A View from Abroad

to Poly list, by DK

Dave Koski and I were on the cell again tonight.

I ranted a bit.

Two points:

(a) Not including much if any Bucky in K-12, where some of his contributions clearly belong, in geometry, history, literature, is a clear signal to the world that some ulterior motive trumps obvious relevance. Study what America teaches its young. What diplomats or politicians put out, is far less of an indicator than what goes down in K-12. We're talking about a positive, hopeful futurist whom many a world leader embraced. To actively exclude this information is a forceful declaration of a commitment to go in a different direction, a commitment renewed daily in schools across the country. Just mentioning geodesic domes in a geometry textbook sidebar is hardly saying much. Where are the whole number volume ratios?

(b) Not making more hay around the minimum space-filler, so-called because it's a tetrahedron without left or right handedness, seems bizarre. Math World doesn't mention it on its page on space-filling, with or without Fuller's terminology (he called it a Mite). True, spatial geometry is somewhat esoteric to begin with, but a minimum space-filler... even K-12 should have room for such a thing.

Here's from Math World. Notice how tetrahedra are not mentioned at all in the paragraph below, even amidst an attempt to be exhaustive:

In the period 1974-1980, Michael Goldberg attempted to exhaustively catalog space-filling polyhedra. According to Goldberg, there are 27 distinct space-filling hexahedra, covering all of the 7 hexahedra except the pentagonal pyramid. Of the 34 heptahedra, 16 are space-fillers, which can fill space in at least 56 distinct ways. Octahedra can fill space in at least 49 different ways. In pre-1980 papers, there are forty 11-hedra, sixteen dodecahedra, four 13-hedra, eight 14-hedra, no 15-hedra, one 16-hedron originally discovered by Föppl (Grünbaum and Shephard 1980; Wells 1991, p. 234), two 17-hedra, one 18-hedron, six icosahedra, two 21-hedra, five 22-hedra, two 23-hedra, one 24-hedron, and a believed maximal 26-hedron. In 1980, P. Engel (Wells 1991, pp. 234-235) then found a total of 172 more space-fillers of 17 to 38 faces, and more space-fillers have been found subsequently.

What you find instead is that Aristotle was wrong: tetrahedra do not fill space. "Although even Aristotle himself proclaimed in his work On the Heavens that the tetrahedron fills space, it in fact does not."

The more nuanced dismissals remember to say he said "regular tetrahedra," however this oft repeated factoid just goes to obscure the fact identical non-regular tetrahedra do fill space.

Mites face-bond to make two more space-filling tetrahedra (called Sytes by Fuller), though these are not as primitive, given their decomposition into component identical space-fillers.

Dave is looking at some Catalans these days, duals to the Archimedeans. The cuboctahedron and icosidodecahedron have the rhombic dodecahedron and rhombic triacontahedron as their respective duals. He's studying the dual concept more generally, going off some of the data in the Robert Williams compendium.

These latter are combinations of duals themselves: the rhombic dodecahedron is a combination of the cube and octahedron; the rhombic triacontahedron is a combination of the icosahedron and pentagonal dodecahedron.

I doubt many people will read a lot of Synergetics with gusto. That's a hard text. Glenn routinely expresses his frustration with it, as do I. Not everyone reads philosophy, period.

Related post to Math Forum: Recap: Letter to Arne Duncan etc. (March 31, 2010)