Message #47

From: mahdeltaphi <mark.hennings@ntlworld.com>
Subject: Re: [MC4D] Orientations of the centre cubes …
Date: Wed, 17 Sep 2003 22:19:19 -0000

>… and then there’s the "very invisible" orientation and
>permutation issues:
>that is, the orientations and permutations of the 0-sticker cubies
>(i.e. internal cubies, which may or may not have physical
>manifestation,
>depending on your faith, since you can never see them),
>on 4x4x4 and larger cubes and hypercubes.

Yes, but these can be ignored for the 3^3 and 3^4 cubes.

In the 3^3 cube, the orientation of the 0-face cubie in the centre is
fixed by the 6 1-face cubies which surround it - each side of the 0-
face cubie always points to one particularly coloured 1-face cubie.
Once the rest of the cube has been solved, therefore, determining the
orientation of the central 0-face cubie is simply a matter of
deciding which coloured face will be uppermost, and so on. Taking the
cube, and rolling it along the table performs rotations of its cubies
which affect the orientation of the 0-face cubie, but does not affect
the relative orientations of the constituent cubies of the cube.

Similarly, the orientation of the central 0-face cubie of the 3^4
cube is fixed by the fact that each of its faces points to one
particular 1-face cubie. Again, then, the orientation of the 0-face
cubie is simply a matter of which (global) 4D rotations you wish to
apply to the cube to pick which colour to display in which position.

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.

Mark