Series
name
exists in
orientable
genera
Schläfli
formula
V / F / E mV, mFspecimen
(genus S2)
underlying
graph
sallSn:{4n,4n}1 / 1 / 2n 4n, 4n2n-fold 1-cycle
jn>1Sn:{4n,4}2n / 2 / 4n 2, 4n
(for n>1)
2-fold 2n-cycle
hn>1Sn:{4,4n}2 / 2n / 4n 4n, 2
(for n>1)
2n-fold 2-cycle
kallSn:{2n+2,2n+2}2 / 2 / 2n+2 2(n+1), 2(n+1)n+1-fold 2-cycle
lallSn:{2n+2,4}2n+2 / 4 / 4n+4 2, n+1
(for n>0)
2-fold (2n+2)-cycle
mallSn:{4,2n+2}4 / 2n+2 / 4n+4 n+1, 2
(for n>0)
n+1-fold 4-cycle
pn%3 not 0Sn:{6,3n}2 / n / 3n 3n, 3
(for n>3)
n-fold 2-cycle
qn%3 not 0Sn:{3n,6}n / 2 / 3n 3, 3n
(for n>3)
3-fold n-cycle
iallSn:{4n+2,2n+1}2 / 1 / 2n+1 2n+1, 2(2n+1)2n+1-fold K2
zallSn:{2n+1,4n+2}1 / 2 / 2n+1 2(2n+1),2n+12n+1-fold 1-cycle

# Series of Regular Maps

### Series in orientable surfaces

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
Any member of series s can be rectified to give the corresponding member of series j. Any member of series k can be rectified to give the corresponding member of series l. Corresponding members of series j and h are dual; also l and m; also p and q; also i and z. This paragraph is summarised in the table to the left.

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   .

### Series in non-orientable surfaces

Two infinite series of non-orientable maps are described by Stephen E. Wilson, in Cantankerous Maps and Rotary Embeddings of Kn, 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 n2-2n+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.

### Series within one genus

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)

### Hexads involving series of regular maps

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.

hexadodd 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
triadeven 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:

• the black items are classes of regular maps. A boldface item is a series of regular maps as documented on this page, "X" and "Y" are unspecified regular maps, and long words designate series of regular maps of genus 0 (orientable) and 1 (non-orientable).
• a long cyan dash, or a pair of shorter cyan dashes at the start and end of a line, connects a pair of duals.
• a long red dash, or a pair of shorter red dashes at the start and end of a line, connects a pair of Petrie duals.
• the word "self" in cyan indicates self-duality
• the word "self" in red indicates self-Petrie-duality