# Regular Maps in the Torus

This page shows some of the regular maps that can be drawn on the genus-1 orientable manifold, the torus. All those with 50 or fewer faces, and their duals, are shown. For the purpose of these pages, a "regular map" is defined here.

For other oriented manifolds, the number of such figures is small, but for the torus, it is infinite. A reason why there are so many for the torus, and a finite number for every other oriented 2-manifold, is that the torus has an Euler characteristic of 0. Thus, once we have found one regular map, we can stitch together several copies of it, to form another which still fits on a torus.

As the "curvature" of the torus is 0, its vertices must be "flat": if they are also fully symmetrical, they must be formed from four squares, or three hexagons, or six triangles. Infinitely many regular maps of each of these three types exist.

### Regular Maps with Four Squares meeting at each Vertex: Schläfli symbol {4,4}(a,b)

There is one regular map with four squares meeting at each vertex for each pair of non-negative integers a,b (except for 0,0). Each has a number of faces equal to a2+b2. Regular maps for integer pairs a,b with a<b exist, but are not shown here; they are the enantiomorphs of those for b,a.

All these regular maps are all self-dual.

The notation {4,4}(a,b) is consistent with that used in ARM, page 18.

ARM disallows (in our notation) {4,4}(1,0), in which the single square shares two edges with itself; and {4,4}(1,1), in which each of the two squares shares each of its vertices (but no edge) with itself.

Any {4,4} can be cantellated, yielding a {4,4} with twice as many vertices, faces and edges.

### Regular Maps with Three Hexagons meeting at each Vertex: Schläfli symbol {6,3}(a,b)

The regular maps with three hexagons meeting at each vertex are more complicated. 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.

### Regular Maps with Six Triangles meeting at each Vertex: Schläfli symbol {3,6}(a,b)

The regular maps with six triangles meeting at each vertex are the duals of those with three hexagons. As for {6,3}(a,b), a and b must be either both odd or both even. The number of faces of these polyhedra is given by (a2+3*b2)/2.

The notation used here {3,6}(a,b) is not consistent with that used in ARM, and is as described above for {6,3}(a,b). Thus our {6,3}(a,b) is the dual of our {3,6}(a,b).

ARM also disallows (in our notation) {3,6}(1,1), in which each of the two triangles shares each of its vertices (but no edge) with itself.

designationno. of
squares
pictureV
F
E
Eu
dual

Petrie dual

rotational
symmetry
group
{4,4}(1,0)11
1
2
0
self-dual

hemi-2-hosohedron

C4
{4,4}(1,1)22
2
4
0
self-dual

4-hosohedron

D8
{4,4}(2,0)44
4
8
0
self-dual

self-Petrie dual

C22⋊C4K4,4
{4,4}(2,1)55
5
10
0
self-dual

Frob20

C5⋊C4
K5

{4,4}(2,2)88
8
16
0
self-dual

self-Petrie dual

?
{4,4}(3,0)99
9
18
0
self-dual

C32⋊C4
{4,4}(3,1)1010
10
20
0
self-dual

(C5⋊C4)×C2
{4,4}(3,2)1313
13
26
0
self-dual

C13⋊C4
{4,4}(4,0)1616
16
32
0
self-dual

C42⋊C4
{4,4}(4,1)1717
17
34
0
self-dual

C17⋊C4
{4,4}(3,3)1818
18
36
0
self-dual

?
{4,4}(4,2)2020
20
40
0
self-dual

?
{4,4}(4,3)2525
25
50
0
self-dual

?
{4,4}(5,0)2525
25
50
0
self-dual

C52⋊C4
{4,4}(5,1)2626
26
52
0
self-dual

?
{4,4}(5,2)2929
29
58
0
self-dual

C29⋊C4
{4,4}(4,4)3232
32
64
0
self-dual

?
{4,4}(5,3)3434
34
68
0
self-dual

?
{4,4}(6,0)3636
36
72
0
self-dual

?
{4,4}(6,1)3737
37
74
0
self-dual

C37⋊C4
{4,4}(6,2)4040
40
80
0
self-dual

?
{4,4}(5,4)4141
41
82
0
self-dual

?
{4,4}(6,3)4545
45
90
0
self-dual

?
{4,4}(7,0)4949
49
98
0
self-dual

C72⋊C4
{4,4}(7,1)5050
50
100
0
self-dual

C50⋊C4
{4,4}(5,5)5050
50
100
0
self-dual

?
no. of
hexagons

{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

Water, gas, and electricity

{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}(8,6)

{6,3}(9,5)

{6,3}(12,2)

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

?

{6,3}(5,7)

{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)

?
no. of
triangles

{3,6}(1,1}

{3,6}(2,0)

21
2
3
0
{6,3}(1,1)

hemi-3-hosohedron

D6
{3,6}(0,2)

{3,6}(3,1)

63
6
9
0
{6,3}(0,2)

C5{6,6}

D6×C3
{3,6}(2,2)

{3,6}(4,0)

84
8
12
0
{6,3}(2,2)

C4{4,6}

S4
{3,6}(1,3)

{3,6}(4,2)

{3,6}(5,1)

147
14
21
0
{6,3}(1,3}

Frob42

C7⋊C6
K7

{3,6}(3,3)

{3,6}(6,0)

189
18
27
0
{6,3}(3,3)

?K3,3,3
{3,6}(0,4)

{3,6}(6,2)

2412
24
36
0
{6,3}(0,4)

?
{3,6}(2,4)

{3,6}(5,3)

{3,6}(7,1)

2613
26
39
0
{6,3}(2,4)

?the Paley order-13 graph

{3,6}(4,4)

{3,6}(8,0)

3216
32
48
0
{6,3}(4,4)

?the Shrikhande graph
{3,6}(1,5)

{3,6}(7,3)

{3,6}(8,2)

3819
38
57
0
{6,3}(1,5)

C19⋊C3
{3,6}(3,5)

{3,6}(6,4)

{3,6}(9,1)

4221
42
63
0
{6,3}(3,5)

?
{3,6}(5,5)

{3,6}(10,0)

5025
50
75
0
{6,3}(5,5)

?

The following figures have more than 50 faces; they are included because their duals are above.

{3,6}(0,6)

{3,6}(9,3)

5427
54
81
0
{6,3}(0,6)

?
{3,6}(2,6)

{3,6}(8,4)

{3,6}(10,2)

5628
56
84
0
{6,3}(2,6)

?
{3,6}(4,6)

{3,6}(7,5)

{3,6}(11,1)

6231
62
93
0
{6,3}(4,6)

?
{3,6}(6,6)

{3,6}(12,0)

7236
72
108
0
{6,3}(6,6)

?
{3,6}(1,7)

{3,6}(10,4)

{3,6}(11,3)

7437
74
111
0
{6,3}(1,7)

?
{3,6}(3,7)

{3,6}(8,6)

{3,6}(9,5)

{3,6}(12,2)

7839
78
117
0
{6,3}(3,7)

?
{3,6}(5,7)

{3,6}(13,1)

8643
86
126
0
{6,3}(5,7)

?
{3,6}(0,8)

{3,6}(12,4)

9648
96
144
0
{6,3}(0,8)

?
{3,6}(7,7)

{3,6}(14,0)

9849
98
147
0
{6,3}(7,7)

?
{3,6}(2,8)

{3,6}(11,5)

{3,6}(13,3)

9849
98
147
0
{6,3}(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