Example with a long side ======================== Do an example 1/119(1,16,34) with 119/16 = [8,2,5,2]. Note that 119 = 7*17, so that this is cyclic, and a,b coprime, but 34 = 2*17. Along the bottom, f0 = (-7.7,0) = 1/17 * vec e1 e2 Propellor f0 = -7,7,0 f1 = -52,1,17 (7) f2 = -3*119, 0, 119 h0 = 119, 0, -119/3 h1 = 67,1,-68/3 (2) h2 15,2,-17/3 (5) h3 = 8,9,-17/3 (2) h4 = 1,16.-17/3 (8) h5 = 0,119, -119/3 g0 = 0, -3*119, 119 g1 = 1, -103, 74 (4) g2 = 4, -55, 17 (2) g3 = 7,-7,0 The tags inside each blade are written. They correspond to changes of basis inside one lattice. The transitions between blades (change of bases between the two different lattices L_rj and L'_j) are h0 = -1/3*f2 h1 = h0 + f1 g0 = -3*h5 g1 = h4 + h4 and f0 + g3 = 0, f1-g2 = 8*f0 (-52,1,17) - (4,-55,17) = 8*(-7,7,0) (but why on earth 8?) [ 1, 16, 34 ], [ 8, 9, 34 ], [ 15, 2, 34 ], [ 4, 64, 17 ], [ 11, 57, 17 ], [ 18, 50, 17 ], [ 25, 43, 17 ], [ 32, 36, 17 ], [ 39, 29, 17 ], [ 46, 22, 17 ], [ 53, 15, 17 ], [ 60, 8, 17 ], [ 67, 1, 17 ], [ 7, 112, 0 ], [ 14, 105, 0 ], [ 21, 98, 0 ], [ 28, 91, 0 ], [ 35, 84, 0 ], [ 42, 77, 0 ], [ 49, 70, 0 ], [ 56, 63, 0 ], [ 63, 56, 0 ], [ 70, 49, 0 ], [ 77, 42, 0 ], [ 84, 35, 0 ], [ 91, 28, 0 ], [ 98, 21, 0 ], [ 105, 14, 0 ], [ 112, 7, 0 ] f0 = -7,7,0 f1 = -52,1,17 (7) f2 = -3*119, 0, 119 h0 = 119, 0, -119/3 <-- h0 = h1 - f1 h1 = 67,1,-68/3 (2) h2 15,2,-17/3 (5) h3 = 8,9,-17/3 (2) h4 = 1,16.-17/3 (8) h5 = 0,119, -119/3 g0 = 0, -3*119, 119 g1 = 1, -103, 74 (4) g2 = 4, -55, 17 (2) g3 = 7,-7,0 Step 1: Delete the one f2 = h0 f0 = -7,7,0 f1 = -52,1,17 (4/3) and x -1 h1 = 67,1,-68/3 <-- h1 = h2 - f1 h2 15,2,-17/3 (5) h3 = 8,9,-17/3 (2) h4 = 1,16.-17/3 (8) h5 = 0,119, -119/3 g0 = 0, -3*119, 119 g1 = 1, -103, 74 (4) g2 = 4, -55, 17 (2) g3 = 7,-7,0 Step 2: Delete the one h1 f0 = -7,7,0 f1 = -52,1,17 (1/3) and x -1 <-- f1 = 3*h2-f0 h2 15,2,-17/3 (4) h3 = 8,9,-17/3 (2) h4 = 1,16.-17/3 (8) h5 = 0,119, -119/3 g0 = 0, -3*119, 119 g1 = 1, -103, 74 (4) g2 = 4, -55, 17 (2) g3 = 7,-7,0 Step 3: Delete the one f1 f0 = -7,7,0 h2 15,2,-17/3 (3) <-- h2 = h3 - f0 h3 = 8,9,-17/3 (2) h4 = 1,16.-17/3 (8) h5 = 0,119, -119/3 g0 = 0, -3*119, 119 g1 = 1, -103, 74 (4) g2 = 4, -55, 17 (2) g3 = 7,-7,0 Step 4: Delete the one h2 f0 = -7,7,0 h3 = 8,9,-17/3 (1) <-- h3 = h4-f0 h4 = 1,16.-17/3 (8) h5 = 0,119, -119/3 g0 = 0, -3*119, 119 g1 = 1, -103, 74 (4) g2 = 4, -55, 17 (2) g3 = 7,-7,0 Step 5: Delete the one h3 f0 = -7,7,0 h4 = 1,16.-17/3 (7) h5 = 0,119, -119/3 g0 = 0, -3*119, 119 g1 = 1, -103, 74 (4) g2 = 4, -55, 17 (2) g3 = 7,-7,0 == We illustrate this on the example of 1/156(121,32,1^) + 1/4(39,0,39^3) that was already mentioned as the inflation by 4 of 1/39(1,8,10^3) in Example 39, Figure~f!39 f0 = -39,39,0 f1 = -35,32,1 f2 = -31,25,2 f3 = -27,18,3 f4 = -23,11,4 f5 = -19,4,5 f6 = -34,1,11 f7 = -117,0,39 corr to tags [2,2,2,2,3,4] = 39/32 h0 = -1/3*f7 = 39,0,-13 N.B. in L' not in L h1 = h0 + f6 = 5,1,-2 h2 = 8*h1 - h0 = 1,8,-3 h3 = 5*h2 - h1 = 0,39,-13 corr. to [8,5] = 39/5. (The condition all a_i == 2 mod n-2 is what ensures that