From dseal@armltd.co.uk Mon Jan 17 14:14:08 1994 Return-Path: Received: from eros.britain.eu.net by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA11921; Mon, 17 Jan 94 14:14:08 EST Received: from armltd.co.uk by eros.britain.eu.net with UUCP id ; Mon, 17 Jan 1994 18:21:12 +0000 Received: by armltd.co.uk (5.51/Am23) id AA07766; Mon, 17 Jan 94 18:16:09 GMT Date: Mon, 17 Jan 94 18:15:33 GMT From: dseal@armltd.co.uk (David Seal) To: (Cube) cube-lovers@ai.mit.edu Subject: Re: Higher Order Cubes Message-Id: <2D3AD5C5@dseal> In-Reply-To: <199401151733.MAA02409@Collatz.McRCIM.McGill.EDU> > 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