Thread: "Checkerboards of Klein's Quartic"

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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),
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

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...

seeya,
Roice

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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
> ), 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
>
>
> 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...
> seeya,
> Roice
>
>
>

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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:
cite="mid:AANLkTincRppqhq6NM38urknVM67pt8ojfTJsqhCP0Vu4@mail.gmail.com"
type="cite">