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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