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Topic: New Topic: Waterman Polyhedra
Replies: 1   Last Post: Jun 19, 2005 1:01 PM

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
New Topic: Waterman Polyhedra
Posted: Jun 17, 2005 7:49 PM
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These have already been described on geometry-research and sci.math, other forums in the Math Forum, so it'd be redundant of me to post there. But here the Waterman series is relevant because of my interest in using polyhedral numbers to bridge algebra with spatial geometry, per 'The Book of Numbers' by Conway and Guy, and other authors (Bucky Fuller of course -- one of my mentors).

Watermans are convex polyhedra the vertices of which are all members of a specific lattice, "popularly known as" (yeah right) the CCP or the FCC, by Fuller as the IVM, and by me as the IVM FM (we might be communicating on some ivm.f in dot notation).

(1) Take all CCP lattice points distance R from the origin (choosing a nice R on purpose) and sweep omni- directionally, picking up all additional lattice points of equal distance R.

(2) Now bring the sweeper in a notch, to the next closest distance where an IVM center might appear (this may be computed), and sweep again.

(3) Keep this up *until* your points collection first becomes non-convex. You've gone too far; back up one step (Watermans must be convex).

This algorithm has been implemented numerous times over the years, by many programmers more talented than I. For example, my waterman generator wouldn't have been possible without Qhull, a free software implementation of convex-hull-finding and other algorithms of great elegance and power. Now we have QuickHull3D in Java, thanks to John Lloyd. Watermans have therefore recently become stars of the Java applet world, thanks to Mark Newbold.

Anyway, if you check Mark's applet, you'll see why these polyhedra work as eye candy, and why they therefore make a good access point into the wonderful world of the CCP aka IVM aka FCC (and the many important connected concepts). And that's a world we want to be in for our early math learning of numeric, rule-expressed polyhedral numbers, such as 1, 12, 42, 92, 162... I fully expect this will be well-known to tomorrow's schoolchildren.


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