From:

Subject:

>>>>> "Jerry" == Jerry Bryan <jbryan@pstcc.cc.tn.us> writes:

Jerry> This has been discussed before on Cube-Lovers, but I am

Jerry> still puzzled or curious about the usage of the word

Jerry> "orbit".

So am I, actually. Dan Hoey just replied with a much more detailed

understanding of the group theory aspects of this word than I have at

present, which I'll have to think about some more. For myself, I mean

no more and no less than a set of cubies which can move into each

other's positions. For a 3x3x3 cube which I imagine to be made of of

3^3=27 smaller cubes (cubies), what I call "orbits" are exactly those

cubies at the 8 corners, 12 edges, 6 faces, and 1 (unseen) at the

center.

Jerry> Secondly, if my understanding of your model is correct, you

Jerry> are treating positions as distinct which cannot be

Jerry> distinguished with normal coloring of a physical cube (even

Jerry> an imaginary physical cube for large N).

Yes, exactly. As Dan just said, he has discussed this vision of the

cube in earlier notes, and called it the "theoretical invisible cube".

When I started thinking about these larger cubes, I built them by

making piles of dice. All the inner cubies were there, and all had

definite orientations, and I could see them every time I tried to

rotate a slice - which required carefully seperating out the layers,

turning one, and putting everything back together. So perhaps that's

why I liked those "invisible" inside pieces. But it also seemed more

elegant. The restricted versions (only the outside, only the

orientations of the corners and edges, etc.) are all special cases.

Jerry> There are several implications of how you treat visibly

Jerry> indistinguishable positions. For example, it impacts your

Jerry> counts of how many positions there are. For another

Jerry> example, it impacts your solutions (e.g., "invisible"

Jerry> incorrect parity on the 4x4x4. "Invisible" bad parity can

Jerry> also occur on the 3x3x3 if you remove the face center color

Jerry> tabs. A slice move will give the edges and corners

Jerry> opposite parity that is not visible.) Perhaps you could

Jerry> discuss these issues with respect to your model.

I'm not sure what there is to say; you seem to understand the issues.

Yes, I am counting "visibly indistinguishable" positions as different,

especially on the larger cubes, if by "visibly" you mean to only look

at the outside. I'm assuming that either the whole thing is

transparent, or that you can take it apart, and see the inside cubies

if you like. There are parity constraints between the different

orbits, including the ones on the inside that are "invisible," but

they turn out to be fairly simple: the parity of each orbit of corners

and the central cubie, from the outer layer all the way down to the

inside, are independent, and can be chosen arbitrarily. And once

they're fixed, the parity of all the other orbits is given.

By "bad" parity I assume you mean a case when the edges and corners

have different parities. Starting from the solved (even parity) 3x3x3

Cube, a slice move definitely does this; four outside edges cycle, and

the corners don't move. However, on the 3x3x3, this *is* visible,

since the face centers will also have odd parity. Moreover, the

central cube (which you can't see, of course, and isn't really there

on a real cube) also has odd parity, in a way: it has undergone an odd

number of quarter turns. On a 4x4x4, a slice move on a solved cube

changes the parity of the inside 2x2x2 corners (which you can't see)

and the edges (which you can). The parity of the outer corners is

left unchanged, since they didn't move, and the parity of the face

centers is also unchanged, since 8 of them move in two cycles of four

cubies. Then the fact that the outside edges are odd while the

outside corners are even simply means that the inside 2x2x2 corners

are also odd. That's all.

Hope that helps,

Dr. Jim Mahoney mahoney@marlboro.edu Physics & Astronomy Marlboro College, Marlboro, VT 05344