This page is obsolete. See the current version of Regular Maps in the Torus

Regular Maps in the Torus

square faces | hexagonal faces | triangular faces

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
comments
{4,4}(1,0)11
1
 2 
0
self-dual


hemi-2-hosohedron

C4 face meets self at vertex face meets self at edge
{4,4}(1,1)22
2
 4 
0
self-dual


4-hosohedron

D8 face meets self at vertex
{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

chiral

{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)×C2chiral
{4,4}(3,2)1313
13
 26 
0
self-dual


C13⋊C4chiral
{4,4}(4,0)1616
16
 32 
0
self-dual


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


C17⋊C4chiral
{4,4}(3,3)1818
18
 36 
0
self-dual


?
{4,4}(4,2)2020
20
 40 
0
self-dual


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


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


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


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


C29⋊C4chiral
{4,4}(4,4)3232
32
 64 
0
self-dual


?
{4,4}(5,3)3434
34
 68 
0
self-dual


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


?
{4,4}(6,1)3737
37
 74 
0
self-dual


C37⋊C4chiral
{4,4}(6,2)4040
40
 80 
0
self-dual


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


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


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


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


C50⋊C4chiral
{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 face meets self at vertex face meets self at edge
{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

The three-rung Möbius ladder S86.

{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
chiral
{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

chiral

{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⋊C3chiral
{6,3}(3,5)

{6,3}(6,4)

{6,3}(9,1)

2142
21
 63 
0
{3,6)


(3,5}

?chiral
{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)


?chiral
{6,3}(4,6)

{6,3}(7,5)

{6,3}(11,1)

3162
31
 93 
0
{3,6}(4,6)


?chiral
{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)


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

{6,3}(8,6)

{6,3}(9,5)

{6,3}(12,2)

3978
39
 117 
0
{3,6}(3,7)


?

chiral

{6,3}(5,7)

{6,3}(13,1)

4386
43
 126 
0
{3,6}(5,7)


?chiral
{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)


?chiral
 no. of
triangles
 
{3,6}(1,1}

{3,6}(2,0)

21
2
 3 
0
{6,3}(1,1)


hemi-3-hosohedron

D6 face meets self at vertex
{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

chiral

{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

chiral

{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⋊C3chiral
{3,6}(3,5)

{3,6}(6,4)

{3,6}(9,1)

4221
42
 63 
0
{6,3}(3,5)


?chiral
{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)


?chiral
{3,6}(4,6)

{3,6}(7,5)

{3,6}(11,1)

6231
62
 93 
0
{6,3}(4,6)


?chiral
{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)


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

{3,6}(8,6)

{3,6}(9,5)

{3,6}(12,2)

7839
78
 117 
0
{6,3}(3,7)


?chiral
{3,6}(5,7)

{3,6}(13,1)

8643
86
 126 
0
{6,3}(5,7)


?chiral
{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)


?chiral

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 S3 S4.
Some Cayley diagrams drawn on the torus.
Some pages on groups

Copyright N.S.Wedd 2009