// 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; }