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
comments
{6,3}(1,1)

{6,3}(2,0)

12
1
 3 
0
{3,6}(1,1)


3-hosohedron

D6 Faces share vertices with themselves Faces share edges with themselves
{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

The three-rung Möbius ladder S86.

replete

{6,3}(2,2)

{6,3}(4,0)

48
4
 12 
0
{3,6}(2,2)


cube

S4replete
{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
Chiral replete
{6,3}(3,3)

{6,3}(6,0)

918
9
 27 
0
{3,6}(3,3)


self-Petrie dual

?the Pappus graph

replete

{6,3}(0,4)

{6,3}(6,2)

1224
12
 36 
0
{3,6}(0,4)


S4{12,3}

?the Nauru graph

replete

{6,3}(2,4)

{6,3}(5,3)

{6,3}(7,1)

1326
13
 39 
0
{3,6}(2,4)


?the F26A graph

Chiral replete

{6,3}(4,4)

{6,3}(8,0)

1632
16
 48 
0
{3,6}(4,4)


S3{8,3}

?The Dyck graph

replete

{6,3}(1,5)

{6,3}(7,3)

{6,3}(8,2)

1938
19
 57 
0
{3,6)(1,5)


C19⋊C3Chiral replete
{6,3}(3,5)

{6,3}(6,4)

{6,3}(9,1)

2142
21
 63 
0
{3,6)


(3,5}

?Chiral replete
{6,3}(5,5)

{6,3}(10,0)

2550
25
 75 
0
{3,6}(5,5)


?replete
{6,3}(0,6)

{6,3}(9,3)

2754
27
 81 
0
{3,6}(0,6)


?replete
{6,3}(2,6)

{6,3}(8,4)

{6,3}(10,2)

2856
28
 84 
0
{3,6}(2,6)


?Chiral replete
{6,3}(4,6)

{6,3}(7,5)

{6,3}(11,1)

3162
31
 93 
0
{3,6}(4,6)


?Chiralreplete
{6,3}(6,6)

{6,3}(12,0)

3672
36
 108 
0
{3,6}(6,6)


?replete
{6,3}(1,7)

{6,3}(10,4)

{6,3}(11,3)

3774
37
 111 
0
{3,6}(1,7)


?Chiral replete
{6,3}(3,7)

{6,3}(9,5)

{6,3}(12,2)

3978
39
 117 
0
{3,6}(3,7)


?

Chiral replete

{6,3}(5,7)

{6,3}(8,6)

{6,3}(13,1)

4386
43
 126 
0
{3,6}(5,7)


?Chiral replete
{6,3}(0,8)

{6,3}(12,4)

4896
48
 144 
0
{3,6}(0,8)


?replete
{6,3}(7,7)

{6,3}(14,0)

4998
49
 147 
0
{3,6}(7,7)


?replete
{6,3}(2,8)

{6,3}(11,5)

{6,3}(13,3)

4998
49
 147 
0
{3,6}(2,8)


?Chiral replete

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

Regular Maps in the Torus, with Square Faces
Regular Maps in the Torus, with Triangular Faces


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

Copyright N.S.Wedd 2009,2010