Thread: "orientations of the centre cubes ..."

From: "mahdeltaphi" <mark.hennings@ntlworld.com>
Date: Tue, 09 Sep 2003 13:38:38 -0000
Subject: Re: orientations of the centre cubes ...




>All I was trying to say was that assigning a stringent orientation
>requirement is a change to the goal of the game, whereas extending
>the cube to four dimensions is a generalization of the same game.

I would argue that asking for a stringent orientation requirement is
not changing the goal of the game, but rather refining/extending it.
One of the great attractions of the cube (in whatever dimension) to
me is that its symmetry group is so very large. All rotation
operations on the 3x3x3 cube rotate the centre pieces on its 3x3
faces, but the effects of those rotations are not normally visible,
since the pieces are (normally) uniformly coloured. Similarly, all
rotations of the tesseract rotate the orientations of the centre
cubes, and move and/or rotate the 2-face pieces as well. Again,
given a uniform colouring system, the rotations of the centre cubes
are invisible, and while the movements of the 2-face pieces are
visible, their rotations are not.

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. It seems to me that the same approach would apply here.
Putting the tesseract's colours together is visually challenging
enough without worrying at that stage about orientations! Ignoring
the orientation problem can then be seen as simply a matter of
choosing to forego the final stage of the solution.

I agree that any system of marking the centre cubes and the 2-face
pieces would detract from the visual appeal of the puzzle to some
degree. However, since you can (probably) solve the colour problem
first, and then go on to consider the colour-preserving subgroup
second, might it be possible to have a menu option which switched
off appropriate orientation indicators until required? People who
did not want to consider the orientation problem could simply keep
that option switched off.

Mark




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



On Tuesday, September 09, "mahdeltaphi" 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.





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