# Regular Maps in the Torus, with Square Faces

### 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.

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.

For any a,b with a≥b, {4,4}(a,b) has a double cover {4,4}(a+b,a-b). This can be regarded as a cantellation.

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⋊C4

{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

?K4,4

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

C5{6,4}

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

?

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