Finding edge-5-colourings of the icosahedron

This page gives guideline for finding edge-5-colourings of the icosahedron; and for showing whether two such colourings are necessarily distinct (after rotations, reflections, and permutations of the colours) or potentially isomorphic.

Lemmas

We start with a set of lemmas. Each is easily proved. No proofs are offered here.

If we have an edge-5-coloured icosahedron, the set of edges of any two colours must form a set of circuits, also called cycles.
This set of circuits must pass through each vertex once. We will call it a "Hamiltonian set" of circuits.
Each such circuit must have an even number of edges.
A region of the surface of the icosahedron bounded by one or more such circuits may not contain a set of five triangles all with a commonn vertex.
In view of the previous lemma, the number of edges forming such a circuit must be 2 greater than the number of triangles which it encloses.

A way to find a new colouring from a known one

If we have an edge-5-colouring of the icosahedron, we can probably find another as follows:

find a pair of the colours such that the edges of those colours form a Hamiltonian set containing more than one circuit.
for some but not all of those circuits, swap the two colours.

This must give rise to an edge-5-colouring. It will probably be different from the one we started with.

How to tell if two colourings are different

The procedure above, applied repeatedly, rapidly gives rise to many edge-5-colourings. But we need a way to tell whether each one we find is different from all those we have found already – an invariant, such that two colourings with diffferent values of the invariant are provably different, despite being rotated, being reflected, and having their colours permuted.

We describe two such invariants:

a list (which we call a casting of the conformations (which we call casts) of the six edges of each of the five colours.
a list of the ways the pairs of colours partition the icosahedron into regions.

The actual colours involved in these invariants are irrelevant. What matters is the list of conformations, and how many colour-pairs are associated with each; and the list of divisions into regions, and how many colours etc. The colours are listed so that the accuracy of the invariants can be checked against the accompanying coloured diagram.

Castings

A typical casting invariant is presented like this,

R	7
G	3
Y	7
B	S
K	3

the invariant itself being the unordered list { 3,3,7,7,S }.

This means that the red edges are in the cast denoted by "7", the green edges in the cast denoted by "3"; etc.

This invariant distinguishes between the mirrored versions of a chiral colouring. To convert an invariant to its mirror-image version, swap 1s with 9s, 3s with 7s, and Ds with Ls.

Castings are discussed and listed here.

Regions

A typical "regions" invariant is presented like this,

RG	P
RB	E-
RY	N+
RK	O
GB	E+
GY	O
GK	N-
BY	E-
BK	E+
YK	P

the invariant itself being the unordered list { E+,E+,E-,E-,N+,N-,O,O,P,P }.

This means that for the colour pairs red-green and yellow-black, the division into regions is the one we have denoted by "P"; for red-blue and blue-yellow it is the one we have denoted by "E-"; etc.

This invariant distinguishes between the mirrored versions of a chiral colouring. To convert an invariant to its mirror-image version, swap +s with -s.

Regions are discussed and listed here.

Numbering of vertices

The list of colourings refers to circuits in the diagrams it shows. These circuits are described by listing their vertices, numbered as shown to the right.