Waterman Polyhedra

Part 1 - Introduction, Root 1 to 24

Written by Paul Bourke
December 2002

Inspired and based upon work by Steve Waterman

Online coordinate generator


Download the 10000000 monster


Root: 1

Radius: sqrt(2)
Spheres: 13
Vertices: 12
Faces: 14
Edges: 24
Area: 18.9282
Volume: 6 2/3
p0001.off
p0001.pov
s0001.pov

Sphere: 0001 Polyhedra: 0001

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.


Root: 2

Radius: sqrt(4)
Spheres: 19
Vertices: 6
Faces: 8
Edges: 12
Area: 27.7128
Volume: 10 2/3
p0002.off
p0002.pov
s0002.pov

Sphere: 0002 Polyhedra: 0002

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.


Root: 3

Radius: sqrt(6)
Spheres: 43
Vertices: 24
Faces: 26
Edges: 48
Area: 64.8693
Volume: 45 1/3
p0003.off
p0003.pov
s0003.pov

Sphere: 0003 Polyhedra: 0003

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.


Root: 4

Radius: sqrt(8)
Spheres: 55
Vertices: 12
Faces: 14
Edges: 24
Area: 75.7128
Volume: 53 1/3
p0004.off
p0004.pov
s0004.pov

Sphere: 0004 Polyhedra: 0004

Note that the convex hull for root = 4 is the same shape as for root = 1, there is a size difference though.


Root: 5

Radius: sqrt(10)
Spheres: 79
Vertices: 24
Faces: 14
Edges: 36
Area: 102.067
Volume: 81 1/3
p0005.off
p0005.pov
s0005.pov

Sphere: 0005 Polyhedra: 0005

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.


Root: 6

Radius: sqrt(12)
Spheres: 87
Vertices: 32
Faces: 42
Edges: 72
Area: 119.682
Volume: 116
p0006.off
p0006.pov
s0006.pov

Sphere: 0006 Polyhedra: 0006
Root: 7

Radius: sqrt(14)
Spheres: 135
Vertices: 48
Faces: 26
Edges: 72
Area: 159.51
Volume: 172
p0007.off
p0007.pov
s0007.pov

Sphere: 0007 Polyhedra: 0007
Root: 8

Radius: sqrt(16)
Spheres: 141
Vertices: 54
Faces: 68
Edges: 120
Area: 168.975
Volume: 200
p0008.off
p0008.pov
s0008.pov

Sphere: 0008 Polyhedra: 0008
Root: 9

Radius: sqrt(18)
Spheres: 177
Vertices: 36
Faces: 38
Edges: 72
Area: 202.373
Volume: 248
p0009.off
p0009.pov
s0009.pov

Sphere: 0009 Polyhedra: 0009
Root: 10

Radius: sqrt(20)
Spheres: 201
Vertices: 24
Faces: 14
Edges: 36
Area: 214.277
Volume: 256
p0010.off
p0010.pov
s0010.pov

Sphere: 0010 Polyhedra: 0010
Root: 11

Radius: sqrt(22)
Spheres: 225
Vertices: 48
Faces: 50
Edges: 96
Area: 242.209
Volume: 338 2/3
p0011.off
p0011.pov
s0011.pov

Sphere: 0011 Polyhedra: 0011
Root: 12

Radius: sqrt(24)
Spheres: 249
Vertices: 24
Faces: 26
Edges: 48
Area: 259.477
Volume: 362 2/3
p0012.off
p0012.pov
s0012.pov

Sphere: 0012 Polyhedra: 0012
Root: 13

Radius: sqrt(26)
Spheres: 321
Vertices: 72
Faces: 74
Edges: 144
Area: 309.072
Volume: 494 2/3
p0013.off
p0013.pov
s0013.pov

Sphere: 0013 Polyhedra: 0013

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 1998

These "missing" polyhedra occur at position (14 + 16n)m2 where n and m are integers greater than or equal to 0. (Steve Waterman).


Root: 14

Radius: sqrt(28)
Spheres: 321
Vertices: 72
Faces: 74
Edges: 144
Area: 309.072
Volume: 494 2/3
p0014.off
p0014.pov
s0014.pov

Sphere: 0014 Polyhedra: 0014
Root: 15

Radius: sqrt(30)
Spheres: 369
Vertices: 48
Faces: 26
Edges: 72
Area: 338.244
Volume: 542 2/3
p0015.off
p0015.pov
s0015.pov

Sphere: 0015 Polyhedra: 0015
Root: 16

Radius: sqrt(32)
Spheres: 381
Vertices: 60
Faces: 38
Edges: 96
Area: 352.44
Volume: 566 2/3
p0016.off
p0016.pov
s0016.pov

Sphere: 0016 Polyhedra: 0016
Root: 17

Radius: sqrt(34)
Spheres: 429
Vertices: 48
Faces: 62
Edges: 108
Area: 391.247
Volume: 697 1/3
p0017.off
p0017.pov
s0017.pov

Sphere: 0017 Polyhedra: 0017
Root: 18

Radius: sqrt(36)
Spheres: 459
Vertices: 54
Faces: 44
Edges: 96
Area: 413.991
Volume: 757 1/3
p0018.off
p0018.pov
s0018.pov

Sphere: 0018 Polyhedra: 0018
Root: 19

Radius: sqrt(38)
Spheres: 531
Vertices: 72
Faces: 74
Edges: 144
Area: 450.628
Volume: 869 1/3
p0019.off
p0019.pov
s0019.pov

Sphere: 0019 Polyhedra: 0019
Root: 20

Radius: sqrt(40)
Spheres: 555
Vertices: 72
Faces: 50
Edges: 120
Area: 461.112
Volume: 893 1/3
p0020.off
p0020.pov
s0020.pov

Sphere: 0020 Polyhedra: 0020
Root: 21

Radius: sqrt(42)
Spheres: 603
Vertices: 72
Faces: 74
Edges: 144
Area: 487.025
Volume: 973 1/3
p0021.off
p0021.pov
s0021.pov

Sphere: 0021 Polyhedra: 0021
Root: 22

Radius: sqrt(44)
Spheres: 627
Vertices: 72
Faces: 50
Edges: 120
Area: 505.712
Volume: 1013 1/3
p0022.off
p0022.pov
s0022.pov

Sphere: 0022 Polyhedra: 0022
Root: 23

Radius: sqrt(46)
Spheres: 675
Vertices: 48
Faces: 26
Edges: 72
Area: 526.167
Volume: 1045 1/3
p0023.off
p0023.pov
s0023.pov

Sphere: 0023 Polyhedra: 0023
Root: 24

Radius: sqrt(48)
Spheres: 683
Vertices: 56
Faces: 66
Edges: 120
Area: 544.319
Volume: 1144
p0024.off
p0024.pov
s0024.pov

Sphere: 0024 Polyhedra: 0024


[Part 1]   [Part 2 (Root 25 to 50)]   [Part 3 (Diversions)]   [Part 4 (Large models)]