In "Transitive Decompositions of Graphs and their Links with Geometry and Origami" which appeared in April's American Mathematical Monthly, Geoffrey Pearce described how to edge-color a unit-origami dodecahedron symmetrically using five colors.

Constructing a model based on Pearce's instructions is pretty straight forward if you use Tom Hull's Phizz Unit. Pearce's instructions make use of the usual planar projection of the dodecahedral graph - you need to color your model based on the diagram below (the dark lines represent a single color, you need to repeat the pattern five times with five distinct colors).

If you do this (hopefully with less garish colors), you end up with a model that looks like the one below. In a single picture, you can't really get an appreciation of the symmetry of the coloring, but if you make it yourself and handle it, you'll see that it is indeed a symmetrical, transitive, coloring.

If you want a minimal coloring, rather than a transitive one, you'll only need three colors. A neat way to generate a three-coloring of the dodecahedron is to first find a Hamiltonian circuit for it. To do this, start with two colors - one color to mark an edge-path that touches each vertex only once, and the other color to mark the edges that are not part of the path. Below is a phizz unit dodecahedron with a Hamiltonian circut in yellow (other edges blue). A question before you begin: how many blue edge units and how many yellow edge units should you fold?

Once you have a Hamiltonian circut on your dodecahedron, replace alternate edges on your circut with your third color. Doing this will give you a dodecahedron whose edges are colored such that every vertex has three distinctly colored edges intersecting at it (can you see why?).

Tom Hull describes how to use this method to color phizz buckyballs in his book Project Origami.

If instead of using Phizz units you use Sonobe units, you will obtain models that have the same characteristics, but look more like icosahedrons than dodecahedrons due to the nice duality between these two unit types.

Thank you for this post. I'm working my way through "Beginner's Book of Modular Origami Polyhedra" by Gurkewitz and Arnstein. In the instructions for the dodecahedron they say, "It is a puzzle, but possible, to use only three colors" with no further explanation. I figured I'd give it a shot but after a few failed attempts I did some searching and found your explanation. Thanks again.

ReplyDeleteThanks Mike - the method for using a hamiltonian path to get a 3-coloring is via Tom Hull's book, which I would really recommend, and the symmetrical 5-coloring is from Geoffrey Pearce's article. If you want fuller descriptions of the methods, check out those sources. Thanks again.

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