From cube-lovers-errors@mc.lcs.mit.edu Sat Aug 9 15:13:50 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id PAA08470; Sat, 9 Aug 1997 15:13:50 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From Hoey@AIC.NRL.Navy.Mil Sat Aug 9 15:09:29 1997 Date: Sat, 9 Aug 1997 15:09:01 -0400 Message-Id: <199708091909.PAA07964@sun30.aic.nrl.navy.mil> 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