# Tables 25.1-25.17 of Conway/Burgiel/Goodman-Strauss, "The Symmetries of Things" (2008), # ch. 25 "The 184 Composite Space Groups", book pp. 370-374 = PDF pp. 389-393. # Identical to Table 1 of Conway, Delgado Friedrichs, Huson, Thurston, # "On three-dimensional space groups", Beitr. Algebra Geom. 42 (2001) 475-507 (arXiv:math/9911185). # # Transcribed by hand from 400 dpi renderings of the book pages and cross-checked # against `pdftotext -layout` of arXiv:math/9911185. # # Coupling encoding: a string like "1/3-" means the vertical map z -> 1/3 - z ; "0+" means z -> z. # 'bracket' is '(' for a circular fibration (K = {n+}) or '[' for an interval one (K = {n+, n-}). from fractions import Fraction as F # ---------------------------------------------------------------- table definitions # Each table: plane group, generator names in column order, relator words, # and the rows (fibrifold name, couplings, point group, IT number). # A relator word is a list of (generator index, exponent) pairs, read LEFT TO RIGHT # as a composition of maps (the word g1 g2 ... gn acts as g1 o g2 o ... o gn). TABLES = [] def T(num, plane, gens, relators, rows): TABLES.append(dict(num=num, plane=plane, gens=gens, relators=relators, rows=rows)) def w(s, gens): """'PQPQ' -> word; supports 'P-' for inverse, e.g. 'PgPg-' """ out = [] i = 0 while i < len(s): g = s[i] e = 1 if i + 1 < len(s) and s[i + 1] == '-': e = -1 i += 1 out.append((gens.index(g), e)) i += 1 return out # ---- 25.1 *632 : * ^P 6 ^Q 3 ^R 2 ; 1 = P^2 = (PQ)^6 = Q^2 = (QR)^3 = R^2 = (RP)^2 T('25.1', '*632', 'PQR', ['PP', 'PQ' * 6, 'QQ', 'QR' * 3, 'RR', 'RP' * 2], [('[*.6.3.2]', '0+ 0+ 0+ 0-', '*226', 191), ('[*:6.3.2]', '1/2+ 0+ 0+ 0-', '*226', 194), ('[*.6:3:2]', '0+ 1/2+ 1/2+ 0-', '*226', 193), ('[*:6:3:2]', '1/2+ 1/2+ 1/2+ 0-', '*226', 192), ('(*.6.3.2)', '0+ 0+ 0+', '*66', 183), ('(*:6.3.2)', '1/2+ 0+ 0+', '*66', 186), ('(*.6:3:2)', '0+ 1/2+ 1/2+', '*66', 185), ('(*:6:3:2)', '1/2+ 1/2+ 1/2+', '*66', 184), ('(*6.3.2)', '0- 0+ 0+', '2*3', 164), ('(*6:3:2)', '0- 1/2+ 1/2+', '2*3', 165), ('(*.6 3_0 2)', '0+ 0- 0-', '2*3', 162), ('(*.6 3_1 2)', '0+ 1/3- 0-', '2*3', 166), ('(*:6 3_0 