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.

rectificationdual pair
sjh
klm
ktltmt
pq
iz
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


Other Regular Maps

General Index