Message #42

From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Re: [MC4D] orientations of the centre cubes …
Date: Tue, 09 Sep 2003 15:28:34 -0500

On Tuesday, September 09, "mahdeltaphi" <mark.hennings@ntlworld.com> wrote:
>Working on a uniformly coloured 3x3x3 cube or tesseract is not
>solving the full symmetry group. The subgroup of the full symmetry
>group which fixes the colours, but ignores orientations, is a normal
>subgroup of the full group, and the quotient group of the full
>symmetry group by this normal subgroup is the group that is being
>studied when working with a uniformly coloured cube/tesseract.
>Although nothing like as big as the full symmetry group, the colour-
>preserving subgroup is nonetheless respectably large, and probably
>deserves some consideration.

>The methods I have always used for solving cubes (of varying
>sizes/dimensions) have always involved getting the colours right,
>and then adjusting the orientations at the end - almost certainly
>not the most efficient approach, but one which gives reliable
>results. …

I think there may be a slight ambiguity in the above
which I would like to understand properly. There are
two sorts of relevant orientation issues for a given
hyper-cubie - that which is forced by the stickers on
it (the obvious orientation issue) and that for the
axes for which the hyper-cubie has no stickers (the
unobvious one applying to 1- and 2-color
hyper-cubies). So there are two levels at which one
may "ignore" orientations.

The second quoted paragraph above sounds much like my
own approach to cubing - namely, get the
(hyper-)cubies into correct position (without regard
to the way their stickers are facing) and then correct
the orientation of the stickers. In particular, I am
always ignoring the dis-orientations you cannot see
because of uniformly colored stickers. But in Mark’s
earlier, "The subgroup of the full symmetry group
which fixes the colours, but ignores orientations
…", because of the phrase "fixes the colours", I get
the impression that the orientations we are ignoring
are only the unobvious ones. Perhaps both quotients
are interesting.

Regards,
David V.