2)', '1/2+ 0- 0-', '2*3', 163), ('(*:6 3_1 2)', '1/2+ 1/3- 0-', '2*3', 167), ('(*6_0 3_0 2_0)', '0- 0- 0-', '226', 177), ('(*6_1 3_1 2_1)', '1/2- 1/3- 0-', '226', 178), ('(*6_2 3_2 2_0)', '0- 2/3- 0-', '226', 180), ('(*6_3 3_0 2_1)', '1/2- 0- 0-', '226', 182)]) # ---- 25.2 632 : ^g 6 ^d 3 ^e 2 ; 1 = g^6 = d^3 = e^2 = gde T('25.2', '632', 'gde', ['g' * 6, 'd' * 3, 'e' * 2, 'gde'], [('[6_0 3_0 2_0]', '0+ 0+ 0+ 0-', '6*', 175), ('[6_3 3_0 2_1]', '1/2+ 0+ 1/2+ 0-', '6*', 176), ('(6_0 3_0 2_0)', '0+ 0+ 0+', '66', 168), ('(6_1 3_1 2_1)', '1/6+ 1/3+ 1/2+', '66', 169), ('(6_2 3_2 2_0)', '1/3+ 2/3+ 0+', '66', 171), ('(6_3 3_0 2_1)', '1/2+ 0+ 1/2+', '66', 173), ('(6 3_0 2)', '0- 0+ 0-', '3x', 147), ('(6 3_1 2)', '1/3- 1/3+ 0-', '3x', 148)]) # ---- 25.3 *442 : 1 = P^2 = (PQ)^4 = Q^2 = (QR)^4 = R^2 = (RP)^2 T('25.3', '*442', 'PQR', ['PP', 'PQ' * 4, 'QQ', 'QR' * 4, 'RR', 'RP' * 2], [('[*.4.4.2]', '0+ 0+ 0+ 0-', '*224', 123), ('[*.4.4:2]', '0+ 0+ 1/2+ 0-', '*224', 139), ('[*.4:4.2]', '0+ 1/2+ 0+ 0-', '*224', 131), ('[*.4:4:2]', '0+ 1/2+ 1/2+ 0-', '*224', 140), ('[*:4.4:2]', '1/2+ 0+ 1/2+ 0-', '*224', 132), ('[*:4:4:2]', '1/2+ 1/2+ 1/2+ 0-', '*224', 124), ('(*.4.4.2)', '0+ 0+ 0+', '*44', 99), ('(*.4.4:2)', '0+ 0+ 1/2+', '*44', 107), ('(*.4:4.2)', '0+ 1/2+ 0+', '*44', 105), ('(*.4:4:2)', '0+ 1/2+ 1/2+', '*44', 108), ('(*:4.4:2)', '1/2+ 0+ 1/2+', '*44', 101), ('(*:4:4:2)', '1/2+ 1/2+ 1/2+', '*44', 103), ('(*4.4.2)', '0- 0+ 0+', '*224', 129), ('(*4.4:2)', '0- 0+ 1/2+', '*224', 137), ('(*4:4.2)', '0- 1/2+ 0+', '*224', 138), ('(*4:4:2)', '0- 1/2+ 1/2+', '*224', 130), ('(*.4 4.2)', '0+ 0- 0+', '2*2', 115), ('(*.4 4:2)', '0+ 0- 1/2+', '2*2', 121), ('(*:4 4:2)', '1/2+ 0- 1/2+', '2*2', 116), ('(*4_0 4.2)', '0- 0- 0+', '*224', 125), ('(*4_1 4.2)', '1/4- 0- 0+', '*224', 141), ('(*4_2 4.2)', '1/2- 0- 0+', '*224', 134), ('(*4_0 4:2)', '0- 0- 1/2+', '*224', 126), ('(*4_1 4:2)', '1/4- 0- 1/2+', '*224', 142), ('(*4_2 4:2)', '1/2- 0- 1/2+', '*224', 133), ('(*4.4 2_0)', '0- 0+ 0-', '2*2', 111), ('(*4.4 2_1)', '1/2- 0+ 0-', '2*2', 119), ('(*4:4 2_0)', '0- 1/2+ 0-', '2*2', 112), ('(*4:4 2_1)', '1/2- 1/2+ 0-', '2*2', 120), ('(*4_0 4_0 2_0)', '0- 0- 0-', '224', 