Message #41

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

>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