# Regular Maps in the Torus, with Hexagonal Faces

### Schläfli symbol {6,3}(a,b)

The regular maps with three hexagons meeting at each vertex are more complicated than those with four squares. They can all be generated from pairs of number of the form a,b where a and b are either both odd or both even. The number of faces of these regular maps is given by (a2+3*b2)/4.

More than one such pair can generate the same regular regular map, for example {6,3}(2,4), {6,3}(5,3), and {6,3}(7,1) are all the same regular map, with 13 faces. I have arbitrarily chosen to list them in ascending order of the first parameter, which is necessarily also descending order of the second parameter.

Most of these regular map are chiral, and so occur as enantiomorphic pairs. Only one (arbitrarily chosen) member of each such pair is shown.

The notation {6,3}(a,b) used here is not consistent with that used in ARM, page 19. Where ARM writes {6,3}(s,0) we write {6,3}(s,s), and where ARM writes {6,3}(s,s) we write {6,3}(0, 2s).

ARM disallows regular maps which (in our notation) are not of either of the forms {6,3}(s,s) and {6,3}(0, 2s), because they lack "full reflexional symmetry", i.e. they are chiral. It also disallows (in our notation) {6,3}(1,1), in which the single hexagon shares three edges with itself.

designationno. of
hexagons
pictureV
F
E
Eu
dual

Petrie dual

rotational
symmetry
group
{6,3}(1,1)

{6,3}(2,0)

12
1
3
0
{3,6}(1,1)

3-hosohedron

D6
{6,3}(0,2)

{6,3}(3,1)

36
3
9
0
{3,6}(0,2)

self-Petrie dual

D6×C3K3,3

The Utility or Thomsen graph

{6,3}(2,2)

{6,3}(4,0)

48
4
12
0
{3,6}(2,2)

cube

S4
{6,3}(1,3)

{6,3}(4,2)

{6,3}(5,1)

714
7
21
0
{3,6}(1,3)

S3{14,3}

Frob42

C7⋊C6
{6,3}(3,3)

{6,3}(6,0)

918
9
27
0
{3,6}(3,3)

self-Petrie dual

?the Pappus graph

{6,3}(0,4)

{6,3}(6,2)

1224
12
36
0
{3,6}(0,4)

S4{12,3}

?the Nauru graph

{6,3}(2,4)

{6,3}(5,3)

{6,3}(7,1)

1326
13
39
0
{3,6}(2,4)

?the F26A graph

{6,3}(4,4)

{6,3}(8,0)

1632
16
48
0
{3,6}(4,4)

S3{8,3}

?The Dyck graph

{6,3}(1,5)

{6,3}(7,3)

{6,3}(8,2)

1938
19
57
0
{3,6)(1,5)

C19⋊C3
{6,3}(3,5)

{6,3}(6,4)

{6,3}(9,1)

2142
21
63
0
{3,6)

(3,5}

?
{6,3}(5,5)

{6,3}(10,0)

2550
25
75
0
{3,6}(5,5)

?
{6,3}(0,6)

{6,3}(9,3)

2754
27
81
0
{3,6}(0,6)

?
{6,3}(2,6)

{6,3}(8,4)

{6,3}(10,2)

2856
28
84
0
{3,6}(2,6)

?
{6,3}(4,6)

{6,3}(7,5)

{6,3}(11,1)

3162
31
93
0
{3,6}(4,6)

?
{6,3}(6,6)

{6,3}(12,0)

3672
36
108
0
{3,6}(6,6)

?
{6,3}(1,7)

{6,3}(10,4)

{6,3}(11,3)

3774
37
111
0
{3,6}(1,7)

?
{6,3}(3,7)

{6,3}(9,5)

{6,3}(12,2)

3978
39
117
0
{3,6}(3,7)

?

{6,3}(5,7)

{6,3}(8,6)

{6,3}(13,1)

4386
43
126
0
{3,6}(5,7)

?
{6,3}(0,8)

{6,3}(12,4)

4896
48
144
0
{3,6}(0,8)

?
{6,3}(7,7)

{6,3}(14,0)

4998
49
147
0
{3,6}(7,7)

?
{6,3}(2,8)

{6,3}(11,5)

{6,3}(13,3)

4998
49
147
0
{3,6}(2,8)

?

The pink lines, arrows, and shading are explained by the page Representation of 2-manifolds.

Index to other pages on regular maps;
indexes to those on S0 C1 S1 S2 C4 C5 S3 C6 S4.
Some pages on groups