From cube-lovers-errors@mc.lcs.mit.edu Sun Jul 27 21:34:49 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id VAA29661; Sun, 27 Jul 1997 21:34:49 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From nbodley@tiac.net Sun Jul 27 12:09:36 1997 Date: Sun, 27 Jul 1997 12:06:18 -0400 (EDT) From: Nicholas Bodley To: Cube Mailing List Subject: 4^3 innards: There's a ball in there, but which way does it point? Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Uniquely to all cubes from the Pocket Cube to the 5^3, only the 4^3 (Rubik's Revenge) has a ball inside it. (I do hope that I don't cause curious people to break their fragile center pieces trying to open up theirs!) This is really a repetition of some old information, but with the current interest in 4^3s, it seems sufficiently interesting to repeat it. However, afaik, the questions about the ball's orientation are probably new. The ball (made of at least 9 pieces plus eight screws, as I recall) consists of a center piece, essentially a sphere (possibly two hemispheres) with "octants" fastened to it; these have the geometry of 90-90-90-degree right spherical triangles, although in practical detail they differ. Gaps between these octants define three circumferential grooves that correspond to the Earth's equator and two orthogonal meridians of longitude. (All 3 grooves are orthogonal.) (Sorry for the redundancies; trying to be clear to everybody). These grooves are "undercut" on one side, so they have an inverted L-shaped cross-section. The cubies (center ones only, as I recall) have feet that tuck under the extended edges of the octants' grooves; this keeps them in place. Machinists know well of the T-slots that work with clamps to hold work in place on machine tools; these are similar, but the cubies are free to slide in the L-slots. If I'm thinking clearly (not too sure!), the ball has a 120-degree rotational symmetry about an unique diameter. One octant has no undercuts; its opposite, I think, has all three edges undercut. Other octants have some edges undercut. Practical details dictate that when one half of the 4^3 is rotated, one half is definitely locked to the internal ball. However, you probably don't know which half it is! (The mathematical folk here might find it fun to predict which half is the locked half for any given configuration; this might even not be a trivial problem. (As well, does the locked part always end up in the same place when the Cube is solved? I suspect so, but am not sure.) (Anybody for a translucent-cubie 4^3?) Is it possible to maneuver the internal ball so that it has effectively revolved by a half turn (or quarter turn?) about any given axis, while preserving the exterior configuration? My best regards to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 1 B.C.E. --------------------------------------------------------------------------