// phelanc.fe // Structure that beats Kelvin's partition of space. // In 1887, Lord Kelvin posed the problem of finding the partition // of space into equal volume cells minimizing the interface area. // He suggested the cell shown in twointor.fe, which is basically // the voronoi cell for a bcc lattice. Now Robert Phelan and Denis // Weaire of Trinity College, Dublin, have found a structure using // two types of cells that has 0.3% less area than Kelvin's. This is // their Evolver datafile. There are 8 cells in a // cubic 2x2x2 flat torus, which start as Voronoi cells on centers // // 0 0 0 // 1 1 1 // 0.5 0 1 // 1.5 0 1 // 0 1 0.5 // 0 1 1.5 // 1 0.5 0 // 1 1.5 0 // Just evolve to get the volumes all to 1, and Kelvin is beat. // With more evolution, the ratio V^2/A^3 beats Kelvin by a // whopping 0.3%. The Weaire-Phelan structure has its tetrakaidecahedra // stacked on their hexagonal faces in three sets of perpendicular, // mutually interlocking columns, with interstices filled by the // dodecahedra. // phelanc.fe with colored bodies TORUS_FILLED // triply periodic, filled with bodies periods 2.000000 0.000000 0.000000 0.000000 2.000000 0.000000 0.000000 0.000000 2.000000 vertices // coordinates from John Sullivan's Voronoi program. 1 1.374833 0.000542 0.313036 2 1.582639 1.583805 0.417091 3 1.999414 1.687884 0.625562 4 0.999778 0.000517 0.500564 5 1.686693 1.374893 1.999381 6 1.999036 0.312928 0.625224 7 0.416118 1.583554 0.417247 8 1.416641 1.417638 0.583002 9 0.999380 1.626008 0.687643 10 1.374528 0.000836 1.688167 11 1.582887 1.582882 1.583776 12 1.583228 0.416633 0.417188 13 0.415660 0.417170 0.416774 14 0.312152 1.375055 0.000199 15 1.999782 1.500468 1.000033 16 1.625132 1.312811 1.000465 17 1.312290 1.000953 0.374434 18 0.999015 1.625244 1.312907 19 0.582337 1.417418 0.583502 20 0.999205 0.000988 1.500509 21 1.582954 0.417290 1.583964 22 1.499589 1.000835 1.999244 23 1.687315 0.625137 1.999668 24 0.624322 0.000725 0.312769 25 0.416475 1.583942 1.583184 26 0.374830 1.313082 0.999521 27 1.624817 0.687664 1.000333 28 0.686621 1.000835 0.374834 29 1.416503 1.416444 1.417442 30 0.624634 0.000964 1.687531 31 1.999937 1.687755 1.375079 32 1.312386 1.000647 1.625031 33 0.499725 1.000658 0.000553 34 0.311830 0.624975 0.000488 35 0.375186 0.688291 0.999389 36 1.416715 0.583171 0.583818 37 0.582556 0.584101 0.583805 38 0.583642 1.417297 1.416440 39 1.999485 0.312616 1.375471 40 0.416489 0.416552 1.584015 41 1.999161 0.500331 0.999822 42 0.999925 0.375397 0.688204 43 0.688262 1.000529 1.624602 44 1.416155 0.584231 1.417123 45 0.584307 0.583998 1.416583 46 0.999499 0.376190 1.313024 edges // defined by vertices and torus wraps 1 1 2 * - * 2 2 3 * * * 3 1 4 * * * 4 2 5 * * - 5 3 6 * + * 6 3 7 + * * 7 2 8 * * * 8 4 9 * - * 9 1 10 * * - 10 5 11 * * * 11 6 12 * * * 12 6 13 + * * 13 7 14 * * * 14 3 15 * * * 15 8 16 * * * 16 8 17 * * * 17 9 8 * * * 18 9 18 * * * 19 9 19 * * * 20 10 20 * * * 21 10 21 * * * 22 11 10 * + * 23 5 22 * * * 24 5 14 + * + 25 12 1 * * * 26 12 23 * * - 27 13 24 * * * 28 7 19 * * * 29 14 25 * * - 30 15 26 + * * 31 16 15 * * * 32 16 27 * * * 33 17 22 * * - 34 17 28 * * * 35 18 29 * * * 36 19 26 * * * 37 20 30 * * * 38 21 23 * * * 39 11 31 * * * 40 11 29 * * * 41 22 32 * * * 42 14 33 * * * 43 23 34 + * + 44 24 7 * - * 45 24 30 * * - 46 19 28 * * * 47 25 30 * + * 48 25 31 - * * 49 26 35 * * * 50 16 29 * * * 51 