all hi + (0,0,13) are points of L.) The to continue with the gs we need g0 = -3*h3 = 0,-117,39, g1 = h2 - h3 = 1,-31,10, after which g2 = 4*g1-g0 = 4,-7,1 g3 = 10*g2-g1 = 39,-39,0 then continue around cyclically as in [CR] with f0 = -g3, f1 = -g3+g2, etc This explains how to construct the cutting axes out of each vertex and give them strength. This draws the propellor picture, where we can say what the internal tags are 4,3,2,2,2,2,1,10,4 at resp f6,f5,f4,f3,f2,f1,f0,g2,g1 but at the edges through A' we have something new, namely h0 = -1/3 f7, h1 = f7 +or- f6, and h3 = -1/3 g0, h2 = g1 + h3. Bigger case n=5 and A = 1/545(1,73,157) ======================================= I want to do a bigger ex. Say n=5 again, and [8,2,8,5] = 545/73. It is constructed from the HJ continued fraction [ 5, 8, 2, 8 ], with the extreme points 1/545(112, 1, 144), 1/545(243, 299, 1) The propellor of vectors out of e1, e2, A' is f0 = [-545, 545, 0] f1 = [-302, 299, 1] // [ 2, 6, 3, 5, 4 ] f2 = [-59, 53, 2] f3 = [-52, 19, 11] f4 = [-97, 4, 31] f5 = [-433, 1, 144] f6 = [-1635, 0, 545] g8 = [545, -545, 0] g7 = [243, -246, 1] // [ 3, 2, 2, 2, 10, 2, 4 ] g6 = [184, -193, 3] g5 = [125, -140, 5] g4 = [66, -87, 7] g3 = [7, -34, 9] g2 = [4, -253, 83] g1 = [1, -472, 157] g0 = [0, -1635, 545] h0 = [545, 0, -545/3] h1 = [112, 1, -113/3] // [ 5, 8, 2, 8 ] h2 = [15, 5, -20/3] h3 = [8, 39, -47/3] h4 = [1, 73, -74/3] h5 = [0, 545, -545/3] One can determine from this every block regular triple, hence the partition of De corresponding to A-Hilb\CC^n. (I just ran a computer search for all triples formed from {fi, gi, hi and 3hi} that close up to a triangle.) It has the following 17 blocks. f0, f1, g7 basic regular triangle e1 e2 P11 along base f1, f2, g7 next up at left, basic regular triangle e1 P1 P2 f2, g6, g7 f2, g5, g6 f2, g4, g5 f2, g3, g4 series of 4 regular triangles of side 2 filling in e2 P2 P18 f2, f3, g3 big regular triangle e1 P18 P99 of side 9 3*h2, f3, g3 regular triangle P55 P99 P135 of side 4, the Meeting of Champions. This fills under the main axis g3. 3*h2, f3, f4 regular triangle e1 P55 P155 of side 5 f4, h1, h2 trap P31 P155 A' of 2 shelves 3*h1, f4, f5 regular triangle e1 P31 P144 of side 1 f5, h0, h1 basic triangle e1 P144 A' along left side g3, h2, h3 trap P72 P135 A' of 2 shelves g3, h3, h4 trap P9 P72 A' of 2 shelves 3*h4, g1, g2 3*h4, g2, g3 regular triangles of side 1 h4, h5, g1 basic triangle e2 P137 Q' 92 Junior points: [1,73,157], [4,292,83], [7,511,9], [8,39,166], [11,258,92], [14,477,18], [15,5,175], [18,224,101], [21,443,27], [25,190,110], [28,409,36], [32,156,119], [35,375,45], [39,122,128], [42,341,54], [46,88,137], [49,307,63], [53,54,146], [56,273,72], [60,20,155], [63,239,81], [66,458,7], [70,205,90], [73,424,16], [77,171,99], [80,390,25], [84,137,108], [87,356,34], [91,103,117], [94,322,43], [98,69,126], [101,288,52], [105,35,135], [108,254,61], [112,1,144], [115,220,70], [122,186,79], [125,405,5], [129,152,88], [132,371,14], [136,118,97], [139,337,23], [143,84,106], [146,303,32], [150,50,115], [153,269