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Date: Sat, 9 Aug 1997 15:09:01 -0400
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From: Dan Hoey
To: kociemba@hrz1.hrz.th-darmstadt.de
Cc: cube-lovers@ai.mit.edu
Subject: Re: Continuous isoglyph patterns
Bravo, Herbert! A very nice list. It's surprising how many of them
are reassembled patterns, too. Only the second and tenth are not
reassembled, and both fail by using a reassembled pattern to
camouflage a small distortion. Pattern #2 is pattern #3 composed with
a two-flip, and pattern #10 is pattern #9 composed with a three-cycle
of edges.
There are four elements of M used to perform the reorientation of the
reassembled patterns. Over half of them use the order-3
major-diagonal rotation, of Plummer's cross: patterns 1, 3, 4, 6, 7,
9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 27, 30, 31, and
32. Several use the order-2 major-diagonal rotation, of Christman's
cross: patterns 5, 8, 14, 20, 26, 28, and 29. The only pattern
reassembled by the major diagonal reflected rotation is the order-6
6-X, pattern 33, And the only pattern reassembled by central
reflection is Pons Asinorum, pattern 34.
I've also classified these patterns by the glyph that appears on the
faces (modulo my clerical errors). Patterns I know traditional names
for are given with an asterisk; I've made up temporary descriptive
names otherwise.
Glyph Type Patterns
X . X 2. Girdle 3-cycle, distorted
X X X 01 3. Girdle 3-cycle
X X X
X . X 8. Christman's girdle
. X X 02 9. Off-girdle 3-cycles
X X X 10. Off-girdle 3-cycles, distorted
11. Girdle 3-cycles
X . X
. X X 03 21. Plummer's C's
X . X
X . X 32. Plummer's X
. X . 04 33. Order-6 X
X . X 34. * Pons Asinorum
. X X
X X X 10 1. * Meson
X X X
. X X
X X . 11 7. Meson & girdle 3-cycle
X X X
. . X 13. Plummer's cluster
. X X 12 14. Christman's cluster
X X X
. X X
X X . 12 16. Meson & girdle 3-cycles
X . X
X . X
X X . 13 24. Plummer's Y's
X . .
. . X 25. Plummer's cluster & girdle 3-cycles
. X . 14 26. Christman's cluster & girdle
X . X
. X .
X X X 30 15. Plummer's rabbits
. X X
. X X
. X X 31 22. Plummer's P's
. X .
X X .
X X . 32 23. * Cube in a cube
. . .
. X . 29. Christman's arrow
X X . 32 30. Plummers's arrow
. . X
X . .
. X X 33 16. Plummer's bend
. . .
X . . 5. Christman's comma
. X . 34 6. Plummer's comma
. . .
. X . 27. * Plummer's Cross
X X X 40 28. * Christman's Cross
. X .
. X .
X X X 41 31. * Four-way street
. . .
. X . 18. Plummer's cube out of cube in a cube
X X . 42 19. * Worm
. . . 20. Christman's cube out of cube in a cube
. X .
. X . 43 12. Plummer's U's
. . .
. . .
. X . 44 4. * Six-spot
. . .
Dan
Hoey@AIC.NRL.Navy.Mil