Series name | exists in orientable genera | Schläfli formula | V / F / E | mV, mF | specimen (genus S2) | underlying graph |
---|---|---|---|---|---|---|

s | all | Sn:{4n,4n} | 1 / 1 / 2n |
4n, 4n | 2n-fold 1-cycle | |

j | n>1 | Sn:{4n,4} | 2n / 2 / 4n |
2, 4n(for n>1) | 2-fold 2n-cycle | |

h | n>1 | Sn:{4,4n} | 2 / 2n / 4n |
4n, 2(for n>1) | 2n-fold 2-cycle | |

k | all | Sn:{2n+2,2n+2} | 2 / 2 / 2n+2 |
2(n+1), 2(n+1) | n+1-fold 2-cycle | |

l | all | Sn:{2n+2,4} | 2n+2 / 4 / 4n+4 |
2, n+1(for n>0) | 2-fold (2n+2)-cycle | |

m | all | Sn:{4,2n+2} | 4 / 2n+2 / 4n+4 |
n+1, 2(for n>0) | n+1-fold 4-cycle | |

p | n%3 not 0 | Sn:{6,3n} | 2 / n / 3n |
3n, 3(for n>3) | n-fold 2-cycle | |

q | n%3 not 0 | Sn:{3n,6} | n / 2 / 3n |
3, 3n(for n>3) | 3-fold n-cycle | |

i | all | Sn:{4n+2,2n+1} | 2 / 1 / 2n+1 |
2n+1, 2(2n+1) | 2n+1-fold K_{2} | |

z | all | Sn:{2n+1,4n+2} | 1 / 2 / 2n+1 |
2(2n+1),2n+1 | 2n+1-fold 1-cycle |

There are several series of regular maps with one member
in each orientable genus from 1 upwards. These series are
listed in the table to the right, in which "*n*"
indicates the genus.

rectification | dual pair | ||||

s | j | h | |||

k | l | m | |||

kt | lt | mt | |||

p | q | ||||

i | z |

Members of series i are
Petrie duals of odd hosohedra.

Members of series k are
Petrie duals of even hosohedra.

Members of series s are
Petrie duals of even hemihosohedra.

Members of series z are
Petrie duals of odd hemihosohedra.

Series p and q have no members in surfaces of genus divisible by 3.

Series kt, lt and mt have members only in surfaces of genus 3 modulo 4.

Pages for each series: **
h
i
j
k
l
m
kt
lt
mt
p
q
s
z
**.

Two infinite series of non-orientable maps are described by
Stephen E. Wilson,
in **Cantankerous Maps and Rotary Embeddings of K _{n}**,
Journal of Combinatorial Theory, series B 47, 262-273 (1989).

A regular map is said to be cantankerous iff any two vertices connected by an
edge are connected by exactly two edges *and* the neighbourhood of the
circuit formed by such a pair of edges is non-orientable.

One of the series has a member in non-orientable genus 3*n*-2, with
Schläfli formula {3*n*,4} and 3*n* vertices, for every
positive integer *n*.
The other series has a member in non-orientable genus
*n*^{2}-2*n*+2, with Schläfli formula {4,2*n*}
and 2*n* vertices, for every positive odd integer *n*.

There are also infinite series of regular maps having all their members in the same genus

- sphere:
- one series of dual pairs (the hosohedra and the di-polygons)

- torus:
- two infinite series of reflexive self-duals with square faces
- one doubly-infinite series of chiral self-duals with square faces
- two infinite series of reflexive dual pairs with hexagonal and triangular faces
- one doubly-infinite series of chiral dual pairs with hexagonal and triangular faces

- projective plane:
- one series of dual pairs (the hemi-hosohedra and the hemi-di-polygons)

Orientable | |

Non-orientable |