,41], [157,16,124], [160,235,50], [167,201,59], [174,167,68], [181,133,77], [184,352,3], [188,99,86], [191,318,12], [195,65,95], [198,284,21], [202,31,104], [205,250,30], [212,216,39], [219,182,48], [226,148,57], [233,114,66], [240,80,75], [243,299,1], [247,46,84], [250,265,10], [254,12,93], [257,231,19], [264,197,28], [271,163,37], [278,129,46], [285,95,55], [292,61,64], [299,27,73], [309,212,8], [316,178,17], [323,144,26], [330,110,35], [337,76,44], [344,42,53], [351,8,62], [368,159,6], [375,125,15], [382,91,24], [389,57,33], [396,23,42], [427,106,4], [434,72,13], [441,38,22], [448,4,31], [486,53,2], [493,19,11] Scrap. Magma for 1/39(1,9,10^3) inflated by 4 $s^{-1}A=\frac1{156}(4,32,120)\oplus\frac14(1,0,1^3)$. 1/156(1,8,49^3) + 1/4(1,0,1^3) 156 x the lattice points are i * [1,8,49] + j * [39,0,39] for i in [0..155] and j in [0,1,2,3] Points := [[ i , (8*i+39*j) mod 156, (49*i + 39*j) mod 156 ] : i in [0..155], j in [0..3]]; #Points; [Points[i] : i in [1..10]]; Jun := [ P : P in Points | P[1] + P[2] + 3*P[3] eq 156]; #Jun; // [Jun[i] : i in [1..10]]; Age2 := [ P : P in Points | P[1] + P[2] + 3*P[3] eq 2*156]; #Age2; [Age2[i] : i in [1..10]]; Sums := [ [P[1]+Q[1], P[2]+Q[2], P[3]+Q[3] ] : P in Jun, Q in Jun ]; &and[P in Sums : P in Age2 ]; Jun; 107 // number of Jun 207 // number of Age2 [ [ 1, 8, 49 ], [ 4, 32, 40 ], [ 7, 56, 31 ], [ 10, 80, 22 ], [ 13, 104, 13 ], [ 16, 128, 4 ], [ 20, 4, 44 ], [ 23, 28, 35 ], [ 26, 52, 26 ], [ 29, 76, 17 ], [ 32, 100, 8 ], [ 39, 0, 39 ], [ 42, 24, 30 ], [ 45, 48, 21 ], [ 48, 72, 12 ], [ 51, 96, 3 ], [ 61, 20, 25 ], [ 64, 44, 16 ], [ 67, 68, 7 ], [ 80, 16, 20 ], [ 83, 40, 11 ], [ 86, 64, 2 ], [ 99, 12, 15 ], [ 102, 36, 6 ], [ 118, 8, 10 ], [ 121, 32, 1 ], [ 137, 4, 5 ], [ 0, 39, 39 ], [ 3, 63, 30 ], [ 6, 87, 21 ], [ 9, 111, 12 ], [ 12, 135, 3 ], [ 16, 11, 43 ], [ 19, 35, 34 ], [ 22, 59, 25 ], [ 25, 83, 16 ], [ 28, 107, 7 ], [ 35, 7, 38 ], [ 38, 31, 29 ], [ 41, 55, 20 ], [ 44, 79, 11 ], [ 47, 103, 2 ], [ 54, 3, 33 ], [ 57, 27, 24 ], [ 60, 51, 15 ], [ 63, 75, 6 ], [ 76, 23, 19 ], [ 79, 47, 10 ], [ 82, 71, 1 ], [ 95, 19, 14 ], [ 98, 43, 5 ], [ 114, 15, 9 ], [ 117, 39, 0 ], [ 133, 11, 4 ], [ 2, 94, 20 ], [ 5, 118, 11 ], [ 8, 142, 2 ], [ 12, 18, 42 ], [ 15, 42, 33 ], [ 18, 66, 24 ], [ 21, 90, 15 ], [ 24, 114, 6 ], [ 31, 14, 37 ], [ 34, 38, 28 ], [ 37, 62, 19 ], [ 40, 86, 10 ], [ 43, 110, 1 ], [ 50, 10, 32 ], [ 53, 34, 23 ], [ 56, 58, 14 ], [ 59, 82, 5 ], [ 69, 6, 27 ], [ 72, 30, 18 ], [ 75, 54, 9 ], [ 78, 78, 0 ], [ 88, 2, 22 ], [ 91, 26, 13 ], [ 94, 50, 4 ], [ 110, 22, 8 ], [ 129, 18, 3 ], [ 1, 125, 10 ], [ 4, 149, 1 ], [ 5, 1, 50 ], [ 8, 25, 41 ], [ 11, 49, 32 ], [ 14, 73, 23 ], [ 17, 97, 14 ], [ 20, 121, 5 ], [ 27, 21, 36 ], [ 30, 45, 27 ], [ 33, 69, 18 ], [ 36, 93, 9 ], [ 39, 117, 0 ], [ 46, 17, 31 ], [ 49, 41, 22 ], [ 52, 65, 13 ], [ 55, 89, 4 ], [ 65, 13, 26 ], [ 68, 37, 17 ], [ 71, 61, 8 ], [ 84, 9, 21 ], [ 87, 33, 12 ], [ 90, 57, 3 ], [ 103, 5, 16 ], [ 106, 29, 7 ], [ 122, 1, 11 ], [ 125, 25, 2 ] ] [ 121, 32, 1 ], [ 82, 71, 1 ], [ 43, 110, 1 ], [ 4, 149, 1 ],