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Waterman polyhedron

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Waterman polyhedra were invented, around 1990, by Steve Waterman.

The Waterman polyhedra are created by packing spheres according cubic close(st) packing (CCP) and sweep away spheres that are farther from the center than a defined radius, the resulting pack of spheres is then convex hulled, and thereby a polyhedron is created.

  • Cubic Close(st) Packed spheres with radius sqrt(2*24)
    Cubic Close(st) Packed spheres with radius sqrt(2*24)
  • Waterman Polyhedron W24 Origin 1
    Waterman Polyhedron W24 Origin 1

Waterman polyhedra form a vast family of polyhedra. Some of them have a number of nice properties like multiple symmetries, or very interesting and regular shapes. Some other are just a bunch of faces formed out of irregular convex polygons. The most popular Waterman polyhedra are those with centers in the point (0,0,0) and build out of hundreds of polygons. Such polyhedra resemble big spheres in 3D. In fact, the more faces has a Waterman polyhedron, the more its shape resembles the sphere circumscribed on it. Its volume and total area are close to those parameters of the circumscribed sphere.

With each point of 3D space we can associate a family of Waterman polyhedra with different values of radii of the circumscribed spheres. Therefore, from mathematical point of view we can consider Waterman polyhedra as a 4D space W(x, y,z, r), where x, y,z are coordinates of a point in 3D, and r is a positive number, and r>1. [1]

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

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

  1. Origin 1: offset 0,0,0, radius sqrt(2*n)
  2. Origin 2: offset 1/2,1/2,0, radius sqrt(2+4*n)/2
  3. Origin 3: offset 1/3,1/3,2/3, radius sqrt(6*(n+1))/3
  4. Origin 3*: offset 1/3,1/3,1/3, radius sqrt(3+6*n)/3
  5. Origin 4: offset 1/2,1/2,1/2, radius sqrt(3+8*(n-1))/2
  6. Origin 5: offset 0,0,1/2, radius sqrt(1+4*n)/2
  7. 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.

Waterman polyhedra and platonic / Archimedean solids

Interestingly some Waterman Polyhedra (WP) actually create platonic solids and archimedean solids. For this comparison of WP they are normalized, e.g. W2 O1 has a different size or volume than W1 O6, but have the same form of an Octahedron.

Platonic solids

  • W1 O3* = W2 O3* = W1 O3 = W1 O4 = tretrahedron
  • W2 O1 = W1 O6 = octahedron
  • W2 O6 = cube
  • Icosahedron and Dodecahedron have no WP representation

Archimedean solids

  • W1 O1 = W4 O1 = cuboctahedron
  • W10 O1 = truncated octahedron
  • W4 O3 = W2 O4 = truncated yetrahedron
  • the others have no WP representation

Note: The W7 O1 might be mistaken for a truncated cuboctahedron, as well W3 O1 = W12 O1 mistaken for a Rhombicuboctahedron, but the those WPs have two edge lengths and therefore don't qualify as archimedean solids.

Generalized Waterman polyhedra (GWP)

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

Notes

  1. ^ Visualizing Waterman Polyhedra with MuPAD by M. Majewski
  2. ^ 7 Origins of CCP Waterman polyhedra by Mark Newbold

Background information

Visualization approaches

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