From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 5 12:36:40 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id MAA24116; Tue, 5 Aug 1997 12:36:40 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Tue Aug 5 12:04:25 1997 Date: Tue, 5 Aug 1997 12:04:08 -0400 Message-Id: <199708051604.MAA13056@sun30.aic.nrl.navy.mil> From: Dan Hoey To: Goyra@iol.ie Cc: cube-lovers@ai.mit.edu In-Reply-To: <199708051031.LAA16445@mail.iol.ie> (Goyra@iol.ie) Subject: Re: Megaminx a.k.a. Supernova Mark Longridge wrote: > I am particularly interested in processes for the 10-spot and > 12-spot. "David Byrden" asked for clarification. Spot patterns are those in which all the corner and edge cubies agree with each other, but not necessarily with all the face centers. They are so named because the non-matching face centers show up as contrasting spots. Mark reported some analysis on them on 31 Oct 95, apparently from GAP. As he reported, there are five conjugacy classes: 0. The identity, 1. The 72-degree twelve-spot, 2. The 144-degree twelve-spot, 3. The 120-degree ten-spot, 4. The 180-degree ten-spot. The angle given is the displacement of the corners-and-edges ensemble from the face-centers ensemble. In cases 1 and 2, the rotation is about an axis through two opposite face centers; in case 3, through opposite corners; in case 4, through opposite edges. Of course, there's no reason to expect optimal processes for these patterns to the same length. Interestingly, while the square of the 72-degree is the 144-degree, it is also the case that the square of the 144-degree is the 72-degree (up to conjugacy). It's also interesting to consider star patterns, in which the edges agree with the face centers, and the corners agree with each other. These come in the same classes as the spots. A third type of pattern is a distorted checkerboard, in which the corners and face centers agree with each other, and the edges agree with each other. These come in the same classes as well. I had hoped to find some in which the edges were apparently reflected with respect to the face centers (as in the Pons Asinorum and the order 6 6-X patterns on the cube) but they seem to be in the wrong orbit for the Megaminx. Dan Hoey@AIC.NRL.Navy.Mil