From @mail.uunet.ca:mark.longridge@canrem.com Fri Feb 10 11:54:22 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA08853; Fri, 10 Feb 95 11:54:22 EST Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <109773-2>; Fri, 10 Feb 1995 11:55:11 -0500 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA26149; Fri, 10 Feb 95 00:10:57 EST Received: by canrem.com (PCB-UUCP 1.1f) id 1CF175; Fri, 10 Feb 95 00:03:07 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: The Pyraminx Lost From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1035.5834.0C1CF175@canrem.com> Date: Thu, 9 Feb 1995 23:55:00 -0500 Organization: CRS Online (Toronto, Ontario) Subgroup Sizes of the Pyraminx Octahedron ------------------------------------------ 8 * 9 = 72 facelets (triangles) The standard Pyraminx Octahedron has 8 faces, 6 vertices, and 12 edges. It's vertices rotate. One may imagine a "Master" Pyraminx Octahedron with edge AND face rotations as well. Christoph Bandelow has a version of the Pyraminx Octahedron (I call it "Octa" for short) which has no tips. Size of Groups without rotating vertex tips: Name Subgroup # of Elements ---- -------- ------------- OCT1 4 OCT2 16 OCT3 116,121,600 OCT4 613,312,204,800 OCT5 502,269,581,721,600 OCT6 2,009,078,326,886,400 Size of Groups with rotating vertex tips: Name Subgroup # of Elements ---- -------- ------------- OCT1 16 OCT2 256 OCT3 7,431,782,400 OCT4 157,007,924,428,800 OCT5 514,324,051,682,918,400 OCT6 8,229,184,826,926,694,400 Approximately 8.2 * 10^18 ..so still less than the 3x3x3 cube The number of elements increases by a factor of 4^N for each successive group if we include the trivial vertex rotations. A Skewb Summary --------------- Without repeating Martin's results on the skewb, (which I concur with) here is a quick summary on Skewb facts: It is impossible for any face piece to turn in place 90 degrees. It is impossible to flip a single face piece 180 degrees. It is impossible to transpose 2 face pieces. The Skewb has no non-trivial centre. The SuperSkewb has non-trivial centre with all 6 face pieces rotated 180 degrees. The Mystery of the Five Pyraminxi --------------------------------- Or perhaps that should be Pyraminxes... but I can not resist comparing the Five Pyraminxes to the Five Wizards of J.R.R Tolkien, due to their mysterious nature. We are probably all familar with the Popular Pyraminx created by Uwe Meffert. What really confounds me is that Dr. Ronald Turner- Smith kepts referring to the 5 pyraminxes in ad literature and his book "The Amazing Pyraminx". The Master Pyraminx I understand, it has all the basic properties of the standard popular pyraminx plus all 6 of it's edges can rotate 180 degrees (which flips one edge, transposes 2 tips, and swaps 2 pairs of interior edge pieces) giving a total number of permutations of 446,965,972,992,000. Then there is the mysterious "Senior Pyraminx" (this is like Tolkien's Blue Wizards no one knows about). I can only speculate on the properties of the Senior Pyraminx having never read a description, and never seen the physical puzzle itself. The only fact on the Senior Pyraminx I am sure about is that it has less permutations than the Master Pyraminx. My theory is that the Senior Pyraminx has all the properties of the standard pyraminx plus it can rotate SOME of it's edges but not all 6 like the Master Pyraminx (perhaps one or two?). Perhaps Mr. Singmaster, who has seen magic solid variants from all over the world, can shed some light on the matter! -> Mark <- Email: mark.longridge@canrem.com