From mark.longridge@canrem.com Sun Nov 12 01:51:15 1995 Return-Path: Received: from itchy.crso.com (itchy.canrem.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA26404; Sun, 12 Nov 95 01:51:15 EST Received: by canrem.com (PCB-UUCP 1.1f) id 1FDFC0; Sun, 12 Nov 95 01:35:04 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Magic Platonic Solids From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1260.5834.0C1FDFC0@canrem.com> Date: Sun, 12 Nov 95 01:33:00 -0500 Organization: CRS Online (Toronto, Ontario) First a correction (sorry Dave!) > # Perhaps David Badley could confirm the following orders: The above should be "David Bagley". I have some further comments on the "Magic Platonic Solids". One can stretch (abuse?) the concept of the slice and anti-slice groups of the cube to include the Megaminx (Magic Dodecahedron). In the case of the Megaminx we can consider one-fifth turns of opposite faces. Unfortunately my experiments with "slice" turns on the Megaminx has not generated any spot patterns as yet. Ben Halpern was not the only one to make a prototype of a tetrahedron with rotating faces, as Kersten Meier made one as well. Only 3 of the 4 generators of the Halpern-Meier Tetrahedron are necessary to generate the 3,732,480 possible states. If we use only 2 generators we only get 19,440 possible states. It is not possible to swap just 1 pair of corners and 1 pair of edges, as is possible with the standard Rubik's cube. The number of possible states of the Halpern-Meier Tetrahedron break down like this: 6! /2 * 2^5 * 4!/2 * 3^3 = 3,732,480 The number of pairs of exchanges of the 6 edges must be even. The number of pairs of exchanges of the 4 corners must be even. 5 of the 6 edges may have any flip, the last edge is forced. 3 of the 4 corners may have any twist, the last corner is forced. The H-M Tetrahedron is roughly comparable to the 2x2x2 cube and the standard Skewb in terms of the number of combinations. Halpern's Tetrahedron 3.7*10^6 Ben Halpern, Kersten Meier Pocket Cube (2x2x2) 3.6*10^6 Erno Rubik Skewb 3.1*10^6 Tony Durham -> Mark <-