From:

~~~ ~~~ Subject:

Jeff Baggett (baggett@mssun7.msi.cornell.edu) asks on the sci.math

newsgroup:

1. I am seeking a description of the group of symmetries associated

with Rubiks cube. I have some ideas but they aren't particularly

elegant. Can someone suggest a paper?

Jeff,

I have looked into this somewhat. As far as I know, the symmetries of

the 3^3 cube are just the symmetries of the cube, but in larger sizes

we can do better. The best way of looking at this is to imagine that

there is a (N-2)^3 cube sitting inside your N^3 cube, and smaller

cubes within, and you are trying to solve them all together.

Suppose we address each cubelet of the N^3 cube using cartesian

coordinates (x,y,z), where (0,0,0) is the center of the cube (for N

odd) and no cubelets have any coordinate zero if N is even. The

maximum absolute value of the coordinates is [N/2].

Then for 1<=I<=[N/2], there is a symmetry F[I]:(x,y,z)->(f(x),f(y),f(z)), where f(I)=-I, f(-I)=I, and f(x)=x otherwise. Then for 1<=I<J<=[N/2], there is a symmetry E[I,J]:(x,y,z)->(e(x),e(y),e(z)), where e(I)=J, e(J)=I, e(-I)=-J, e(-J)=-I, and e(x)=x otherwise.

These are symmetries of the cube group, and they map elementary moves

to elementary moves (provided we take an elementary move to be a

rotation of the slab of N^2 cubelets that have a particular nonzero

value of a particular coordinate). Symmetries of the cube group that

preserve elementary moves are useful in the study of local minima in

the cube group.

It turns out that if you only want to consider the outside of the cube

(ignoring the (N-2)^3 cube inside) all of these symmetries are still

present except F[[N/2]] and E[I,[N/2]].

I mentioned these symmetries in a note to the Cube-Lovers mailing list

in 1983. I called E[I,J] evisceration, F[1] inflection, and F[[N/2]]

exflection in that note (where I was dealing explicitly with only the

4^3). The discussion of the relation to local minima took place in

1980. Let me know if you'd like a copy of these messages.

I ran into these symmetries earlier, though. They are symmetries of

the N^3 tic-tac-toe board! I would not be surprised if they arise in

some other connection in mathematics, but I have never run into them.

They generalize into larger dimensions, as well.

I've also taken the liberty of Cc'ing the Cube-Lovers list with this

note. If you'd like to be on that list, you may ask of

"Cube-Lovers-Request@AI.AI.MIT.Edu".

Dan Hoey

Hoey@AIC.NRL.Navy.Mil