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Date: Mon, 24 Aug 1998 10:05:52 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: minimal maneuvers for E symmetric positions
In-Reply-To: <199808220307.XAA10899@cauchy.math.brown.edu>
To: michael reid
Cc: cube-lovers@ai.mit.edu
Message-Id:
On Fri, 21 Aug 1998 23:07:24 -0400 michael reid
wrote:
> E is the subgroup of cube symmetries consisting of rotations
> (no reflections) that preserve the tetrad of corners UFR,
> UBL, DFL and DBR. of course it preserves the other tetrad as
> well. there are 72 positions that have E symmetry:
>
> each corner must remain in place, but can be twisted.
> corners in the same tetrad must be twisted in the same
> direction; therefore, by conservation of twist, adjacent
> corners are twisted in opposite directions.
>
> the UR edge can go in any location in any orientation.
> this determines the location and orientation of all edges.
>
There are generally several different (equivalent) ways to
characterize a subgroup of the cube symmetries. For example, of
the 48 symmetries, 24 of them are even and 24 of them are odd,
and 24 of them are rotations and 24 of them are reflections.
The E symmetries may be characterized as the intersection of the
even symmetries with the rotational symmetries, and hence
consist of the 12 even rotations.
The 12 even rotations consist of the identity, the three 180
degree rotations around the face axes (c_u2 around the U-D axis,
c_f2 around the F-B axis, and c_r2 around the R-L axis), and the
eight 120 degree rotations around the four major diagonal axes
(c_urf and c_ufr; c_ufl and c_ulf; c_ulb and c_ubl; and c_ubr
and c_urb). It is the eight major axis rotations which give E
its tetradic nature.
In addition to the characterizations of the E positions which
Mike gave (the corners must stay home, perhaps twisted, etc.),
we can describe the E positions informally by the appearance of
the faces. Each face must have the same pattern as its opposite
face, and each pattern must have the 180 rotational symmetry of
the square.
The hardest part (to me, at least) in thinking about what a
position x with Symm(x)=E must look like is to subtract out or
ignore those positions which are E-symmetric but which have more
symmetry. Indeed, many of the Symm(x)=E positions look very
much like slightly broken versions of positions with stronger
symmetry. For example, #3 and #6 look like slightly broken
6-spots. #7, #10, and #12 look like slightly broken 6-H's.
#1, #2, and #4 look like slightly broken Pons Asinorums. Etc.
This visual effect is the strongest if your cube adopts the
"opposite faces differ by yellow" convention, so that white is
opposite yellow, green is opposite blue, and red is opposite
orange. Your eye will then tend to identify white with yellow,
green with blue, and red with orange. With these
identifications having taken place, most (if not all) of the
Symm(x)=E positions look exactly like positions with more
symmetry.
----------------------
Jerry Bryan
jbryan@pstcc.cc.tn.us