89), ('(*4_1 4_1 2_1)', '1/2- 1/4- 0-', '224', 91), ('(*4_2 4_2 2_0)', '0- 1/2- 0-', '224', 93), ('(*4_2 4_0 2_1)', '1/2- 0- 0-', '224', 97), ('(*4_3 4_1 2_0)', '0- 1/4- 0-', '224', 98)]) # ---- 25.4 4*2 : ^g 4 * ^P 2 ; 1 = g^4 = P^2 = [P,g]^2 T('25.4', '4*2', 'gP', ['g' * 4, 'PP', 'PgP-g-' * 2], [('[4_0*.2]', '0+ 0+ 0-', '*224', 127), ('[4_2*.2]', '1/2+ 0+ 0-', '*224', 136), ('[4_0*:2]', '0+ 1/2+ 0-', '*224', 128), ('[4_2*:2]', '1/2+ 1/2+ 0-', '*224', 135), ('(4_0*.2)', '0+ 0+', '*44', 100), ('(4_1*.2)', '1/4+ 0+', '*44', 109), ('(4_2*.2)', '1/2+ 0+', '*44', 102), ('(4_0*:2)', '0+ 1/2+', '*44', 104), ('(4_1*:2)', '1/4+ 1/2+', '*44', 110), ('(4_2*:2)', '1/2+ 1/2+', '*44', 106), ('(4~*.2)', '0- 0+', '2*2', 113), ('(4~*:2)', '0- 1/2+', '2*2', 114), ('(4_0*2_0)', '0+ 0-', '224', 90), ('(4_1*2_1)', '1/4+ 0-', '224', 92), ('(4_2*2_0)', '1/2+ 0-', '224', 94), ('(4~*_0 2_0)', '0- 0-', '2*2', 117), ('(4~*_1 2_0)', '0- 1/2-', '2*2', 118), ('(4~*2_1)', '0- 1/4-', '2*2', 122)]) # ---- 25.5 442 : 1 = g^4 = d^4 = e^2 = gde T('25.5', '442', 'gde', ['g' * 4, 'd' * 4, 'ee', 'gde'], [('[4_0 4_0 2_0]', '0+ 0+ 0+ 0-', '4*', 83), ('[4_2 4_2 2_0]', '1/2+ 1/2+ 0+ 0-', '4*', 84), ('[4_2 4_0 2_1]', '1/2+ 0+ 1/2+ 0-', '4*', 87), ('(4_0 4_0 2_0)', '0+ 0+ 0+', '44', 75), ('(4_1 4_1 2_1)', '1/4+ 1/4+ 1/2+', '44', 76), ('(4_2 4_2 2_0)', '1/2+ 1/2+ 0+', '44', 77), ('(4_2 4_0 2_1)', '1/2+ 0+ 1/2+', '44', 79), ('(4_3 4_1 2_0)', '3/4+ 1/4+ 0+', '44', 80), ('(4 4_0 2)', '0+ 0- 0-', '4*', 85), ('(4 4_1 2)', '1/4+ 0- 1/4-', '4*', 88), ('(4 4_2 2)', '1/2+ 0- 1/2-', '4*', 86), ('(4 4 2_0)', '0- 0- 0+', '2x', 81), ('(4 4 2_1)', '1/2- 0- 1/2+', '2x', 82)]) # ---- 25.6 *333 : 1 = P^2 = (PQ)^3 = Q^2 = (QR)^3 = R^2 = (RP)^3 T('25.6', '*333', 'PQR', ['PP', 'PQ' * 3, 'QQ', 'QR' * 3, 'RR', 'RP' * 3], [('[*.3.3.3]', '0+ 0+ 0+ 0-', '*223', 187), ('[*:3:3:3]', '1/2+ 1/2+ 1/2+ 0-', '*223', 188), ('(*.3.3.3)', '0+ 0+ 0+', '*33', 156), ('(*:3:3:3)', '1/2+ 1/2+ 1/2+', '*33', 158), ('(*3_0 3_0 3_0)', '0- 0- 0-', '223', 149), ('(*3_1 3_1 3_1)', '2/3- 1/3- 0-', '223', 151), ('(*3_0 3_1 3_2)', '1/3- 1/3- 0-', '223', 155)]) # ---- 25.7 3*3 : ^g 3 * ^P 3 ; 1 = g^3 = P^2 = [P,g]^3 