Saturday, April 03, 2010

Suggesting a Change

I proposed augmenting a page at MathWorld yesterday, using the contribution form.

This morning, Dave Koski phoned to let me know I'd misused the term "orthoscheme" in my initial submission. The text below is from today's newly revised version.

I posted the text of both submissions to Synergeo, as echoed back to me from the MathWorld web server.

The page I am talking about is the one on Space-Filling Polyhedra.
Existing text:

"A space-filling polyhedron, sometimes called a a plesiohedron (Gr├╝nbaum and Shephard 1980), is a polyhedron which can be used to generate a tessellation of space. Although even Aristotle himself proclaimed in his work On the Heavens that the tetrahedron fills space, it in fact does not."

Proposed addition:

However an irregular tri-rectangular tetrahedron, named the Mite by R. Buckminster Fuller (for "minimum tetrahedron"), does fill space by itself, with no need of left and right handed complements.

Discussion:

The space-filling polyhedra page curiously avoids any mention of space-filling tetrahedra, simpler than hexahedra.

Fuller's contribution is more nuanced than just offering nomenclature (Mite). In Fig 950.12 he cites pg. 71 of 'Regular Polytopes' to complement Coxeter's treatment of this shape with original modular dissections thereof.

Two additional space-filling tetrahedra may be assembled from Mites.

Presumably Michael Goldberg's treatment enumerates space-filling tetrahedra, suggesting another sentence in the 2nd to last paragraph summarizing his findings?

Coxeter, D. \titleisbn{Regular Polytopes.}{0486614808} New York, New York: Dover Publications, p.~71, 1973.\par

Fuller, R. B. \titleisbn{Synergetics: explorations in the geometry of thinking.}{0020653204} New York, New York: Macmillan, Fig.~950.12, 1975.\par

Inchbald, Guy. ''The "cells" of the Archimedean honeycomb duals'' 1996. \url{http://www.steelpillow.com/polyhedra/AHD/chart_lg.gif}.\par

You'll find I was ranting a little recently (not for the first time) about the fact that so few mathematicians include any mention of Fuller's Mite, or minimum tetrahedron, in their catalogs of space-fillers.

I continue registering my more generic consternation on Synergeo this afternoon, in the context of discussing nuclear disarmament with other contributors. I write in terms of a "global university" as that's what Spaceship Earth has shaped up to become. Fuller's "final exam time" metaphor, for this time in history, is consistent with this imagery. I also point back to this journal entry from math-teach @ Math Forum, where I explicitly tie my concerns to those of the current administration.