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Date: Tue, 5 Aug 1997 19:11:12 -0400
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From: Dan Hoey
To: Cube-lovers@ai.mit.edu
Subject: Glyph patterns
Years ago I thought for a while on a taxonomy of some of the pretty
patterns. Mark's bringing some of them up on the Megaminx has
reminded me of them.
My favorite class of pretty patterns is the "glyph" patterns. These
are the patterns on which each face of the cube has facets of at most
two colors. They include most of the pretty patterns we've discussed
on the list. The "glyphs" here are the partition of the nine facets
into two colors, where we aren't concerned with which colors but with
the partition. I call the part of the glyph that includes the face
center the "figure" and its complement the "ground".
There are only 51 glyphs up to the symmetry of the square, or 70 if we
distinguish chiral pairs. Some of the common ones we have discussed
are blank, X, plus, dot, bar, T, slash, and H. I recall seeing a
cubing book that assigns 26 of the glyphs to letters of the alphabet,
where you try to place all the letters of your favorite six-letter
word on the cube, or something like that.
Classification and analysis of glyph patterns is often simplified by
separating out the corner-glyph from the edge-glyph. There are only 6
each of these subglyphs (up to symmetry), mostly determined by how
many "figure" facets of each type there are.
Name 0 1 2 D 3 4
+-----+-----+-----+-----+-----+-----+
|. .|X .|X X|X .|X X|X X|
Corner | . | . | . | . | . | . |
|. .|. .|. .|. X|. X|X X|
+-----+-----+-----+-----+-----+-----+
| . | X | X | X | X | X |
Edge |. . .|. . .|X . .|. . .|X . X|X . X|
| . | . | . | X | . | X |
+-----+-----+-----+-----+-----+-----+
So a type-2D glyph would have the corner-glyph 2 and the edge-glyph D.
There are two type-2D glyphs, called T and U.
An important subclass of the glyph patterns are the "isoglyphs", which
have the same glyph on all six faces. We've talked about the 6-T,
6-plus, 6-X, 6-H, and 6-spot patterns. Recall that you can twist just
two opposite corners of the cube--I think Hofstadter called this a
boson or something. I was amused to find that there is just one other
6-corner isoglyph of the cube.
Another subclass are the "continuous" glyph patterns, in which the
glyphs on neighboring faces match along the edge. That is to say, a
facet of an edge cubie and an adjacent facet of a corner cubie have
the same color if and only if the other facet of the edge cubie and
the adjacent facet of the corner cubie have the same color. This
matching condition gives the 6-plus patterns much of their charm.
When every cubie of a continuous glyph pattern has either all "figure"
facets or all "ground" facets, we call the pattern a "reassembled"
glyph pattern. In this case, we can envision the cube having been cut
apart into figure and ground cubies and put back together in a
different orientation. Note that the reorientation may include a
reflection, as we see in the Pons Asinorum.
Some of the prettiest reassembled glyph patterns have corner type 4 on
all faces--I call them "path patterns", because you can consider them
a road map going around the cube. In 1981 Dave Ackley found one he
called the "four-way street", which is the unique continuous type-41
isoglyph. If you can find it, you know what I'm talking about.
I've been considering writing a program (or perhaps sparking someone
else's interest in writing a program) to count and classify all the
glyph patterns, possibly by using corner-edge reduction. It might be
interesting to see if there is a set of nine cubes that exhibits all
51 glyphs, or if not what the smallest panglyphic set is.
Dan Hoey
Hoey@AIC.NRL.Navy.Mil