The icosahedron, drawn flat
The graph of the icosahedron, shown to the right, can have its edges coloured with
five colours in such a way no two edges of the same colour meet at any vertex. Such
a colouring is called an "edge-5-colouring". The purpose of these pages is to list
all possible edge-5-colourings of the icosahedral graph.
A remarkably symmetrical edge-5-colouring of the icosahedron
The most symmetrical 5-edge-colouring of the icosahedron
One way to edge-5-colour the icosahedron is as follows. Regard it as embedded
symmetrically in 3-space (you were probably doing that already), and assign the
same colour to two edges iff they are (orthogonal or parallel). This colouring
is shown to the left.
This colouring has some remarkable properties:
Each of the 20 cyclically-ordered
A cyclical ordering of a set can be regarded as an arrangement of its elements in a
circle (rather than a list, as for a standard ordering). Thus (a,b,c), (b,c,a) and
(c,a,b) are all the same cyclic ordering; but (a,c,b) is a different cyclic
ordering. three-element subsets of the set of colours
appears once around one of the 20 faces.
Each of the 12 even permutations
Just as permutations of ordered sets can be classified as "even" (having any number
of odd-length cycles and an even number of even-length cycles) and "odd" (having any
number of odd-length cycles and an odd number of even-length cycles), so can
cyclic permutations of sets of odd size. of the set of five colours
appears around one of the 12 vertices.
The partition of the set of edges into five subsets (one for each colour) is
preserved by rotations and reflections of the icosahedron in 3-space.
Each of the 24 rotations whose axes passes through two vertices does a 5-cycle
on the colours.
Each of the 20 rotations whose axes passes through two face-centes does a 3-cycle
on three colours and fixes the other two.
Each of the 15 rotations whose axes passes through two edge-centres, and each of
the 15 reflections does two 2-cycles on the colours and fixes the fifth.
Other edge-5-colourings of the icosahedron
There are 17 other distinct edge-5-colourings of the icosahedron, harder to find and
less interesting than the one shown and described above. Here is a list of
all the distinct
edge-5-colourings of the icosahedron. They are shown below, numbered as in
that paper.
The seven reflexive colourings are shown first, and then the eleven dual pairs.
Number 29 is the colouring shown above, with the colours permuted. In this
enumeration, colourings which are mirror images of one another are counted as
distinct; so, for example, 16 and 19 below are mirror images (with the colours
permuted).
