From cube-lovers-errors@mc.lcs.mit.edu Thu Oct 8 19:04:05 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.8/mc) with SMTP id TAA24808; Thu, 8 Oct 1998 19:04:04 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Date: Thu, 8 Oct 1998 15:11:54 +0100 From: David Singmaster To: cube-lovers@ai.mit.edu Message-Id: <009CD656.72A81016.35@ice.sbu.ac.uk> Subject: Davenport's pattern The pattern given by Jacob Davenport is what I called a cube in a cube in a cube. I discovered this in 1979 or 1980 and was very pleased with it. Indeed, I used the cube in a cube as the logo of the late and much lamented David Singmaster Ltd. in 1980-1982 (approx. dates since I'm not where my records are). The pattern is in my Notes. There are various ways to generate the pattern, but the one that I can remember uses what Roger Penrose called the Y-commutator, which has the form FR'F'R. The reason this is the Y-commutator is that it affect the three edges adjacent to a corner and the corner and its three adjacent corners. I.e. the affected pieces form a Y, while the pieces affected by the ordinary commutator FRF'R' form a Z. Combining three Y-commutators as follows: FR'F'R RU'R'U UF'U'F gives a process that twists the corner and the three adjacent edges as a unit and twists an adjacent corner the opposite direction. NOTE - I'm doing this from memory and I have a suspicion that the middle group may need to be inverted?? By moving the odd corner to the right place adjacent to the opposite corner and applying the inverse of the above, one gets the same sort of pattern at the opposite corner and the odd corner has been restored. Now one 3-cycles the centers, as is easily done by a commutator of slice moves, and one has the cube in a cube. Now one can twist the two opposite corners to get the cube in a cube in a cube, though I find this not as visually dramatic as the cube in a cube. Someone - Mike Reid ? - sent me a minimal method for one of these patterns, but it's not very memorable. DAVID SINGMASTER, Professor of Mathematics and Metagrobologist School of Computing, Information Systems and Mathematics Southbank University, London, SE1 0AA, UK. Tel: 0171-815 7411; fax: 0171-815 7499; email: zingmast or David.Singmaster @sbu.ac.uk