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
**a ^{2}+b^{2}**.

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.

designation | no. of squares | picture | V F E Eu | dual | rotational symmetry group | comments |
---|---|---|---|---|---|---|

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

{4,4}_{(1,1)} | 2 | 2 2 4 0 | self-dual | D8 | ||

{4,4}_{(2,0)} | 4 | 4 4 8 0 | self-dual self-Petrie dual | C_{2}^{2}⋊C4 | ||

{4,4}_{(2,1)} | 5 | 5 5 10 0 | self-dual | Frob20 ≅ C5⋊C4 | K_{5} | |

{4,4}_{(2,2)} | 8 | 8 8 16 0 | self-dual self-Petrie dual | ? | K_{4,4} | |

{4,4}_{(3,0)} | 9 | 9 9 18 0 | self-dual | C_{3}^{2}⋊C4 | ||

{4,4}_{(3,1)} | 10 | 10 10 20 0 | self-dual | (C5⋊C4)×C2 | ||

{4,4}_{(3,2)} | 13 | 13 13 26 0 | self-dual | C13⋊C4 | ||

{4,4}_{(4,0)} | 16 | 16 16 32 0 | self-dual | C_{4}^{2}⋊C4 | ||

{4,4}_{(4,1)} | 17 | 17 17 34 0 | self-dual | C17⋊C4 | ||

{4,4}_{(3,3)} | 18 | 18 18 36 0 | self-dual | ? | ||

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

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

{4,4}_{(5,0)} | 25 | 25 25 50 0 | self-dual | C_{5}^{2}⋊C4 | ||

{4,4}_{(5,1)} | 26 | 26 26 52 0 | self-dual | ? | ||

{4,4}_{(5,2)} | 29 | 29 29 58 0 | self-dual | C29⋊C4 | ||

{4,4}_{(4,4)} | 32 | 32 32 64 0 | self-dual | ? | ||

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

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

{4,4}_{(6,1)} | 37 | 37 37 74 0 | self-dual | C37⋊C4 | ||

{4,4}_{(6,2)} | 40 | 40 40 80 0 | self-dual | ? | ||

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

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

{4,4}_{(7,0)} | 49 | 49 49 98 0 | self-dual | C_{7}^{2}⋊C4 | ||

{4,4}_{(7,1)} | 50 | 50 50 100 0 | self-dual | C50⋊C4 | ||

{4,4}_{(5,5)} | 50 | 50 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