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. 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.

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


?

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

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


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