Message #53

From: rbreiten <rbreiten@yahoo.com>
Subject: Re: [MC4D] Orientations of the centre cubes …
Date: Fri, 21 Nov 2003 22:31:15 -0000

— In 4D_Cubing@yahoogroups.com, "mahdeltaphi" <mark.hennings@n…>
wrote:

>[…]

> There is, probably, a real problem to be considered with the 4^3
> cube, and similar. The 4^3 cube contains 8 internal cubies, which
> themselves represent a 2^3 cube in their own right. I have not
> considered to what extent the central 2^3 cube can be manipulated
> independently of the external faces of the 4^3 cube (physical
> implementations of the 4^3 cube do not display these central
> cubies, so tracking them is more difficult). However, a computer
> simulation of the 4^3 cube could do this.

OK I finally got around to thinking about this (and playing with my
5^3 cube since I couldn’t find my 4^3). Let’s pretend that my
missing 4^3 cube is actually 64 little (fully distinguishable and
orientable) cubes stacked into a 4^3 cube that magically can be moved
only according to the regular rules. We are interested in the group
of permutations G of the internal 2^3 sub-cube that leave the outer
shell fixed.

Rotating an outer face a quarter turn is an 4-cycle on its corners,
two parallel 4-cycles on its edges, and a 4-cycle on its center
faces. Rotating an inner layer a quarter turn is a 4-cycle on its
four edges and a 4-cycle on a face of the internal cube (4-cycles are
odd permutations of course).

If the outer shell is solved, all the edges are in an even
permutation (identity). This restricts the internal cube to an even
number of quarter turns on its faces. Depending on your algorithm
for solving 4^3, this may be a familiar fact (oops, got an edge
pair "flipped").

Sloppily last night I decided that G was generated by half turns of
the internal cube’s faces; clearly this is a subgroup of G as it is
easy to construct a sequence of moves that leaves the outer shell
fixed but rotates a face of the internal cube a half turn.

Today, though, I think that by using conjugations I can probably
construct a sequence of moves that fixes the shell but rotates two
faces of the internal cube a quarter turn each, which means that the
only restriction on the corners of the internal cube would be that
they are in an even permutation.

Intuitively this makes sense since it is a very similar restriction
to that of the supergroup problem of the 1-color faces on 3^3.

I will write down the first sequence and try to construct the second
one over the weekend.