**The Principle of the Waterman Polyhedron**

Steve Waterman's polyhedra are derived from a "Cubic Close Packing" ("CCP") of equal-radius balls.

The polyhedra themselves do not show these balls, but instead are actual "polyhedra" (terminology: geometry).

The cubic close packing of equal radius balls is that same packing used by grocers to stack round fruits (ex: oranges), or by cannoneers to stack cannonballs, and is that packing used by R. Buckminster Fuller in deriving his tetrahedrally based geometric system exemplified by the "IVM" --the "Isotropic Vector Matrix".

It is also that packing hypothesized by Johannes Kepler, and recently provisionally proven by Hales and Ferguson, to be the tightest/closest **possible** packing for a large number of equal radius balls.

(But, see my exception[L], concerning only 13 balls)

Steve, interested in how a **sphere** might interact with this cubic close packing, employs a technique in which he selects a ball from within an in principle infinitely extended CCP to use as the center of an hypothetical sphere, then defines a radius for that sphere, then records the locations (and hence, patterns of distribution) of all balls in the CCP whose centers lie upon (are "congruent with") the surface of that sphere.

The resulting patterns of distribution of the ball-centers (employed as vertices), when polygons defined by them are filled in as planes, are the "Waterman Polyhedra." Many --very many-- of these are visible in Steve's collection, at this website (same site as the Waterman's Polyhedra link above).

Here is a PovRay image I made, showing in stereo 3D (crossed-eyes freeviewing version), the general principle of the Waterman polyhedra's origins in the CCP.

The green balls show the locations of balls in the CCP, from a point of view selected for visibility down into it, and the grey 'glass' sphere shows how some of the CCP's balls' centers touch the surface of the sphere.

The balls have also been reduced in radius so that you can see down into the CCP. Were they full size, this would look like a closed cubic pack of big Jade beads and you couldn't see the 'glass' sphere at all.

This image is not a precise one (the 'glass' sphere has no special radius); it is intended only to show the **principle** involved.

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