Hexads of Regular Maps

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

Example of a hexad

QδπC4:{6,4}3634C4K4 .2S4×C2
C4:{4,6}34646K4 .2K2,2,2
{6,3}(2,2)6384crown4K4 .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

Degenerate hexads

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.

Three distinct members in a hexad

Q =Qδtetrahedron336446S0K4K4A4S4
Qπ =Qδπhemicube43343C1K3 .2S4
Qπδ =Qκhemioctahedron3433 4K3 .2K4

Two distinct members in a hexad

Q =Qδπ =QπδN72.9′999 2828126 C72???
Qδ =Qπ =QκN72.9

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.

One distinct member in a hexad

Q =Qδ =Qδπ =Qκ =Qπδ =Qπ2-hosohedron222222 S0C2C2C2×C2C2×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.

Hexads of non-regular maps

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:

Q =Qδ{4,4}(2,1)44105510S1K5K5Frob20Frob20
Qπ =QδπC5:{10,4}104452C5C2 .5D10
Qπδ =QκC5:{4,10}41042 5C2 .5K5

More on Regular Maps