27 36 * * * 52 17 36 * * * 53 28 33 * * * 54 28 37 * * * 55 18 38 * * * 56 29 32 * * * 57 26 38 * * * 58 20 18 * - * 59 31 39 * + * 60 22 23 * * * 61 33 34 * * * 62 34 13 * * * 63 34 40 * * - 64 30 40 * * * 65 35 41 - * * 66 35 37 * * * 67 36 12 * * * 68 36 42 * * * 69 33 43 * * - 70 37 42 * * * 71 37 13 * * * 72 38 25 * * * 73 32 43 * * * 74 32 44 * * * 75 24 4 * * * 76 35 45 * * * 77 21 39 * * * 78 39 40 + * * 79 41 27 * * * 80 42 46 * * * 81 43 38 * * * 82 42 4 * * * 83 44 27 * * * 84 44 46 * * * 85 15 31 * * * 86 45 43 * * * 87 45 46 * * * 88 41 39 * * * 89 21 44 * * * 90 6 41 * * * 91 46 20 * * * 92 45 40 * * * faces // colored according to body 1 1 2 5 11 25 color 1 backcolor 4 2 -1 3 8 17 -7 color 8 backcolor 4 3 2 6 13 -24 -4 color 5 backcolor 1 4 5 12 27 44 -6 color 3 backcolor 1 5 11 26 43 62 -12 color 5 backcolor 1 6 1 4 10 22 -9 color 8 backcolor 1 7 17 16 34 -46 -19 color 2 backcolor 8 8 7 16 33 -23 -4 color 8 backcolor 5 9 -2 7 15 31 -14 color 5 backcolor 4 10 -6 14 30 -36 -28 color 5 backcolor 3 11 -13 28 46 53 -42 color 5 backcolor 8 12 24 29 48 -39 -10 color 6 backcolor 1 13 44 13 29 47 -45 color 1 backcolor 8 14 62 27 45 64 -63 color 1 backcolor 7 15 25 9 21 38 -26 color 7 backcolor 1 16 -3 9 20 37 -45 75 color 8 backcolor 7 17 -10 23 41 -56 -40 color 8 backcolor 6 18 -22 39 59 -77 -21 color 4 backcolor 1 19 8 19 -28 -44 75 color 3 backcolor 8 20 -18 19 36 57 -55 color 2 backcolor 3 21 -17 18 35 -50 -15 color 2 backcolor 4 22 34 53 69 -73 -41 -33 color 8 backcolor 7 23 -46 36 49 66 -54 color 5 backcolor 2 24 -16 15 32 51 -52 color 2 backcolor 5 25 31 30 49 65 79 -32 color 6 backcolor 5 26 -53 54 71 -62 -61 color 5 backcolor 7 27 42 61 -43 -60 -23 24 color 5 backcolor 6 28 48 59 78 -64 -47 color 1 backcolor 3 29 43 63 -78 -77 38 color 1 backcolor 6 30 -41 60 -38 89 -74 color 7 backcolor 6 31 -56 -35 55 -81 -73 color 2 backcolor 8 32 40 -35 -58 -20 -22 color 8 backcolor 4 33 -57 -30 85 -48 -72 color 6 backcolor 3 34 50 56 74 83 -32 color 2 backcolor 6 35 34 54 70 -68 -52 color 7 backcolor 2 36 69 81 72 -29 42 color 6 backcolor 8 37 -33 52 67 26 -60 color 7 backcolor 5 38 49 76 86 81 -57 color 2 backcolor 6 39 66 71 -12 90 -65 color 3 backcolor 5 40 51 67 -11 90 79 color 5 backcolor 4 41 -37 58 55 72 47 color 8 backcolor 3 42 -89 -21 20 -91 -84 color 7 backcolor 4 43 74 84 -87 86 -73 color 7 backcolor 2 44 83 51 68 80 -84 color 4 backcolor 2 45 70 82 -75 -27 -71 color 3 backcolor 7 46 -68 67 25 3 -82 color 4 backcolor 7 47 -69 61 63 -92 86 color 6 backcolor 7 48 -76 65 88 78 -92 color 3 backcolor 6 49 -66 76 87 -80 -70 color 3 backcolor 2 50 90 88 -59 -85 -14 5 color 4 backcolor 3 51 79 -83 -89 77 -88 color 4 backcolor 6 52 -91 -87 92 -64 -37 color 3 backcolor 7 53 -85 -31 50 -40 39 color 6 backcolor 4 54 -82 80 91 58 -18 -8 color 3 backcolor 4 bodies 1 1 -3 -4 -5 -6 -12 13 14 -15 -18 28 29 volume 1 2 7 20 21 -23 24 31 34 -35 38 -43 -44 -49 volume 1 3 39 45 48 49 -50 4 54 -28 52 -33 19 -20 -41 -10 volume 1 4 -40 44 46 50 51 -1 -42 -54 -53 -9 18 -2 -32 -21 volume 1 5 3 -8 9 10 11 -24 -25 23 26 27 -37 40 -39 5 volume 1 6 33 -38 25 12 36 -48 53 -51 -34 -27 47 -17 -30 -29 volume 1 7 -22 30 35 -26 37 42 43 -45 -46 -47 15 -52 -16 -14 volume 1 8 2 6 -7 8 16 17 -19 22 -31 32 -36 -11 41 -13 volume 1 read hessian_normal // typical evolution gogo := { g 5; V; r; g 5; r; g 5; convert_to_quantities; hessian; hessian; } 