This is probably well-known, but recently I noticed that it's possible to create a highly-symmetrical 4-coloring of the icosahedron's vertices (resp. dodecahedron's faces). This coloring partitions the icosahedron's vertices into 4 sets of 3, that sit at the vertices of 4 equilateral triangles. These triangles have edges that lie at the chords of pentagonal cross-sections of the icosahedron -- I'm guessing they are subsets of some Kepler-Poinsot solid? No adjacent vertices have the same color (being a 4-coloring). It would appear that these 4 equilateral triangles have tetrahedral symmetry (though they do not themselves form a tetrahedron since they are disjoint).
Is there a similar highly-symmetrical coloring of the dodecahedron's vertices (resp. icosahedron's faces)? In particular, is there a 3-coloring that exhibits a high degree of symmetry?