============================================================================== PART A relators + decorations, Tables 25.1-25.17 ============================================================================== Table 25.1 *632 18 rows Table 25.2 632 8 rows Table 25.3 *442 34 rows Table 25.4 4*2 18 rows Table 25.5 442 13 rows Table 25.6 *333 7 rows Table 25.7 3*3 8 rows Table 25.8 333 4 rows Table 25.9 *2222 37 rows Table 25.10 2*22 29 rows Table 25.11 22* 27 rows Table 25.12 22x 10 rows Table 25.13 2222 10 rows Table 25.14 ** 25 rows Table 25.15 *x 12 rows Table 25.16 xx 8 rows Table 25.17 o 5 rows 273 rows checked, 0 relator failures. 131 rows had punctuation/subscripts recomputed, 0 mismatches. ============================================================================== PART B 4x4 affine matrices for three worked examples ============================================================================== (1) *632 in the hexagonal basis a1,a2 (120 degrees apart). P = mirror at 0 deg through the origin, coupled 0+ Q = mirror at 30 deg through the origin, coupled 1/3- R = mirror at 90 deg through (1/2,0), coupled 0- P^2 -> z -> z + 0 OK (PQ)^6 -> z -> z + 0 OK Q^2 -> z -> z + 0 OK (QR)^3 -> z -> z + 1 OK R^2 -> z -> z + 0 OK (RP)^2 -> z -> z + 0 OK PQ 6-fold rotation about (0,0) with z-reversal; z -> 1/3-z ==> rotoreflection QR 3-fold rotation about (2/3,1/3); z -> 1/3+z ==> screw axis 3_1 RP 2-fold rotation about (1/2,0) with z-reversal; z -> 0-z ==> 2-fold axis perpendicular to z (1b) same three mirrors, coupling 0+ 0+ 0+ -> (*.6.3.2), IT 183 = P6mm P^2 -> z -> z + 0 OK (PQ)^6 -> z -> z + 0 OK Q^2 -> z -> z + 0 OK (QR)^3 -> z -> z + 0 OK R^2 -> z -> z + 0 OK (RP)^2 -> z -> z + 0 OK PQ 6-fold rotation about (0,0); z -> 0+z ==> pure rotation axis QR 3-fold rotation about (2/3,1/3); z -> 0+z ==> pure rotation axis RP 2-fold rotation about (1/2,0); z -> 0+z ==> pure rotation axis (2) 442 = p4: g = quarter turn about (0,0), d = quarter turn about (1/2,1/2), e = half turn about (0,1/2). Coupling 1/4+ 1/4+ 1/2+. g^4 -> z -> z + 1 OK d^4 -> z -> z + 1 OK e^2 -> z -> z + 1 OK g d e -> z -> z + 1 OK g 4-fold rotation about (0,0); z -> 1/4+z ==> screw axis 4_1 d 4-fold rotation about (1/2,1/2); z -> 1/4+z ==> screw axis 4_1 e 2-fold rotation about (0,1/2); z -> 1/2+z ==> screw axis 2_1 (3) 22x = pgg: a = half turn about (0,0), b = half turn about (1/2,0), Z = glide (x,y) -> (x+1/2, 1/2-y). Relations 1 = a^2 = b^2 = a b Z^2. (2_0 2_0 x_0) 0+ 0+ 0+ IT 32 Pba2 a^2 -> z+0; b^2 -> z+0; a b Z^2 -> z+0 (2_0 2_0 x_1) 0+ 0+ 1/2+ IT 34 Pnn2 a^2 -> z+0; b^2 -> z+0; a b Z^2 -> z+1 (2_0 2_1 x) 0+ 1/2+ 1/4+ IT 43 Fdd2 a^2 -> z+0; b^2 -> z+1; a b Z^2 -> z+1 (2_1 2_1 x) 1/2+ 1/2+ 0+ IT 33 Pna2_1 a^2 -> z+1; b^2 -> z+1; a b Z^2 -> z+1 the vertical components of the pgg generators, coupling 0+ 1/2+ 1/4+ (Fdd2): a 2-fold rotation about (0,0); z -> 0+z ==> pure rotation axis b 2-fold rotation about (1/2,0); z -> 1/2+z ==> screw axis 2_1 Z horizontal reflection/glide; z -> 1/4+z Z^2 horizontal translation (1,0); z -> 1/2+z ============================================================================== PART C the polar census ============================================================================== 67 names have round brackets and only '+' couplings. 3-D point group (orbifold -> Hermann-Mauguin) and number of names: *22 = mm2 22 *44 = 4mm 12 * = m 7 *33 = 3m 6 44 = 4 5 *66 = 6mm 4 66 = 6 4 33 = 3 3 22 = 2 3 1 = 1 1 distinct IT numbers: 64 1, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 75, 76, 77, 79, 80, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 143, 144, 146, 156, 157, 158, 159, 160, 161, 168, 169, 171, 173, 183, 184, 185, 186 IT numbers with more than one name (aliases: >1 invariant direction): IT 7: (*:*:) over **, (x x_0) over xx IT 8: (*.*:) over **, (*.x) over *x IT 9: (*:x) over *x, (x x_1) over xx the ten polar crystal classes 1, 2, m, mm2, 3, 3m, 4, 4mm, 6, 6mm contain 68 space groups. polar IT numbers with no all-plus name : [78, 145, 170, 172] all-plus names outside the polar classes: [] 67 names -> 64 groups; 68 - 64 = 4 missing, all of them enantiomorphic partners reported under the lower IT number.