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Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 29 18:56:43 1998
Date: Sun, 29 Mar 1998 18:57:08 -0400 (EDT)
From: Jerry Bryan
Subject: Re: All the Partial Isoglyphs
In-Reply-To:
To: Cube-Lovers
Message-Id:
On Sun, 29 Mar 1998, Jerry Bryan wrote:
> Here is a breakdown of how the solid faces can be arranged.
>
> 97 - two solid faces, opposite to each other
> 11 - two solid faces, adjacent to each other
> 25 - one solid face
> 1 - three solid faces, mutually adjacent to each other
> 2 - three solid faces, not mutually adjacent to each other
> 3 - four solid faces, other two opposite to each other
> 1 - four solid faces, other two adjacent to each other
> ---
> 140
>
As this table shows, the vast majority of partial isoglyphs involve two
solid faces opposite to each other. The basic reason for this is the
corners. If the corners are not fixed, then the only partial isoglyphs
which are possible have two solid faces opposite to each other.
Conversely, the 43 partial isoglyphs which do not have two solid faces
opposite to each other do fix the corners.
In fact, 67 of the partial isoglyphs derive from just 5 of the glyphs,
namely those which fix the corners. If the corners of the partial
isoglyph are fixed, you can think of the edges as consisting of a set of
strongly constrained edge flips and swaps. (Be careful -- if the corners
are fixed, then *any* resultant position can be thought of as just a bunch
of edge flips and swaps. But for partial isoglyphs, the possible edge
flips and swaps are strongly constrained.)
The glyph which yields the most partial isoglyphs is the one my charts
call BF, whick looks like the following.
X0X
XXX
XXX
With this glyph, each face of a partial isoglyph can have at most one edge
cubie which is swapped or flipped, but on a cube-wide basis there are
quite a few different ways to arrange for this to happen.
Another interesting glyph which fixes the corners is called BD on my
charts, and which appears as follows.
X0X
XXX
X0X
As an isoglyph, this glyph yields five different patterns on the 6-H
theme. As a partial isoglyph, this glyph yields a number of pretty 2-H,
3-H, 4-H, and 5-H patterns. You may also think of the H patterns as
complicated edge swappers/flippers, with exactly zero or two edges
swapped/flipped on each face, and with the coloring requirements for
partial isoglyphs being maintained.
The following two glyphs (A7 and AF in my charts) are in the same spirit
as the H, except that the configuration of the edges on each face which
are swapped/flipped is slightly different than for the H.
X0X X0X
0X0 0XX
XXX XXX
Finally, for completeness in the list of glyphs which fix the corners, the
glyph called A5 on my charts appears as follows.
X0X
0X0
X0X
However, this glyph only yields two partial isoglyphs.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990