T('25.7', '3*3', 'gP', ['ggg', 'PP', 'PgP-g-' * 3], [('[3_0*.3]', '0+ 0+ 0-', '*223', 189), ('[3_0*:3]', '0+ 1/2+ 0-', '*223', 190), ('(3_0*.3)', '0+ 0+', '*33', 157), ('(3_1*.3)', '1/3+ 0+', '*33', 160), ('(3_0*:3)', '0+ 1/2+', '*33', 159), ('(3_1*:3)', '1/3+ 1/2+', '*33', 161), ('(3_0*3_0)', '0+ 0-', '223', 150), ('(3_1*3_1)', '1/3+ 0-', '223', 152)]) # ---- 25.8 333 : 1 = g^3 = d^3 = e^3 = gde T('25.8', '333', 'gde', ['ggg', 'ddd', 'eee', 'gde'], [('[3_0 3_0 3_0]', '0+ 0+ 0+ 0-', '3*', 174), ('(3_0 3_0 3_0)', '0+ 0+ 0+', '33', 143), ('(3_1 3_1 3_1)', '1/3+ 1/3+ 1/3+', '33', 144), ('(3_0 3_1 3_2)', '0+ 1/3+ 2/3+', '33', 146)]) # ---- 25.9 *2222 : 1 = P^2 = (PQ)^2 = Q^2 = (QR)^2 = R^2 = (RS)^2 = S^2 = (SP)^2 T('25.9', '*2222', 'PQRS', ['PP', 'PQPQ', 'QQ', 'QRQR', 'RR', 'RSRS', 'SS', 'SPSP'], [('[*.2.2.2.2]', '0+ 0+ 0+ 0+ 0-', '*222', 47), ('[*.2.2.2:2]', '0+ 0+ 0+ 1/2+ 0-', '*222', 65), ('[*.2.2:2:2]', '0+ 0+ 1/2+ 1/2+ 0-', '*222', 69), ('[*.2:2.2:2]', '0+ 1/2+ 0+ 1/2+ 0-', '*222', 51), ('[*.2:2:2:2]', '0+ 1/2+ 1/2+ 1/2+ 0-', '*222', 67), ('[*:2:2:2:2]', '1/2+ 1/2+ 1/2+ 1/2+ 0-', '*222', 49), ('(*.2.2.2.2)', '0+ 0+ 0+ 0+', '*22', 25), ('(*.2.2.2:2)', '0+ 0+ 0+ 1/2+', '*22', 38), ('(*.2.2:2:2)', '0+ 0+ 1/2+ 1/2+', '*22', 42), ('(*.2:2.2:2)', '0+ 1/2+ 0+ 1/2+', '*22', 26), ('(*.2:2:2:2)', '0+ 1/2+ 1/2+ 1/2+', '*22', 39), ('(*:2:2:2:2)', '1/2+ 1/2+ 1/2+ 1/2+', '*22', 27), ('(*2.2.2.2)', '0+ 0+ 0+ 0-', '*222', 51), ('(*2.2.2:2)', '0+ 0+ 1/2+ 0-', '*222', 63), ('(*2.2:2.2)', '0+ 1/2+ 0+ 0-', '*222', 55), ('(*2.2:2:2)', '0+ 1/2+ 1/2+ 0-', '*222', 64), ('(*2:2.2:2)', '1/2+ 0+ 1/2+ 0-', '*222', 57), ('(*2:2:2:2)', '1/2+ 1/2+ 1/2+ 0-', '*222', 54), ('(*2_0 2.2.2)', '0- 0- 0+ 0+', '*222', 67), ('(*2_1 2.2.2)', '1/2- 0- 0+ 0+', '*222', 74), ('(*2_0 2.2:2)', '0- 0- 0+ 1/2+', '*222', 72), ('(*2_1 2.2:2)', '1/2- 0- 0+ 1/2+', '*222', 64), ('(*2_0 2:2:2)', '0- 0- 1/2+ 1/2+', '*222', 68), ('(*2_1 2:2:2)', '1/2- 0- 1/2+ 1/2+', '*222', 73), ('(*2.2 2.2)', '0- 0+ 0- 0+', '2*', 10), ('(*2.2 2:2)', '0- 0+ 0- 1/2+', '2*', 12), ('(*2:2 2:2)', '0- 1/2+ 0- 1/2+', '2*', 13), ('(*2_0 2_0 2.2)', '0- 0- 0- 0+', '*222', 49), ('(*2_0 2_1 2.2)', '0- 0- 1/2- 