# 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