Monday, August 19, 2013

Patterned icosahedron, and symmetric origami colorings

Meenakshi Mukerji's "Patterned Icosahedron", from Ornamental Origami.  I was in desperate need of a d20, so I put a number of dots on each face.  Yes, I use my statmech textbook as a folding surface.

One of the things I will talk about repeatedly in this origami series is coloring.  After all, coloring is one thing I can easily change without changing the form or function of the design.  So even if I exactly follow the recipe out of a book, the colors still offer some creative choice.

This icosahedron is made of thirty units (also called modules).  They are called "edge units" because each unit corresponds to an edge of the icosahedron.  I only used three colors: red, green, and brown.  The colors are not placed haphazardly.  I tried to place them in a symmetrical pattern.

But what does it mean for a coloring to be symmetrical?

I don't think there is any standard definition, but one math blogger offered the following definition:
The group of rotations of the polyhedron should act transitively on the set of colors, and the stabilizer of each color should act transitively on the edges of that color.
Here is my translation.  A symmetrical coloring has two conditions:
1. For every pair of colors, it is possible to rotate the first color exactly onto the second.  That is, the set of locations of the second color before rotation are exactly equal to the locations of the first color after rotation.
2. It is possible to rotate any unit onto any other unit of the same color, without changing the complete set of locations occupied by that color.

It's not possible to fulfill both of these conditions for an icosahedron with three colors.  However, I fulfilled the first condition.  Here is a sketch of my coloring:

With apologies to the colorblind

This is also called a "proper" coloring, because no color appears more than once on each face.

Figuring out symmetric colorings is very fun, although that's not always how I choose colors.  I find that in practice, symmetric colorings are pretty subtle, and aren't really obvious in a 2D photo.