written by Rene K. Mueller, Copyright (c) 2007, last updated Wed, January 9, 2008
Waterman polyhedra exist in mathematics since about 1990, a rather recent discovery.
The concept of Waterman polyhedra was developed by Steve Waterman and relate to "Cubic Close Packing" (CCP) of spheres,
also known as IVM (Isotropic Vector Matrix) by R. Buckminster Fuller.
Spheres are packed in a cubic manner:
x, y and z are integers,
sphere diameter is √2 thereby,
at 0,0,0 resides the first sphere,
and then those removed which have longer vector than √(2*n) from the center (0,0,0) and then a convex hull is calculated which gives a polyhedron:
So the nice thing is, the creation of the polyhedra is done parametrically, in other words, systematically.
Needless to say, there are infinite of Waterman polyhedra, and interestingly there are doubles to be encountered, where:
the same amount of packed spheres result in the same polyhedra (14 + 16 n) m2 (whereas n & m integers >= 0)
the final polyhedra from packed spheres result in a previous discovered polyhedra when normalized (e.g. W3 O1 = W24 O1)
Those are not yet removed in the overview which follows.
Platonic & Archimedean Solids vs Waterman Polyhedra
The Waterman Polyhedra (WP) covers also some of the platonic and archimedean solids.
For sake of this comparison the WP are normalized, as W2 O1 has a different size/volume than W1 O6, but the same form of a Octahedron.
Waterman W1 O4
Waterman W2 O1
Waterman W2 O6
W1 O3* = W2 O3* = W1 O3 = W1 O4 = Tretrahedron
W2 O1 = W1 O6 = Octahedron
W2 O6 = Cube
Icosahedron and Dodecahedron have no WP representation.
Waterman W1 O1
Waterman W10 O1
Waterman W2 O4
W1 O1 = W4 O1 = Cuboctahedron
W10 O1 = Truncated Octahedron
W4 O3 = W2 O4 = Truncated Tetrahedron
The following look like archimedean solids, but they are not:
Waterman W7 O1
Waterman W3 O1
W7 O1 != Truncated Cuboctahedron
W3 O1 = W12 O1 != Rhombicuboctahedron
as they are not having one length of edge, but two lengths of edges.
The others have no WP representation.
The Generalized Waterman Polyhedra (GWP) are currently still sorted by Steve Waterman, and I will include them later as they may cover more platonic and archimedean solids.
Waterman Polyhedron, very informative site by Steve Waterman, the inventor of the Waterman polyhedra
Paul Bourke has also a comprehensive website (I highly recommend to visit), his waterman.c I used to calculate those rendered Waterman polyhedra.
3D Junkyard, vast & interesting collection of links by David Eppstein