Two of the simpler relationships between pairs of regular maps are
duality and
Petrie duality.
In this page I follow the notation of ^{ARM}
page 192 in using δ to denote duality and
π to denote Petrie duality.

Consider a regular map A with Schläfli symbol
{p,q}_{r}.

δ permutes p and
q, so that Aδ has
Schläfli symbol {q,p}_{r}, and

π permutes p and
r, so that Aπ has
Schläfli symbol {r,q}_{p}.

Thus δ and π both permute the set (p,q,r). Between them they generate all six possible permutations, and πδπ = δπδ. We shall denote πδπ and δπδ by κ. κ permutes q and r.

In general, where A = {p,q}_{r} is a regular map,
Aδ = {q,p}_{r},
Aπ = {r,q}_{p},
Aδπ = {r,p}_{q},
Aπδ = {q,r}_{p}, and
Aκ = {p,r}_{q}
all exist and are all regular maps. They form a hexad.
An example of a hexad is shown in the following table.

Name | p | q | r | V | F | E | genus | UG | FAG | RSG | FSG | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Q | cube | 4 | 3 | 6 | 8 | 6 | 12 | S0 | crown_{4} | K_{2,2,2} | S4 | S4×C2 |

Qδ | octahedron | 3 | 4 | 6 | 8 | K_{2,2,2} | crown_{4} | |||||

Qδπ | C4:{6,4}_{3} | 6 | 3 | 4 | C4 | K_{4} .2 | S4×C2 | |||||

Qκ | C4:{4,6}_{3} | 4 | 6 | 4 | 6 | K_{4} .2 | K_{2,2,2} | |||||

Qπδ | {3,6}_{(2,2)} | 3 | 4 | 8 | S1 | crown_{4} | A4×C2 | |||||

Qπ | {6,3}_{(2,2)} | 6 | 3 | 8 | 4 | crown_{4} | K_{4} .2 |

In this table, p,q,r are the number of edges per face, the number of edges per vertex, and the number of edges per Petrie polygon. V,F,E are the numbers of vertices, faces and edges in the whole regular map. UG is its underlying graph, and FAG is its face adjacency graph. RSG is its rotational symmetry group, and FSG is its full symmetry group; for regular maps in non-orientable surfaces, RSG and FSG must be the same.

We see that

**δ**conserves r, RSG, and the genus of the regular map. It swaps p with q, V with F, mV with mF, and mV with mF.**π**conserves UG, and therefore q, V and mV. It swaps p with r.**κ**conserves FAG, and therefore p, F and mF. It swaps q with r.

The six members of a hexad need not all be distinct. The number of distinct members must divide six, and may be six, three, two, or one. We give an example of each.

Name | p | q | r | V | F | E | genus | UG | FAG | RSG | FSG | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Q =Qδ | tetrahedron | 3 | 3 | 6 | 4 | 4 | 6 | S0 | K_{4} | K_{4} | A4 | S4 |

Qπ =Qδπ | hemicube | 4 | 3 | 3 | 4 | 3 | C1 | K_{3} .2 | S4 | |||

Qπδ =Qκ | hemioctahedron | 3 | 4 | 3 | 3 | 4 | K_{3} .2 | K_{4} |

Name | p | q | r | V | F | E | genus | UG | FAG | RSG | FSG | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Q =Qδπ =Qπδ | N72.9′ | 9 | 9 | 9 | 28 | 28 | 126 | C72 | ? | ? | ? | |

Qδ =Qπ =Qκ | N72.9 |

Whereas the other degenerate hexads are common, it is unusual to find degenerate
hexads with two members; hence the obscure example above. This example is due to
Wilson ^{W79} page 565.

Name | p | q | r | V | F | E | genus | UG | FAG | RSG | FSG | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Q =Qδ =Qδπ =Qκ =Qπδ =Qπ | 2-hosohedron | 2 | 2 | 2 | 2 | 2 | 2 | S0 | C2 | C2 | C2×C2 | C2×C2×C2 |

We can illustrate the relationships among the members of full and degenerate hexads by the diagrams above. This shows four hexads, with 6, 3, 2, and 1 members. Red lines represent δ, green lines π, and blue lines κ; or any other assignment of colours, if you prefer.

π and δ are still meaningful for things that are not regular maps. Thus we can form hexads, not only for regular maps with mirror symmetry as above as above, but, for example, for chiral regular maps, whose duals are not regular, as here:

Name | p | q | r | V | F | E | genus | UG | FAG | RSG | FSG | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Q =Qδ | {4,4}_{(2,1)} | 4 | 4 | 10 | 5 | 5 | 10 | S1 | K_{5} | K_{5} | Frob20 | Frob20 |

Qπ =Qδπ | C5:{10,4} | 10 | 4 | 4 | 5 | 2 | C5 | C_{2} .5 | D10 | |||

Qπδ =Qκ | C5:{4,10} | 4 | 10 | 4 | 2 | 5 | C_{2} .5 | K_{5} |

More on Regular Maps

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