Message #39

From: mahdeltaphi <mark.hennings@ntlworld.com>
Subject: Re: Orientations of the centre cubes
Date: Tue, 09 Sep 2003 09:13:01 -0000

Thinking further, it is also clear that only the pieces which are
common to 3 or 4 cubical faces have their positions
absolutely fixed by their colours alone. The pieces that are common
to 2 cubical faces can also be correctly
positioned, colourwise, but be so in differing orientations.

This makes the full number of possible orientations even greater
than I first said (even allowing for the fact that I foolishly put 6
instead of 24 in my first message).

Below I offer some thoughts about moves which alter the orientations
of these pieces.

I am sure that it should be possible to mark the 1-face and 2-face
pieces in a manner which was not too ugly aesthetically, but which
nevertheless indicated their orientations uniquely. For example, I
simply used to cut a notch on one side of each centre face of my
3x3x3 cube, and a mathing notch on one edge piece for each 3x3 face.
The cube was in the correct orientation when all the notches matched.

I never saw a "good" 3x3x3 cube with pictures on the sides. I did
once have a Charles&Di wedding cube, but that was in appallingly
cheap and nasty plastic, and was next to impossible to use. Marking
a standard cube worked much better!

Moves similar to those which permute the centre faces of
the 3x3x3 cube will alter these pieces’ orientations. For example,
the sequence:
Right 15 Left
Left 15 Right
Up 11 Left
Right 15 Right
Left 15 Left
Up 11 Twice
Right 15 Left
Left 15 Right
Up 11 Left
Right 15 Right
Left 15 Left
Up 11 Twice

[or R(15) L(15)^{-1} U(11) R(15)^{-1} L(15) U(11)^2 R(15) L(15)^{-1}
U(11) R(15)^{-1} L(15) U(11)^2]

gives a half-twist to the Top/Up centre piece, as well as giving a
matching half-twist to the Up/Bottom centre piece.
Additionally, I think that it gives a matching half-twist to the
centre piece of the Up face. In other words, it
gives a half-twist to the central spine of three pieces running
through the Up face from Top to Bottom.

Other moves which permute the orientations of the 3x3x3 centre faces
can be used similarly.

More interestingly, consider the sequence:
Up 11 Left
Down 11 Left
Right 15 Twice
Left 15 Twice
Back 15 Right
Right 5 Left
Front 11 Left
Right 15 Left (holding down the "2" button)
Front 11 Right
Right 5 Right
Back 15 Left
Left 15 Twice
Right 15 Twice
Down 11 Right
Up 11 Right

[ or U(11) D(11) R(15)^2 L(15)^2 B(15)^{-1} R(5) F(11) R(15,2) F(11)^
{-1} R(5)^{-1} B(15) L(15)^2 R(15)^2 D(11)^{-1}
U(11)^{-1} ]

Call this sequence Q.

Q is made up of three parts. The first part is the sequence
U(11) D(11) R(15)^2 L(15)^2 B(15)^{-1} R(5) F(11)
which moves all the pieces of the Front face away from the Front
face, placing them where the second part of Q,
namely the move
R(15,2)
does not affect them. The remaining part of Q is the inverse of the
first part. Thus the only part of the Front face
that is affected by Q is the centre cube, which has been replaced by
the centre cube from the Up face. Of course, the
rest of the tesseract has been messed up!

Applying a rotation to the Front face will not affect the relative
(within the Front face) positions of the components of that
face, and will not affect the way in which the rest of the tesseract
has been muddled up. Undoing Q then returns the
centre of the the Up face to the middle of the Up face (but with an
altered orientation, due to the Front rotation),
and undoes all the other moves that have been performed on that
cube. The result is that the Front face has been
restored (with its centre face in its original orientation, but the
rest of that face rotated). Applying an
appropriate rotation to the Front face should restore the tesseract
to its original colour status, but the Front and
Up centre cubes will have been rotated!

Thus the sequence

Q F(11) Q^{-1} F(11)^{-1}

should be one such move.

I would be very interested in an implementation of the 4D cube which
enabled me to think about moves like this more easily!

Mark