This is a follow-up to Coloring Sonobe origami, part 1: proper coloring.
The most natural first polyhedra to make from the Sonobe unit are, like the octahedron, regular solids. As a result, they have lots of symmetries. Thus, a natural goal is to color a Sonobe construction in a way that exhibits symmetry, as well. It’s not completely obvious to me what this should mean, but here are two ideas:
- Each color should be interchangeable with every other.
- Among units of a given color, each unit should essentially be interchangeable with every other of the same color.
(For the three-coloring, the first symmetry is easy to check: just rotate the picture 120 degrees. In the four-coloring, we get from rotation by 120 degrees that green, red and black are symmetric; in this particular drawing it’s not completely clear that the blue units are also arranged in an identical pattern, but redrawing the graph in various ways (or putting the octahedron back in 3 dimensions) would make this clear.) Incidentally, here are what these look like “for real”:
Here’s the precise mathematical statement that seems to best capture the two intuitions I have about what symmetry should mean:
Or maybe there are better, non-equivalent notions of symmetry here; for example, should one require that every rotation of the polyhedron acts as a permutation on the colors? (All thoughts welcome!) In any case, given a notion of symmetry like this, it’s natural to ask what sorts of colorings there are that exhibit symmetry. Although it seems like a very nice question, I don’t know much in general about this, but I’ll share what I know (or perhaps more properly what I believe to be true) about symmetric proper colorings of small polyhedra:
- The first point above in my definition of symmetry enforces that there should be the same number of edges of each color. Thus, the number of colors must divide the number of edges.
- Of course, using exactly one edge of every color will always work.
- For a tetrahedron (which, recall, looks like a cube when actually constructed from Sonobe units), this means 1, 2, 3 or 6 colors. We throw out 1 and 2 because we’re looking for proper colorings; this leaves 3 colors (2 units of each); there’s a unique way to color the tetrahedron like this, and it exhibits nice symmetry.
- For an octahedron, similar considerations show that we consider 3, 4 or 6 colors. With 6 colors, the second condition requires that the two edges of the same color be opposite, and this gives a symmetric proper coloring. With 3 and 4 colors, one can show that the only symmetric proper colorings are those in the image above. (Actually the four-coloring has two mirror-image versions.) Note that there are plenty of other, nonsymmetric, proper colorings with these numbers of colors.
- The icosahedron has 30 edges, so we have nontrivial possibilities with 3, 5, 6, 10 and 15 colors. Under my definition of symmetry, 3 colors can’t yield a proper symmetric coloring: each vertex would (by the pigeonhole principle) have two edges adjacent to it of the same color, and applying the rotation that carries one of these edges onto the other quickly leads to a contradiction. The five- and six-colorings are in my origami gallery. 15 colors is easy (take opposite pairs of edges), and 10 is doable: The 6- and 10-colorings are different under reflection (see Puzzler 3), while the others are unique. Interestingly, to get 5 colors from 15, we merge (well-chosen) triples of colors, but one can’t get the 5-coloring from the 10-coloring by merging pairs of colors. (Challenge: explain this in terms of the group theory of the icosahedron.) In the 5-coloring, the centers of the six units of a given color are the vertices of a regular octahedron; in the 6-coloring, a regular pentagon; in the 10-coloring, an equilateral triangle.
Of course, there exist proper colorings with any number of colors larger than 3, and many of these exhibit some amount of symmetry (just not precisely the symmetry I have requested). Also, there exist non-proper colorings that exhibit lots of symmetry; of course this trivially includes the 1-coloring, but also (for example) one can choose a subgroup of the symmetry group and color the orbits under the action of this subgroup to get nice effects.
For semi-regular polyhedra, the definition I offered above for symmetric coloring doesn’t apply. On the other hand, it’s obvious that there are colorings of semi-regular polyhedra that exhibit symmetry. For example, see the six-colored cross in my first origami gallery. There is also a nice three-coloring of the cross:
(Note that the three-coloring doesn’t arise from the six-coloring by identifying pairs of colors; rather than try to prove this, I’ve placed two pictures of the colorings side-by-side from which perhaps you will be able to convince yourself.)
What’s the right definition of symmetry to fit these cases? Perhaps the color classes should be the same up to rotations of the polyhedron, but the stabilizer of a color class only needs to act transitively on subsets of edges on which the whole symmetry group acts transitively. (This, I believe, is true of the six-colored cross.) Or perhaps the stabilizer of each color should act transitively on the edges of that color, and we should forget about symmetry between color classes. (This is true for the three-colored cross: two colors are equivalent, and the third is totally different.) Or, perhaps there’s some other definition that includes both of these (obviously symmetric-looking) cases.