# Rainbow Icosahedra

This puzzle was inspired by playing with a
Magz magnetic ball-and-stick toy:
how to color (in the usual mathematical sense) the
edges of an icosahedron with 6 colors?
Why use six colors, when
it's easy with five?
[MathPages.com]
Notice that *C*(6,3), the number of combinations of 6 colors
taken 3 at a time, is 20--the same as the number of faces.
Now the problem gets interesting!
You may enjoy trying it before looking at my solutions.

**Problem.**
In how many ways, distinct under symmetries of the icosahedron
and renaming of the colors, can the edges of
a regular icosahedron be colored with 6 colors so that

all 5 edges meeting at any vertex have different colors, and

each combination of 3 distinct colors occurs on the boundary
of some face?

**Observation 1.**
In any such coloring each
color must appear on exactly 5 of the 30 edges
of the icosahedron.
For consider one color, say red.
*C*(5,2)=10 is the number of pairs of non-red colors, and hence
the number of distinctly colored triangles with one red edge.
Since each red edge borders two triangles, there must be 5 red edges.

**Observation 2.**
Think of the 5 edges of one color as a "tile",
albeit a completely disconnected one. Then we can address the problem
without attention to colors:

find all the distinct shapes of tiles;

find all the ways of tiling the icosahedron with these shapes; and

discard tilings in which 2 faces touch the same 3 tiles.

A Haskell program to carry out this plan
revealed just 18 tile shapes and 12 tilings.

One tiling has 3-fold rotational symmetry: PostScript
or PDF or HTML (lower quality).

Four tilings have 5-fold rotational symmetry:
PostScript
or PDF.

The other seven tilings lack symmetry: PostScript
or PDF.

#### Some properties of the tilings

No tiling has a symmetry group of even order,
despite the abundance of even-order subgroups in the
icosahedral group.
In particular, no tiling is invariant under
reflection, 180-degree rotation, or central inversion coupled
with renaming of colors.
Every tiling uses tiles of more than one shape.
However the 3-fold symmetric tiling can be
seen as 3 congruent bicolor supertiles.

Each of the 18 tile shapes occurs in some tiling.

#### Related 5-colorings

Branko Grunbaum suggested looking for 5-colorings in
which each combination of colors occurs on exactly
two faces with the colors in opposite
sequence on the two faces.
All 13 such colorings have even-order symmetry groups.
One example
has the full symmetry of the icosahedron:
PostScript,
PDF.
#### Dual problems

By duality, one can color the edges of a dodecahedron
with 6 colors so that every combination of 3 colors
meets at exactly one vertex, or with 5 colors so that
every sequence of three colors occurs clockwise about exactly
one vertex.
John Conway found there are three
5-colorings of the edges of a
dodecahedron such that each of the twelve
5-color necklaces, distinct under rotation and reversal,
occurs around some face.
These colorings are dual to 5-colorings of the
edges of an icosahedron in which every sequence
of 5 colors occurs
clockwise or else counterclockwise about exactly one vertex.

*Modified February 11, 2006; March 24, 2008.*