Waterman Polyhedra
Part 1 - Introduction, Root 1 to 24
Written by Paul BourkeDecember 2002
Inspired and based upon work by Steve Waterman
New (April 2004): Online coordinate generator
Download the 10000000 monster
Waterman PolyhedraPart 1 - Introduction, Root 1 to 24Written by Paul BourkeDecember 2002 Inspired and based upon work by Steve Waterman New (April 2004): Online coordinate generator |
Download the 10000000 monster |
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Root: 1
Radius: sqrt(2) |
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Consider spheres of radius 1/sqrt(2) arranged such that the centers are only at integer coordinate values and only where the sum of the coordinates x+y+z is even. Such a packing is often referred to as CCP (Cubic Closest Packing), also known as the IVM (Isotropic Vector Matrix) by R Fuller. The odd and even layers of such a packing is illustrated below. 
Note that the gaps between the spheres are at coordinates where x+y+z is odd. Each sphere is surrounded by 12 closest neighbours in a cuboctahedral arrangement. |
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Root: 2
Radius: sqrt(4) |
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A subset of the infinite CCP packing can formed by including only those sphere centers within a certain radius from the origin. If a integer is used to index these subsets, call it root, then if the allowed radii are integer multiples of sqrt(2 root) then the convex hull formed from the set of sphere centers is known as a Waterman polyhedra. |
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Root: 3
Radius: sqrt(6) |
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The polyhedra along with the sphere packings for root from 1 to 50 are shown here along with various statistics. In addition, the polyhedra is given for each root in the OFF format. While the derivation of the polyhedra requires the determination of the convex hull, the CCP subsets from which the convex hull is derived can be written as a PovRay script, see: waterman.pov. |
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Root: 4
Radius: sqrt(8) |
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Note that the convex hull for root = 4 is the same shape as for root = 1, there is a size difference though. |
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Root: 5
Radius: sqrt(10) |
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A C program that creates the PovRay files for the sphere packings shown here is: waterman.c. A PovRay scene file that can be used to render the models is: scene.pov. |
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Root: 6
Radius: sqrt(12) |
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Root: 7
Radius: sqrt(14) |
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Root: 8
Radius: sqrt(16) |
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Root: 9
Radius: sqrt(18) |
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Root: 10
Radius: sqrt(20) |
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Root: 11
Radius: sqrt(22) |
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Root: 12
Radius: sqrt(24) |
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Root: 13
Radius: sqrt(26) |
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In the CCP subsets shown here the blue spheres are exactly at the integer multiple radius of sqrt(2 root). Note that some CCP subsets don't have any spheres at that distance, for example, see root 14, 30, and 46. In these cases the polyhedra are the same as the earlier one, so root 13 is the same as root 14, root 29 is the same as root 30, etc. The longer list of roots (up to root 2000) when this occurs is: 14 30 46 56 62 78 94 110 120 126 142 158 174 184 190 206 222 224 238 248 254 270 286 302 312 318 334 350 366 376 382 398 414 430 440 446 462 478 480 494 504 510 526 542 558 568 574 590 606 622 632 638 654 670 686 696 702 718 734 736 750 760 766 782 798 814 824 830 846 862 878 888 894 896 910 926 942 952 958 974 990 992 1006 1016 1022 1038 1054 1070 1080 1086 1102 1118 1134 1144 1150 1166 1182 1198 1208 1214 1230 1246 1248 1262 1272 1278 1294 1310 1326 1336 1342 1358 1374 1390 1400 1406 1422 1438 1454 1464 1470 1486 1502 1504 1518 1528 1534 1550 1566 1582 1592 1598 1614 1630 1646 1656 1662 1678 1694 1710 1720 1726 1742 1758 1760 1774 1784 1790 1806 1822 1838 1848 1854 1870 1886 1902 1912 1918 1920 1934 1950 1966 1976 1982 1998These "missing" polyhedra occur at position (14 + 16n)m2 where n and m are integers greater than or equal to 0. (Steve Waterman). |
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Root: 14
Radius: sqrt(28) |
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Root: 15
Radius: sqrt(30) |
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Root: 16
Radius: sqrt(32) |
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Root: 17
Radius: sqrt(34) |
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Root: 18
Radius: sqrt(36) |
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Root: 19
Radius: sqrt(38) |
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Root: 20
Radius: sqrt(40) |
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Root: 21
Radius: sqrt(42) |
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Root: 22
Radius: sqrt(44) |
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Root: 23
Radius: sqrt(46) |
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Root: 24
Radius: sqrt(48) |
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