0+', '*222', 66), ('(*2_1 2_1 2.2)', '1/2- 0- 1/2- 0+', '*222', 53), ('(*2_0 2_0 2:2)', '0- 0- 0- 1/2+', '*222', 50), ('(*2_0 2_1 2:2)', '0- 0- 1/2- 1/2+', '*222', 68), ('(*2_1 2_1 2:2)', '1/2- 0- 1/2- 1/2+', '*222', 54), ('(*2_0 2_0 2_0 2_0)', '0- 0- 0- 0-', '222', 16), ('(*2_0 2_0 2_1 2_1)', '0- 0- 0- 1/2-', '222', 21), ('(*2_0 2_1 2_0 2_1)', '0- 0- 1/2- 1/2-', '222', 22), ('(*2_1 2_1 2_1 2_1)', '0- 1/2- 0- 1/2-', '222', 17)]) # ---- 25.10 2*22 : ^g 2 * ^P 2 ^Q 2 ; 1 = g^2 = P^2 = (PQ)^2 = Q^2 = (Q g P g^-1)^2 T('25.10', '2*22', 'gPQ', ['gg', 'PP', 'PQPQ', 'QQ', 'QgPg-' * 2], [('[2_0*.2.2]', '0+ 0+ 0+ 0-', '*222', 65), ('[2_1*.2.2]', '1/2+ 0+ 0+ 0-', '*222', 71), ('[2_0*.2:2]', '0+ 0+ 1/2+ 0-', '*222', 74), ('[2_1*.2:2]', '1/2+ 0+ 1/2+ 0-', '*222', 63), ('[2_0*:2:2]', '0+ 1/2+ 1/2+ 0-', '*222', 66), ('[2_1*:2:2]', '1/2+ 1/2+ 1/2+ 0-', '*222', 72), ('(2_0*.2.2)', '0+ 0+ 0+', '*22', 35), ('(2_1*.2.2)', '1/2+ 0+ 0+', '*22', 44), ('(2_0*.2:2)', '0+ 0+ 1/2+', '*22', 46), ('(2_1*.2:2)', '1/2+ 0+ 1/2+', '*22', 36), ('(2_0*:2:2)', '0+ 1/2+ 1/2+', '*22', 37), ('(2_1*:2:2)', '1/2+ 1/2+ 1/2+', '*22', 45), ('(2_0*2.2)', '0+ 0+ 0-', '*222', 53), ('(2_1*2.2)', '1/2+ 0+ 0-', '*222', 58), ('(2_0*2:2)', '0+ 1/2+ 0-', '*222', 52), ('(2_1*2:2)', '1/2+ 1/2+ 0-', '*222', 60), ('(2~*.2.2)', '0- 0+ 0+', '*222', 59), ('(2~*.2:2)', '0- 0+ 1/2+', '*222', 62), ('(2~*:2:2)', '0- 1/2+ 1/2+', '*222', 56), ('(2~*2.2)', '0- 0- 0+', '2*', 12), ('(2~*2:2)', '0- 0- 1/2+', '2*', 15), ('(2_0*2_0 2_0)', '0+ 0- 0-', '222', 21), ('(2_1*2_0 2_0)', '1/2+ 0- 0-', '222', 23), ('(2_0*2_1 2_1)', '0+ 1/2- 0-', '222', 24), ('(2_1*2_1 2_1)', '1/2+ 1/2- 0-', '222', 20), ('(2~*_0 2_0 2_0)', '0- 0- 0-', '*222', 50), ('(2~*_1 2_0 2_0)', '0- 1/2- 1/2-', '*222', 48), ('(2~*2_0 2_1)', '0- 1/4- 1/4-', '*222', 70), ('(2~*2_1 2_1)', '0- 0- 1/2-', '*222', 52)]) # ---- 25.11 22* : ^g 2 ^d 2 * ^P ; 1 = g^2 = d^2 = P^2 = [P, g d] T('25.11', '22*', 'gdP', ['gg', 'dd', 'PP', 'PgdP-d-g-'], [('[2_0 2_0*.]', '0+ 0+ 0+ 0-', '*222', 51), ('[2_0 2_1*.]', '0+ 1/2+ 0+ 0-', '*222', 63), ('[2_1 2_1*.]', '1/2+ 1/2+ 0+ 0-', '*222', 59), ('[2_0 2_0*:]', '0+ 0+ 1/2+ 0-', '*222', 53), ('[2_0 2_1*:]', '0+ 