Series name	exists in orientable genera	Schläfli formula	V / F / E	mV, mF	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 K₂
z	all	Sn:{2n+1,4n+2}	1 / 2 / 2n+1	2(2n+1),2n+1	2n+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 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 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.

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
	even	self k — hosohedron — dipolygon self
	multiple of 8	self j — h — kt self
	multiple of 8	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

Other Regular Maps

General Index