Waterman polyhedron

Seven origins of cubic close(st) packing (CCP)

There can be seven origins defined in CCP, where n = {1, 2, 3, …}:

• Origin 1: offset 0,0,0, radius \sqrt{2n}
• Origin 2: offset ,,0, radius \tfrac12\sqrt{2+4n}
• Origin 3: offset ,,, radius \tfrac13\sqrt{6(n+1)}
• Origin 3*: offset ,,, radius \tfrac13\sqrt{3+6n}
• Origin 4: offset ,,, radius \tfrac12\sqrt{3+8(n-1)}
• Origin 5: offset 0,0,, radius \tfrac12\sqrt{1+4n}
• Origin 6: offset 1,0,0, radius \sqrt{1+2(n-1)}

Depending on the origin of the sweeping, a different shape and resulting polyhedron are obtained.

Relation to Platonic and Archimedean solids

Some Waterman polyhedra create Platonic solids and Archimedean solids. For this comparison of Waterman polyhedra they are normalized, e.g. W2 O1 has a different size or volume than W1 O6, but has the same form as an octahedron.

Platonic solids

• Tetrahedron: W1 O3*, W2 O3*, W1 O3, W1 O4
• Octahedron: W2 O1, W1 O6
• Cube: W2 O6
• Icosahedron and dodecahedron have no representation as Waterman polyhedra.

Archimedean solids

• Cuboctahedron: W1 O1, W4 O1
• Truncated octahedron: W10 O1
• Truncated tetrahedron: W4 O3, W2 O4
• The other Archimedean solids have no representation as Waterman polyhedra.

The W7 O1 might be mistaken for a truncated cuboctahedron, as well W3 O1 = W12 O1 mistaken for a rhombicuboctahedron, but those Waterman polyhedra have two edge lengths and therefore do not qualify as Archimedean solids.

Generalized Waterman polyhedra

Generalized Waterman polyhedra are defined as the convex hull derived from the point set of any spherical extraction from a regular lattice.

Included is a detailed analysis of the following 10 lattices – bcc, cuboctahedron, diamond, fcc, hcp, truncated octahedron, rhombic dodecahedron, simple cubic, truncated tet tet, truncated tet truncated octahedron cuboctahedron.

Each of the 10 lattices were examined to isolate those particular origin points that manifested a unique polyhedron, as well as possessing some minimal symmetry requirement. From a viable origin point, within a lattice, there exists an unlimited series of polyhedra. Given its proper sweep interval, then there is a one-to-one correspondence between each integer value and a generalized Waterman polyhedron.

Notes

References

1. Popko, Edward S.. (2012). "Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere". CRC Press.
2. [http://www.mupad.com/mathpad/2006/majewski/index.php Visualizing Waterman Polyhedra with MuPAD] {{Webarchive. link. (2008-01-12 by M. Majewski)
3. [http://dogfeathers.com/java/ccppoly.html 7 Origins of CCP Waterman polyhedra] by Mark Newbold