1/2+ 1/2+ 0-', '*222', 64), ('[2_1 2_1*:]', '1/2+ 1/2+ 1/2+ 0-', '*222', 57), ('(2_0 2_0*.)', '0+ 0+ 0+', '*22', 28), ('(2_0 2_1*.)', '0+ 1/2+ 0+', '*22', 40), ('(2_1 2_1*.)', '1/2+ 1/2+ 0+', '*22', 31), ('(2_0 2_0*:)', '0+ 0+ 1/2+', '*22', 30), ('(2_0 2_1*:)', '0+ 1/2+ 1/2+', '*22', 41), ('(2_1 2_1*:)', '1/2+ 1/2+ 1/2+', '*22', 29), ('(2_0 2_0*)', '0+ 0+ 0-', '222', 17), ('(2_0 2_1*)', '0+ 1/2+ 0-', '222', 20), ('(2_1 2_1*)', '1/2+ 1/2+ 0-', '222', 18), ('(2_0 2~*.)', '0+ 0- 0+', '*222', 57), ('(2_1 2~*.)', '1/2+ 0- 0+', '*222', 62), ('(2_0 2~*:)', '0+ 0- 1/2+', '*222', 60), ('(2_1 2~*:)', '1/2+ 0- 1/2+', '*222', 61), ('(2_0 2~*_0)', '0+ 0- 0-', '*222', 54), ('(2_0 2~*_1)', '0+ 1/2- 0-', '*222', 52), ('(2_1 2~*_0)', '1/2+ 1/2- 0-', '*222', 56), ('(2_1 2~*_1)', '1/2+ 0- 0-', '*222', 60), ('(2 2*.)', '0- 0- 0+', '2*', 11), ('(2 2*:)', '0- 0- 1/2+', '2*', 14), ('(2 2*_0)', '0- 0- 0-', '2*', 13), ('(2 2*_1)', '0- 1/2- 0-', '2*', 15)]) # ---- 25.12 22x : ^g 2 ^d 2 x ^Z ; 1 = g^2 = d^2 = g d Z^2 T('25.12', '22x', 'gdZ', ['gg', 'dd', 'gdZZ'], [('[2_0 2_0 x_0]', '0+ 0+ 0+ 0-', '*222', 55), ('[2_0 2_0 x_1]', '0+ 0+ 1/2+ 0-', '*222', 58), ('[2_1 2_1 x]', '1/2+ 1/2+ 0+ 0-', '*222', 62), ('(2_0 2_0 x_0)', '0+ 0+ 0+', '*22', 32), ('(2_0 2_0 x_1)', '0+ 0+ 1/2+', '*22', 34), ('(2_0 2_1 x)', '0+ 1/2+ 1/4+', '*22', 43), ('(2_1 2_1 x)', '1/2+ 1/2+ 0+', '*22', 33), ('(2_0 2_0 x~)', '0+ 0+ 0-', '222', 18), ('(2_1 2_1 x~)', '1/2+ 1/2+ 0-', '222', 19), ('(2 2 x)', '0- 0- 0+', '2*', 14)]) # ---- 25.13 2222 : 1 = g^2 = d^2 = e^2 = z^2 = g d e z T('25.13', '2222', 'gdez', ['gg', 'dd', 'ee', 'zz', 'gdez'], [('[2_0 2_0 2_0 2_0]', '0+ 0+ 0+ 0+ 0-', '2*', 10), ('[2_0 2_0 2_1 2_1]', '0+ 0+ 1/2+ 1/2+ 0-', '2*', 12), ('[2_1 2_1 2_1 2_1]', '1/2+ 1/2+ 1/2+ 1/2+ 0-', '2*', 11), ('(2_0 2_0 2_0 2_0)', '0+ 0+ 0+ 0+', '22', 3), ('(2_0 2_0 2_1 2_1)', '0+ 0+ 1/2+ 1/2+', '22', 5), ('(2_1 2_1 2_1 2_1)', '1/2+ 1/2+ 1/2+ 1/2+', '22', 4), ('(2_0 2_0 2 2)', '0+ 0+ 0- 0-', '2*', 13), ('(2_0 2_1 2 2)', '0+ 1/2+ 0- 1/2-', '2*', 15), ('(2_1 2_1 2 2)', '1/2+ 1/2+ 0- 0-', '2*', 14), ('(2 2 2 2)', '0- 0- 0- 0-', 'x', 2)]) # ---- 25.14 ** : ^L * ^P * ^Q ; 1 = P^2 = Q^2 = [L,P] = [L,Q] T('25.14', '**', 'LPQ', ['PP', 'QQ', 'LPL-P-', 'LQL-Q-'], [('[*_0.