From BRYAN@wvnvm.wvnet.edu Sun Nov 20 12:56:36 1994 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA13248; Sun, 20 Nov 94 12:56:36 EST Message-Id: <9411201756.AA13248@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 2796; Sun, 20 Nov 94 12:56:30 EST Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 7514; Sun, 20 Nov 1994 12:56:30 -0500 X-Acknowledge-To: Date: Sun, 20 Nov 1994 12:56:26 -0500 (EST) From: "Jerry Bryan" To: Cc: "Cube Lovers List" Subject: Re: Antipode In-Reply-To: Message of 11/14/94 at 13:54:31 from dlitwin@geoworks.com On 11/14/94 at 13:54:31 dlitwin@geoworks.com said: > Despite having read all of the archives, I still don't know what an >antipode is. I suspect I'd have to know more about group theory, but can >you briefly describe what one is (you may want to CC the cube-lovers list >as well, in case more don't understand the term). I guess the most limited definition is two points on the opposite sides of a sphere, at the ends of a diameter -- e.g., the north pole and the south pole. However, the definition need not be limited to three dimensions (points on the opposite ends of a diameter of a circle are sometimes referred to as antipodes, I think) nor to circles and spheres (I have seen opposite corners of a square referred to as antipodes). Generalizing further, antipodes are "opposite" or "maximally distant" points of any sort of structure, depending on what "opposite" or "maximally distant" mean in the context at hand. With respect to Rubik's cube, antipodes of Start are states which are maximally distant from Start, and it is a matter of great interest what that maximal distance might be. I have to admit to a certain discomforture with one aspect of the way we tend to refer to antipodes in the Rubik's cube. Most Rubik structures that have been investigated do not have a single point which is maximally distant from Start; rather, they have several or many maximally distant points, and all the maximally distant points are called antipodes. I would be more comfortable using "antipode" only when the maximally distant point is unique. One example where the maximally distant point is unique is the subgroup consisting of edges only (no corners or centers) where only Q-turns are allowed. In this case, the maximally distant point has been called the "unique antipode". The description "unique antipode" seems redundant somehow -- "antipode" ought to imply "unique", but that has not been the custom on Cube-Lovers. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU If you don't have time to do it right today, what makes you think you are going to have time to do it over again tomorrow?