From cube-lovers-errors@mc.lcs.mit.edu Wed Apr 1 10:55:58 1998 Return-Path: Received: from sun28.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id KAA10617; Wed, 1 Apr 1998 10:55:57 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Mar 29 18:20:22 1998 Date: Sun, 29 Mar 1998 18:20:42 -0400 (EDT) From: Jerry Bryan Subject: All the Partial Isoglyphs To: Cube-Lovers Message-Id: I have been able to calculate all the partial isoglyphs a little more quickly than expected. I can report that there are 10 continuous partial isoglyphs and 130 non-continuous partial isoglyphs, unique up to symmetry. 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 The partial isoglyphs are all included in the chart which follows. If nothing is listed with respect to the manner in which the solid faces are arranged, then there are two solid faces opposite to each other. Otherwise, the arrangement of the solid faces is listed explicitly. This chart follows the same format as the previous one I posted for all the isoglyphs, except that this time I included only a single representative glyph for each partial isoglyph, rather than the complete equivalence class of glyphs. 000 0X0 000 00 D B2 F2 D U' L2 R2 U' (8) continuous 000 0X0 00X 01 (none) 000 0X0 0X0 02 D B2 L2 R2 F2 U' L2 B2 F2 R2 (10) not D' B2 F2 D U L2 R2 U' (8) not F2 D2 B2 D U L2 D' U' (8) not B2 D U' L2 D U' (6) * not D B2 D L2 . B F' D2 R' B2 R2 D' U F L R' (15) not 000 0X0 0XX 03 L2 U2 B2 D' B2 R2 D2 B2 R2 U' B2 L2 U2 R2 U' (15) not 000 0X0 X0X 05 D U F2 D' L2 . B' F U2 L' D U' F2 R2 F' L R' (16) not D B2 L2 R2 F2 U' L2 B2 F2 R2 U2 (11) not L2 F2 U B2 F2 U2 B2 F2 U' F2 R2 (11) not 000 0X0 XXX 07 D' B2 F2 D' U L2 R2 U' (8) continuous 000 0XX 0X0 0A L2 D2 R2 F2 U2 R2 F2 U2 F2 U2 (10) not D B2 D' U' L2 R2 U2 R2 U' (9) not 000 0XX 0XX 0B B2 L2 U2 L2 U' L2 B2 D2 F2 U F2 R2 U2 R2 U' (15) not 000 0XX X00 0C F2 L2 U2 L2 U L2 F2 D2 B2 U' B2 R2 U2 R2 U' (15) not 000 0XX X0X 0D L2 D2 R2 F2 U2 R2 F2 U2 F2 (9) not B2 R2 F2 D U B2 L2 F2 L2 D U (11) not F2 R2 B2 D' U' B2 R2 F2 R2 D' U' (11) not 000 0XX XX0 0E L2 U2 F2 D F2 R2 D2 F2 R2 U F2 L2 U2 R2 U' (15) not R2 U2 F2 D' F2 L2 U2 B2 R2 U' B2 R2 U2 R2 U' (15) not F2 U2 R2 U F2 D2 F2 R2 D' F2 R2 D2 F2 R2 U' (15) not 000 0XX XXX 0F B2 D U' L2 D U (6) * not 000 XXX 000 18 D2 U2 (2) * continuous D' U (2) * continuous L2 F2 L2 R2 F2 R2 (6) * not F2 U2 B2 F2 U2 F2 (6) * not 000 XXX 00X 19 (none) 000 XXX 0X0 1A D F2 R2 B2 F2 R2 D' U R2 U' (10) not D' B2 U2 B2 L2 R2 D2 F2 L2 R2 D' (11) not L2 B2 F2 R2 D' U2 L2 B2 F2 R2 U' (11) not U' L2 R2 U2 L2 R2 U' (7) * not 000 XXX 0XX 1B B2 D2 L2 U' F2 D2 F2 L2 D F2 L2 U2 B2 R2 U' (15) not 000 XXX X0X 1D L2 B2 F2 R2 D U2 L2 B2 F2 R2 U' (11) not F2 U L2 R2 D2 L2 R2 U' F2 (9) not 000 XXX XXX 1F D (1) * continuous D2 (1) * continuous 00X 0X0 X00 24 D' L2 F2 U2 B2 R2 U2 L2 D U' R2 U' (12) not 00X 0X0 