TITLE: Coloring the pentagonal hexecontahedron
QUESTION [7 upvotes]: So, I'd like to color the pentagonal hexecontahedron in a way that is satisfying aesthetically and mathematically. For me this equates to, in order of priority -
1. No same-colored faces can share an edge
2. We must use as few colors as possible
3. We must have as much symmetry as possible
Here's a
5-colored pentagonal hexecontahedron - 5 colors is a lot, but it's nice that there are apparently no adjacent faces with the same color. Not clearly very symmetrical though (if it is, it's obscured by the large number of colors).
By the four-color theorem, we know it can be done with 4. I know it can't be done with 3, proof below. So, the aesthetics question boils down to "how symmetrical a 4-coloring can we make?".
By the way, it is possible to color the deltoid hexecontahedron with only three colors, without same colored faces touching, and with chiral octahedral symmetry. Here's a lovely piece of origami designed by Tomoko Fuse that does this.
Answer this question and you will be credited in a documentary about virus structure! I'm coloring a Satellite Tobacco Mosaic Virus
EDIT so the origami one I show is actually quite symmetrical! But still not hugely useable since you (I) have to stare at it for a while to see it.

REPLY [3 votes]: Here's one 4-colouring:
Front:

Back:

For convenience, here is a list of vertices with xyz coordinates, and 
here is a list giving 
the vertex numbers of each face and the colour of the face.