Challenge of the Waterman Polyhedron

( Below two images are "crosseyed" stereo pairs )

Wat1157s0033.gif (66K)

Waterman Polyhedron Root 33; 1157 balls in sphere radius 5.7446
Data files :
(Look for "root 33")

Tv1157.gif (78K)

Tetrahedraverse engine product, 1157 balls in sphere radius 5.7335

(Please note: 'containing' sphere passes through the centers of the surface balls in both images, not around their outer surfaces.)

Data files available here:
Coordinates of the balls in the above pack :SnapShot.3d
Coordinates of the initial random distribution of the balls, before packing began: apntfld.3d
(A .3d file is Graph3D's native format. It is a straight ASCII text file, 3 XYZ coordinates per line; treat it as such.)

The Waterman polyhedra are spherical sections "cut" from an extended Buckminster Fuller-type Cubic Close Packing of balls.
The CCP packing is accepted by most geometers as the densest possible packing arrangement for equal-radius balls.
It's density (space occupied by ball vs. space between balls, collectively) is said to be about 74%.
The random-jammed packing, on the other hand, is said to have a maximum pack density of around 68%.
A question was raised on Synergeo mailing list, whether a random-jammed packing could fit as many balls
into a sphere the same size as that sphere which is used to cut out a Waterman polyhedron from the CCP.
This is the result of my first attempt to do that.

Viewers please note that just because I was able to random-jam-pack the same numnber of balls into
a smaller sphere than it takes to contain the balls of Waterman polyhedron "Root 33,"
does not mean that all Waterman polys can have their balls rearranged to achieve a similar result.
The more balls there are in a Waterman polyhedron, the more closely the whole sphere-approximate pack
approaches a "perfect" sphericity, hence the more closely the polyhedron approaches the theoretical pack-density limit of around 74 percent.

My thanks to Steve Waterman, Kirby Urner, John Braley, Alan Michaelson, and Dick Fischbeck for interaction during this experimental packing.

content copyright © 2003 by John Brawley

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