X0X 25 (none) 00X 0X0 XX0 26 B2 L2 D U F2 L2 D U F2 R2 (10) not L2 D' U' F2 D U . L R' U2 L R (11) not D' U' F2 D' U . L R' U2 L R (10) not U' B2 L2 D2 R2 F2 U2 F2 R2 U' (10) not 00X 0X0 XXX 27 L2 D2 R2 B2 U R2 B2 D2 L2 F2 U' F2 D2 R2 U' (15) not 00X 0XX XX0 2E (none) 00X 0XX XXX 2F L2 D2 B2 D B2 L2 U2 B2 L2 D' B2 R2 U2 R2 U' (15) not 00X XX0 00X 31 L2 F2 L2 R2 F2 R2 U2 (7) * not 00X XX0 0X0 32 F2 D2 L2 U B2 D2 B2 L2 D' B2 L2 U2 F2 R2 U' (15) not 00X XX0 0XX 33 D F2 R2 B2 F2 R2 D' U R2 U (10) not 00X XX0 X0X 35 F2 L2 D2 R2 D' R2 F2 D2 B2 D' B2 R2 U2 R2 U' (15) not 00X XX0 XX0 36 (none) 00X XX0 XXX 37 L2 U2 F2 D F2 R2 U2 F2 R2 D' F2 L2 D2 R2 U' (15) not B2 U2 F2 L2 D L2 D2 R2 B2 D B2 D2 F2 R2 U' (15) not F2 R2 D2 L2 D L2 F2 U2 F2 D F2 R2 U2 R2 U' (15) not 00X XXX 00X 39 U2 F2 U L2 . B' F U2 R' F2 R2 D U' B L R' (15) not B2 L2 R2 F2 D' L2 B2 F2 R2 U' (10) not B2 U' B2 L2 R2 F2 D' F2 (8) not 00X XXX 0X0 3A B2 U2 R2 U' B2 D2 B2 R2 D B2 R2 D2 B2 R2 U' (15) not 00X XXX 0XX 3B L2 D2 L2 F2 U2 R2 B2 U2 F2 (9) not U' R2 B2 R2 D F2 D' R2 B2 R2 U (11) not B2 R2 F2 D' U' B2 L2 F2 R2 D' U' (11) not 00X XXX X00 3C D' L2 B2 U2 F2 R2 U2 L2 D' U R2 U' (12) not F2 R2 D U L2 B2 R2 B2 D' U' F2 R2 (12) not D' R2 B2 U2 B2 R2 U2 R2 D' U R2 U' (12) not 00X XXX X0X 3D (none) 00X XXX XX0 3E B2 L2 D2 B2 R2 F2 L2 U2 F2 R2 (10) not D' U' L2 D' U . B F' D2 B F (10) not U2 L2 . B' L2 D2 U2 R2 F' R2 (9) not D2 L2 . B' D2 L2 R2 U2 F' R2 (9) not 00X XXX XXX 3F R2 U2 F2 D' F2 L2 D2 B2 R2 D B2 L2 D2 R2 U' (15) not 0X0 XXX 0X0 5A D F2 R2 F2 D' U R2 F2 R2 U' (10) not B2 F2 D2 L2 R2 D B2 F2 U2 L2 R2 U' (12) not 0X0 XXX 0XX 5B (none) 0X0 XXX X0X 5D B2 F2 L2 R2 D B2 F2 L2 R2 (9) not B2 F2 L2 R2 D2 B2 F2 L2 R2 (9) not 0X0 XXX XXX 5F D' L2 B2 F2 R2 U' L2 B2 F2 R2 (10) not L2 B2 D' B2 L2 R2 F2 U' F2 R2 (10) not 0XX XX0 X0X 75 F2 D2 B2 L2 D L2 U2 L2 F2 D F2 U2 F2 R2 U' (15) not 0XX XX0 XXX 77 R2 D2 B2 D' B2 R2 D2 F2 L2 D F2 R2 U2 R2 U' (15) not 0XX XXX XX0 7E D' R2 F2 U2 F2 R2 U2 R2 D U' R2 U' (12) not 0XX XXX XXX 7F (none) X0X 0X0 X0X A5 D2 R2 U2 L2 R2 U2 R2 U2 (8) not (four solid, other two opposite) D2 L2 F2 L2 R2 F2 R2 U2 (8) not X0X 0X0 XXX A7 D' B2 U' L2 . B' F U2 R' D U' B2 L2 B L R' (15) not B2 L2 R2 F2 D' U2 L2 B2 F2 R2 U' (11) not U2 F2 D' U' R2 U2 . R B2 F L' R D L' B2 F2 R B (17) not *1 D2 L2 B2 R2 U' F2 L2 D U' . R B2 U' F' D2 U' R F2 D L' R2 (20) not *2 B2 R2 F2 U' L2 U . R B D2 B' R' D' R' F2 L R2 B2 U' (18) not *2 D F2 L2 F2 D' U' R2 D' R2 . B' D2 B' D' L' U L2 R' U' R' (19) not *2 D' L2 R2 D' U' B2 F2 U' (8) not F2 L2 D2 B2 U2 B2 F2 R2 F2 U2 (10) not F2 D' F2 D B2 U B2 F2 U2 L2 U F2 . R B U' B2 U B' R' (19) not D' F2 R2 B2 F2 R2 D' U R2 U (10) not *1 two solid faces, adjacent *2 one solid face X0X 0XX XXX AF D . F' D2 U2 B R B' D2 U2 F L' D' (12) * continuous *1 R2 U2 . L B L U R' U R' D' F' D' (12) * not *2 L2 U2 R2 D2 R2 U2 (6) * not *4 U F2 D U2 L2 U' F2 . L' U' F D2 U L' F2 U2 B' R (17) not *3 D B2 D' U2 . F D F U' R' U F' U' R' U' B2 (15) not *2 F2 U' L2 U L2 U' B2 . L' U' B U2 B' U L' F2 D' R2 (17) not R2 D' R2 U' R2 U . R U L F2 D2 L' U' B' D B D (17) not *2 L2 U2 R2 D2 R2 . B' L' B' U' F U' F D R D F2 (16) not *2 B2 D2 B2 U2 F2 L2 D' . F' D' L' U L' U R B R (16) not *2 L2 U2 R2 F2 D2 R2 F2 D2 F2 U2 (10) not F2 L2 U2 B2 D R2 . B' L' D' L' D' B' U' B' F U' B U2 (18) not R2 D2 B2 D2 F2 U . F' D' L' D' B D L D F' R2 (16) not *3 U F2 L2 U2 L2 F2 R2 D' U' . B' D' L' B' R' B' R B L (18) not *2 B2 L2 . B' D' L' U B' R2 U' L F' D' F (13) not *2 B2 R2 F2 L2 U2 F2 R2 U2 F2 U2 (10) not *1 three solid faces mutually adjacent *2 one solid face *3 two solid faces adjacent *4 four solid, other faces opposite X0X XXX X0X BD D B2 F2 D' U L2 R2 U' (8) continuous D2 B2 D2 U2 F2 U2 (6) * not *4 R2 . F' U2 L2 D2 B2 L2 U2 R2 F' R2 (11) * not *2 F2 D2 L2 . F D2 U2 B' R2 U2 F2 (10) * not *3 F2 R2 U2 . B' D2 U2 F D2 R2 F2 (10) * not *1 D2 U2 B2 U2 R2 . F' D2 U2 B L2 U2 F2 (12) not *5 L2 D' U' B2 F2 D' U' R2 (8) not L2 D2 R2 . B' U2 F U2 L2 U2 F R2 F' (12) not *5 B2 F2 L2 R2 U' B2 F2 D2 L2 R2 U' (11) not L2 R2 U2 B2 F2 U B2 F2 U2 L2 R2 U' (12) not L2 R2 D2 L2 B2 U B2 F2 D' F2 R2 U2 (12) not D B2 L2 B2 D U' R2 F2 R2 U' (10) not *1 - three solid faces, not mutually adjacent *2 - four solid faces, other two faces adjacent *3 - two solid faces, adjacent *4 - four solid faces, other two faces opposite *5 - one solid face X0X XXX XXX BF D' U R2 F2 D U' . R' D U' B' L2 B D' U R' (15) continuous L2 D U . B D' B' U' L2 D L D L' D2 (13) continuous *1 B2 D U' L2 D' U (6) * not D B2 U2 . L' U2 B2 D2 R' D (9) * not *2 R2 D U' . B D' B' D' U R D R (11) * not *2 F L R' D2 L' R F (7) * not *2 L2 U2 . B U2 L2 D2 F D2 (8) * not *1 L2 . F L R' D2 L' R F L2 (9) * not *2 U2 L2 D2 . B' L2 U2 R2 F' (8) * not *1 L2 D U' . F' L F D' U L' B' L' (11) * not *2 F2 U2 L2 D2 . B' L2 U2 R2 F (9) * not *2 D2 . B' L' R D2 L R' B' D2 (9) * not *3 R2 U2 . B D2 L2 U2 F D2 (8) * not *1 U2 B2 U2 L2 U2 . B D2 R2 U2 F' D2 (11) not *2 D . R B2 F2 L' U' L B2 F2 R' (10) * not *2 D . R' B F' U R' U' B' F R (10) * not *2 D . F' R' B' L' D' L B R F (10) * not *1 B2 D L2 U . R U R' F U2 L D' L B' (13) not *2 R2 D . F D' F' R2 D' B' D B (10) * not *1 D F2 D R2 . F R2 D2 R2 F R2 D F2 D' (13) not *2 D' F2 U2 B2 U2 F2 D' (7) * not F2 R2 U2 . B' U2 R2 U2 B' U2 F2 (10) * not *2 F2 D2 . F D2 R2 D2 F D2 R2 F2 (10) * not *2 B2 R2 U' L2 U R2 B2 R2 U F2 U' R2 (12) not D L2 B2 F2 R2 U' L2 B2 F2 R2 (10) not F' L2 R2 B2 L2 R2 F' (7) * not L2 . B L' B' D2 R' B' R B D2 L' (11) * not *1 D U' . B F' U' B' F R2 D' U F' (11) * not *1 - 2 solid, adjacent *2 - 1 solid *3 - 3 solid, not mutually adjacent XXX XXX XXX FF (none) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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