*_0.]', '0+ 0+ 0+ 0-', '*22', 25), ('[*_1.*_1.]', '1/2+ 0+ 0+ 0-', '*22', 38), ('[*_0.*_0:]', '0+ 0+ 1/2+ 0-', '*22', 35), ('[*_1.*_1:]', '1/2+ 0+ 1/2+ 0-', '*22', 42), ('[*_0:*_0:]', '0+ 1/2+ 1/2+ 0-', '*22', 28), ('[*_1:*_1:]', '1/2+ 1/2+ 1/2+ 0-', '*22', 39), ('(*.*.)', '0+ 0+ 0+', '*', 6), ('(*.*:)', '0+ 0+ 1/2+', '*', 8), ('(*:*:)', '0+ 1/2+ 1/2+', '*', 7), ('(*~.*~.)', '0- 0+ 0+', '*22', 26), ('(*~.*~:)', '0- 0+ 1/2+', '*22', 36), ('(*~:*~:)', '0- 1/2+ 1/2+', '*22', 29), ('(*.*_0)', '0+ 0+ 0-', '*22', 28), ('(*.*_1)', '1/2+ 0+ 0-', '*22', 40), ('(*:*_0)', '0+ 1/2+ 0-', '*22', 32), ('(*:*_1)', '1/2+ 1/2+ 0-', '*22', 41), ('(*~.*~_0)', '0- 0+ 0-', '*22', 39), ('(*~.*~_1)', '1/2- 0+ 0-', '*22', 46), ('(*~:*~_0)', '0- 1/2+ 0-', '*22', 45), ('(*~:*~_1)', '1/2- 1/2+ 0-', '*22', 41), ('(*_0*_0)', '0+ 0- 0-', '22', 3), ('(*_1*_1)', '1/2+ 0- 0-', '22', 5), ('(*~_0*~_0)', '0- 0- 0-', '*22', 27), ('(*~_0*~_1)', '0- 0- 1/2-', '*22', 37), ('(*~_1*~_1)', '0- 1/2- 1/2-', '*22', 30)]) # ---- 25.15 *x : * ^P x ^Z ; 1 = P^2 = [P, Z^2] T('25.15', '*x', 'PZ', ['PP', 'PZZP-Z-Z-'], [('[*.x_0]', '0+ 0+ 0-', '*22', 38), ('[*.x_1]', '0+ 1/2+ 0-', '*22', 44), ('[*:x_0]', '1/2+ 0+ 0-', '*22', 46), ('[*:x_1]', '1/2+ 1/2+ 0-', '*22', 40), ('(*.x)', '0+ 0+', '*', 8), ('(*:x)', '1/2+ 0+', '*', 9), ('(*.x~)', '0+ 0-', '*22', 31), ('(*:x~)', '1/2+ 0-', '*22', 33), ('(*_0 x_0)', '0- 0+', '*22', 30), ('(*_0 x_1)', '0- 1/2+', '*22', 34), ('(*_1 x)', '0- 1/4+', '*22', 43), ('(*x~)', '0- 0-', '22', 5)]) # ---- 25.16 xx : x ^Y x ^Z ; 1 = Y^2 Z^2 T('25.16', 'xx', 'YZ', ['YYZZ'], [('[x_0 x_0]', '0+ 0+ 0-', '*22', 26), ('[x_0 x_1]', '0+ 1/2+ 0-', '*22', 36), ('[x_1 x_1]', '1/2+ 1/2+ 0-', '*22', 31), ('(x x_0)', '0+ 0+', '*', 7), ('(x x_1)', '0+ 1/2+', '*', 9), ('(x~ x_0)', '0- 0+', '*22', 29), ('(x~ x_1)', '0- 1/2+', '*22', 33), ('(x~ x~)', '0- 0-', '22', 4)]) # ---- 25.17 o : o ^{X,Y} ; 1 = [X,Y] T('25.17', 'o', 'XY', ['XYX-Y-'], [('[o_0]', '0+ 0+ 0-', '*', 6), ('[o_1]', '1/2+ 1/2+ 0-', '*', 8), ('(o)', '0+ 0+', '1', 1), ('(o~_0)', '0- 0-', '*', 7), ('(o~_1)', '0- 1/2-', '*', 9)])