From cube-lovers-errors@curry.epilogue.com Tue Jun 11 13:24:32 1996
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Date: Tue, 11 Jun 1996 16:56:52 BST
From: David Singmaster Computing & Maths South Bank Univ
To: cube-lovers@ai.mit.edu
Message-Id: <009A3B3F.74098380.1922@vax.sbu.ac.uk>
Subject: Orbits, re Jerry Bryan's message of 5 Jun
First apologies if this has already been discussed - I'm behind on my
email.
Orbit is used in the context of the action of a group on a set. The
orbit of a point in the set is the set of all points that the original point
can be carried to by the actions of the group.
Jerry mentioned that I said there are 12 orbits of the entire cube.
This is correct in that one is thinking of the group of the cube as acting on
patterns or configurations of the entire cube. Martin Scho"nert is correct
in saying that these are cosets of the cube group in the larger group of
assemblies of the cube, or permutations of the parts.
However, it is not always the case that the set being acted upon can be
given a group structure. E.g. when one considers the action of the cube group
on the individual pieces, then the orbit of a corner piece is the set of
8 corners.
Perhaps the astronomical imagery can help. Think of a planet (or
what have you). There is a group of physical motions of this and the orbit
is the set of positions which these motions can carry the planet to. The
case of the orbit of a corner piece is quite easy to visualise. The more
general contexts of the orbits of achievable positions are less easy to
visualize.
Perhaps another example may help. Consider the 14-15 or 1 puzzle.
For a given position of the blank, only the even permutations are achievable,
so we can speak of two orbits, each of which has 15!/2 positions. If we
permit the blank to move about, we again get half the possibilities, i.e.
16!/2 positions, and two orbits.
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 @vax.sbu.ac.uk