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 |
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).
One of these series has a member in non-orientable genus 3n-2, with Schläfli formula {3n,4} and 3n vertices, for every positive integer n. The other series has a member in non-orientable genus n^{2}-2n+2, with Schläfli formula {4,2n} and 2n vertices, for every positive odd integer n.
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.
There are also infinite series of regular maps having all their members in the same genus
Hexads are described at Hexads of Regular Maps. Every regular map is in exactly one hexad, being related to the other members of the hexad by the operations of duality and Petrie duality. Hexads may be degenerate, with only three (or more rarely two or one) members.
Type of hexad | Number of edges of hexad members | Hexad |
---|---|---|
hexad | odd | – i — z — hemihosohedron — hemidipolygon — dipolygon – hosohedron – |
multiple of 3 | – p — q — X' — X — Y' – Y – | |
odd multiple of 4 | – j — h — X' — X — Y' – Y – | |
triad | even | self s — hemihosohedron — hemidipolygon self |
self k — hosohedron — dipolygon self | ||
multiple of 8 | self j — h — kt self | |
self l — m — X self | ||
multiple of 16 | self lt — mt — X self |
In the table above:
Orientable | |
Non-orientable |