecahedra and glue them together five to an edge, you will find that 57 of them form a regular polytope, the 57-cell, Schläfli symbol {5,3,5}. Its symmetry group is PSL(2,19). | 3 | |

hemi-icosahedron | {3,5} | 6 10 15 1 | hemi-dodecahedron C | S5 | If you take a hemi-icosahedron and glue another one to each face, and bend them round so that three meet at each edge, you will find that the 11 of them form a regular polytope, the 11-cell, Schläfli symbol {3,5,3}. Its symmetry group is PSL(2,11). | 3 | |

hemi-hosohedron | {2,2n} | 1n n 1 | hemi-dihedron
| D4n |
The image uses 7 as an example value for n. | ½ | |

hemi-dihedron | {2n,2} | n1 n 1 | hemi-hosohedron
| D4n |
The images use 7 as an example value for n. The second image only works for odd n. | ½ | |

hemi-digonal hosohedron | {2,2} | 1 1 1 1 | self-dual | C2×C2 |
Cantellation of the hemi-digonal hosohedron yields the hemi-4-hosohedron. | ½ |

Index to other pages on regular maps;

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

Some pages on groups

Copyright N.S.Wedd 2009