Date: Mon, 17 Jan 94 18:15:33 +0000 (GMT)
~~~ From: David Seal <dseal@armltd.co.uk >
Subject: Re: Higher Order Cubes

This brings up an interesting point. Perhaps it would be possible to
build a 4-Cube that was internally a 5-Cube but for which the middle
slice was not actually visible on the surface? Or a 2-Cube that's
internally a 3-Cube? I wonder if it might make for smoother-turning
cubes.

Yes, I think you could build such a 4-Cube. Likewise, you could build a
2-Cube as a 3-Cube with invisible middle slices. But I don't believe you'd
want one: it could get completely jammed much too easily.

The reason: If you take a 3-Cube and rotate its left and right slices 45
degrees each, you cannot rotate any of its other faces. This isn't
surprising, since you don't expect to be able to perform one rotation
halfway through another. If its middle slices were hidden, however, it would
*appear* to be a 2-Cube which is not halfway through a rotation, and the
fact that you couldn't move any faces but the left and right ones would be
surprising - and undesirable.

Unfortunately, I believe such a situation could probably arise quite easily.
If you were to take the 2-Cube concerned and rotate its right face 90
degrees relative to its left face, you're going to be OK if the hidden
middle layer rotates 0 or 90 degrees relative to the left face, but not OK
if it rotates any other amount. I suspect most mechanisms would be more
liable to rotate it an intermediate amount!

There may be a way out, though. If you can anchor the place where the three
axes meet to one of the corner cubelets in some way, the problem is solved:
if the "anchor cubelet" is in the left face, then the hidden layer will
rotate 0 decrees; if it is in the right face, then 90 degrees.

David Seal
dseal@armltd.co.uk