Message #1375

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Checkerboards of Klein’s Quartic
Date: Mon, 31 Jan 2011 21:11:57 -0800

Roice,

I like the way you describe checkerboards as twists of only some sets of
piece types. That is a really clear description that makes it easy to
understand. Best of all the resulting images are beautiful! We need no
better reason than that.

-Melinda

On 1/31/2011 8:48 PM, Roice Nelson wrote:
>
>
> 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 here
> <http://library.msri.org/books/Book35/files/thurston.pdf>), 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 at:
>
> www.gravitation3d.com/magictile/checkerboards
> <http://www.gravitation3d.com/magictile/checkerboards>
>
> 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
> <http://games.groups.yahoo.com/group/4D_Cubing/message/837>". 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…
> seeya,
> Roice
>
>
>