I'm borrowing a couple technical tomes from the Flextegrity company archives. Extended Icosahedral Structures, edited by Marko V. Jaric and Denis Gratias (Academic Press, 1989), and Rene Motro's Tensegrity, Structural Systems for the Future (Kogan Page Science, 2003).
Last night we coasted along at the high school level, sharing elementary geometry. My write-up argues that our literature base provides a more secure grounding for tackling such science readings, however the proof is in the pudding. What do I myself, for example, comprehend from these readings? In a way I have an unfair advantage though, as I've chosen from a neighboring domain.
Linus Pauling's contribution in the first tome is on the skeptical side, advances the hypothesis, backed up with experimental evidence, that "icosahedral twinning of cubic crystals" explains a lot of what we were seeing in the raw intelligence.
Background: X-ray diffraction and electron micrographs are disclosing more information about crystalline structures and the readouts provide startling evidence of 5-fold symmetry (icosahedra, rhombic triacontahedra etc.) for some of the metallic alloys (aluminum with magnesium is especially implicated but we've seen it before elsewhere for sure). Given icosahedra don't pack to fill space, there'd never been a strong expectation that crystals, which do extend omni-directionally, would have such a symmetry. Scientific instruments were proving otherwise and the heat was on to come up with explanations. The papers were flying thick and fast (1980s).
Pauling takes the conservative line that individual components might (obviously do) have icosahedral symmetry, but if you zoom back far enough, you'll discover these mega-groupings that obey known rules of the non-5-fold kind. Buckminsterfullerenes packed in an fcc, or flextegrity itself would be good examples, but Pauling's groupings are more complex. Mg32(An, Al)49, for example, is analyzed in Figure 1 as being a bcc, but inside you have these nested 5-fold clusters (icosahedron, dodecahedron, truncated icosahedron). Figure 4 shows the icosahedron's relationship with a cube (like Sam's tensegrity). The 82o-atom primitive cubic crystal, consisting of clusters of 104 (Figure 3), provides a great climax for this paper. There's lots about "unit cubes" which a knee-jerk buckaneer might worry about. Think of a 2F cube and stop worrying. Then watch the chemistry channel around 30 angstroms (on your frequency spectrum dial).
This article is a great compendium of already-found structures, going back to the 1920s. It's heartening to see all this knowledge in one place, makes me wanna look at the VRML worlds and toons, cines, claymations (Portland Knowledge Lab stuff).
On the other hand, in a higher dimension you can have an icosahedron of 20 regular tetrahedra, other magic tricks impermissible in ordinary dimensions. Sadoc and Mosseri do a great job on this, starting on page 163. Check out Figure 1 from Pearce (I've just been citing him), and the impossibility of having 20 regular tetrahedra make a regular icosahedron on page 169 (Figure 2).
Go from R3 to S3 and problem solved (!), with the caveat (page 173) that "the template itself lives in an unphysical space (S3)." We see Coxeter's name a lot, plus this number sequence 1, 13, 45, 117, .... (not finding that in OEIS though).
Riffing off the above... Another explanation for the readouts has been the aperiodic one: whereas "a five-fold-symmetric lattice" is something of an oxymoron, aperiodicity is compatible with quasi-uniform density and plenty of structural conformity. "Distributed individuality", as manifest by the Penrose tiles, analogous spatial modules, results in multiple unique 3D tessellations of the same space. There's much more one could say (and Pauling does). The number of Barlow packings (CCP, HCP both subtypes) is likewise huge (if finite) and those all have density 0.74 -- so let that be a reminder of the geometry, if not the chemistry. There's an essay on glass I still need to look at (Figure 2 is revealing).
Regarding the second title, I'm another player with a personal angle on the Fuller-Snelson rift, which R. Motro spends some time on, the consummate diplomat. All I'm gonna add at the moment is I think the patent system is somewhat cruelly exclusive, although I can't blame those who rush into it, but there's a Zeitgeist phenomenon we should credit as well. These ideas are in the wind and not everyone is within shouting distance of a patent attorney, might be too busy starving in India someplace. For a few minutes, imagine a patent office with Holy Ghost on every single filing. No need for a patent office then. My point exactly. Back to reality: I celebrate human genius in all its forms and am happy to count my friends as true blessings.
OK, some source code from Patrick to look at (buzzbot). Gotta boogey.
Followup: Dave Koski, upon reading this blog post, sent me the following update: "Not sure if the Pauling institute has a gun to your head or if the flextegrity dynamic is getting to you. Pauling was long ago dismissed with his assertion that quasi-crystalline structures was a trompe l'oiel of being icosahedral in nature."