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.

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   .

### 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).

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 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.

### 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)