spheres and polyhedra

packing spheres

If we want to pack identical spheres efficiently we intuitively choose a periodic arrangement like the cubic close packing or CCP.
There are three ways to look at this arrangement: the two first are classical and the third has first be highlighted by Steve Waterman.

Be patient during the initialization!
(different views of the same block show how the layers are arranged)

Waterman's polyhedra

Steve Waterman focused his interest on the convex hulls of sets of spheres' centers in a CCP; a set is bounded by the sphere centered at the origin and with radius N*sqrt(2). The integer N is called the "root".
Examples: for N=1 we get 13 centers, and the convex hull is a cuboctahedron (12 vertices); for N=2 we get 6 more centers (19 in all) which build the second convex hull, a regular octahedron.
If we increment N generally (but not always) new points are added to the previous set and become vertices of the new convex hull; a point can however be a vertex of two or more successive convex hulls. As N grows the Waterman's polyhedra become more "spherical". They have all octahedral symmetry (like the cube).

Among them there are the cuboctahedron (W1 and W4), the regular octahedron (W2) and the truncated octahedron (W10).
Here are a few other examples: (to see more visit the references)

root 6

root 12

root 24

root 48

root 96

root 192

Steve Waterman studied more polyhedra related to the CCP by moving the origin to other interesting locations (six in all: the centers of the basic clusters build with 1, 2, 3, 4, 5 or 6 spheres). For example with the origin at a "void" we get successively the regular octahedron, the cube, the truncated octahedron... One may also use other ways to define the sets of spheres.

This technique to build polyhedra may also be used with other packings: HCP, cubic...  Nobody can guess how interesting the results of such studies would be; nevertheless, besides the three ways to build it, two properties unique to the CCP explain why Steve Waterman focused his attention on this packing: spheres with diameter 1 are at square root of an integer distances, and spheres with diameter sqrt(2) have their center at integer (x,y,z) coordinates.

Waterman's projection

It is well known that for every representation of the earth globe on a flat map distortions occur in the distances, the angles, the areas and the shapes, no matter which projection is used. "All maps have a point of view and a map's quality is a function of its purpose" says Bob Abramms, one of the authors of Many ways to see the world  (book and DVD). The best known is Mercator's cylindrical projection (first world map published in 1569) which produces a rectangular parallels/meridians grid; shortest ways are represented by line segments and the angles are correct, but shapes are distorted, especially in high latitudes. The Robinson projection (1960) reduces the extreme distortions of the Mercator.
An other way to solve the problem is to project the sphere on a polyhedron (central projection); more faces give a more accurate result, but quickly the map (the polyhedron's net) becomes ugly. R. Buckminster Fuller used a regular icosahedron for his "Dymaxion Sky-Ocean projection" (1943) but the equator is not easy to see on it.
Steve M. Waterman used one of the polyhedra he defined, the W5 which is a regular octahedron with the vertices truncated at the quarter of its edges. The result is accurate and land breaks can be avoided; since the map is a net of the polyhedron, different arrangements of the faces, and thus of the "water breaks", are possible.
Steve's "butterfly map" is definitely one of the most beautiful - and most accurate! - world maps.

the W5 (root 5)

the "butterfly map" (Atlantic view)

the "Waterman globe" (Izidor Hafner)

Thanks to Steve for his kind help and pertinent suggestions during the draw up of this page and the CCP page.

references: •  the pages by Steve Waterman
•  Paul Bourke's pages with a Waterman's polyhedra generator
•  Waterman's polyhedra by Kirby Urner  -  Waterman applet by Mark Newbold
•  Great Stella by Robert Webb (label "Melbourne" on the link globe) generates Waterman polyhedra
•  "World's maps on polyhedra nets" by Izidor Hafner and Tomislav Zitko - LiveGraphics3D animations
•  Cartographical Map Projections (with a Polyhedral Maps page) by Carlos A.Furuti

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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects April 2005
updated 19-06-2005