Message #1374

From: Roice Nelson <>
Subject: Checkerboards of Klein’s Quartic
Date: Mon, 31 Jan 2011 22:48:35 -0600

Hi All,

I recently spent some time studying checkerboards of Klein’s Quartic and the
Megaminx/Cube, and it turned out to be an illuminating exercise.

The result is that I now think of a puzzle checkerboard as a superposition
of two symmetries in the full symmetry group of the puzzle shape. The
identity symmetry colors the 1C/3C pieces (centers and corners), and the
other symmetry colors the 2C pieces (edges). So if you look at a pristine
puzzle, imagine applying rotations and/or reflections to the entirety of a
ghost copy of the puzzle to yield a second symmetry, then use the ghost to
color all the edge pieces on the original puzzle. Not all checkerboards
found this way will be possible, due to parity restrictions on the permuted
edges. But the mental model of symmetry superposition combined with parity
checking covers all the potential checkerboards. (Also, some checkerboards
will be partial, meaning at least some solid colored faces will remain.)

Because of the above, looking for possible checkerboards on a puzzle leads
to getting a feel for the object’s symmetry group, which in the case of KQ
has some surprises! You can definitely checkerboard the KQ puzzle, but my
initial guess about the nature of the checkerboard was wrong. I suspected
it would consist of eight 3-cycles among members of face "affinity groups"
(that term is described
but it turned out this is not a possible symmetry of KQ. I’ve placed images
and log files for the two styles of full KQ checkerboards I found instead

They result from applying rotational symmetries about an edge or a vertex,
and are analogous to the two kinds of checkerboards you can do on a Megaminx
(also due to rotational symmetries about an edge or a vertex).

One of the more interesting symmetry observations was simply how many
elements are left unmoved after a given rotation of KQ. Rotating a
dodecahedron about a face leaves two faces unmoved (a "face stabilizer").
Likewise, its edge stabilizer fixes two edges, and vertex stabilizer two
vertices. But on KQ, the face/edge/vertex stabilizers fix 3/4/2 elements,
respectively. The face stabilizer observation is what Nelson discovered
about the "two bottoms<>".
I would have guessed this would be the same for all elements, so it
surprised me to see there are "three bottom" edges and only one opposing
vertex. I didn’t make a KQ checkerboard based on a face stabilizer because
it would have been partial (the result would be three 7-cycles + three solid
faces). It is fun to find the stabilized elements in the checkerboard
pictures and observe relationships among them. For example, on the vertex
stabilized checkerboard, the edges that rotate around the two unmoved
vertices do so in an opposite sense, one CW and one CCW (also true for
Megaminx when you are looking at it as a surface in MagicTile).

Another unexpected result for me was that the simple checkerboard of
2-cycles on a Rubik’s Cube stems from a cube symmetry that involves a
reflection. So if you make this checkerboard and study the colors of edge
pieces, you’ll see that their relationship includes a mirroring relative to
the colors of center pieces. I thought this was pretty cool. Reflection
symmetries unfortunately don’t lead to valid checkerboards on Megaminx or
KQ, since the number of 2-cycles in the result is odd instead of even.

I hope you enjoy the pictures, and can provide further gory details if
anyone is interested…