:: flatworms by M.C. Escher ::(geometry of nature)
Some of you have maybe joined our book circle around King of Infinite Space, the well-told story of "the man who saved geometry", Donald Coxeter.
As I was discussing in Pointful Gossip (gossip with a point), Donald had an uneasy relationship with Bucky Fuller, on the one hand okaying the latter's choice to dedicate Synergetics to the master, on the other hand wanting to keep Bucky in his place, as a great man maybe, but not a great mathematician.
Dr. Fuller was a troublemaker for the settled disciplines. He got a lot of notoriety for those domes, but even here, the game was to discredit him if at all possible, deflate the guy's brief for having "the right stuff" (paraphrase).
Bucky was subversive of academic authority because his philosophy ran counter to the conventional wisdom of his day, and within earshot of students. Bucky was a dangerous maverick whose influence needed to be checked, or at least channeled.
In terms of channeling, an original discovery of Bucky's to which Coxeter was willing to give his stamp of approval, was 10 * F * F + 2: the number of balls per each layer of a growing cuboctahedron of balls i.e. a key access point to the CCP and/or face-centered cubic lattice (FCC), what Bucky himself called the "isotropic vector matrix" (with a focus on edges, more like Zome's).
This formula could be proved using high school level math. Geometry could afford a small accolade to the maverick polymath at this juncture.
As Siobahn Roberts tells it:
When a reporter from LIFE magazine called in 1970, Coxeter gave Fuller a somewhat backhanded -- but then accidentally glowing compliment.... Coxeter sent back a letter saying that one equation would be 'a remarkable discovery, justifying Bucky's evident pride,' if only it weren't too good to be true. The next day, Coxeter called: 'On further reflection, I see that it is true.'This very same formula was turning out to have applications in virology, as it explained aspects of the viral capsid, the hard shell containing the RNA coil, relating its protein capsomere counts to the icosahedral numbers.
X-Ray diffraction had newly revealed the geodesic dome pattern in nature's micro-architecture. This was years prior to the discovery of buckminsterfullernene, the naturally occurring carbon cages with similar properties (i.e. five fold rotational symmetry).
However, Fuller himself had a bigger discovery in mind, when he hooked up with Coxeter as a leading light, and that was in connection with his modular dissection work in a related sub-branch of geometry.
Fuller goes into meticulous detail in 950.12, with related Figure, noting how page 71 of Coxeter's Regular Polytopes gives three tetrahedral space-fillers, one being a component of the other two and hence the more primitive.
Fuller emphasizes in Synergetics that this more primitive space-filler is his own discovery, the MITE, whereas Coxeter is merely describing and renaming it, while at the same time omitting all mention of the A & B modules (two cellular automata, both left and right handed). The MITE consists of two As and one B and has a volume of 1/8 in the prefrequency concentric hierarchy.
:: MITEs Cube with
pg. 71 of Regular Polytopes ::
pg. 71 of Regular Polytopes ::
Donald lived long enough to see this priority claim in print, and apparently published no objection to this characterization, which came without accusation of plagiarism or anything so undiplomatic. Fuller was playing some masterful chess here, dedicating the work to this great geometer, a mentor, while upping the ante with some chips of his own.
If 10 * F * F + 2 were a lucky break, then this MITE would seal the deal: Fuller was no charlatan and keeping him safely at bay, pigeon-holed as some renegade architect, would no longer be feasible. Geometers would need to acknowledge him for more than "just domes".
As it happened, academia continued to draw a line in the sand, by refusing to adopt any of Fuller's nomenclature, which, in effect, would be to accredit his discoveries, make them mainstream.
Scientific American had succeeded in cutting 10 * F * F + 2 out of the virus story, by finding an alternative telling, wiring through Michael Goldberg's studies in Japan. The Scientific American later made up for this somewhat, by publishing about Yashushi Kajikawa's synergetics-inspired modular dissections in the five-fold realm (likewise David Koski's focus), but only in its Japanese edition.
As of 2009, the intelligent reading lay public is still safely clueless, not connecting the dots between this Medal of Freedom winning radome architect, so admired by U.S. Marines and Russian intelligence services (and proponent of the global grid), and high caliber geometry breakthroughs outside of dome design, in a realm closer to the cellular automata.
When A New Kind of Science was published, obvious associations were not made, and to this day the MITE is missing from Wolfram's MathWorld on the web, while on Wikipedia, Synergetics is confused with some Springer-Verlag publication by the same title [note: we've done subsequent disambiguation work].
"Obfuscating Fuller" continues to be the name of the game in some circles, some thirty years later. Fortunately for students though, we have this Coffee Shops Network, other venues, for sharing this esoterica. Plus not everyone in academia has been trying to keep the lid on this story.
Arthur Loeb, the MIT crystallographer, dared to speak openly about A&B modules, came forward with the brilliant Amy Edmondson as his protege and author of A Fuller Explanation, seeking validation through Birkhäuser (a respected academic publisher, and subsidiary of Springer-Verlag). This book drew heavy fire from some corners, but helped a lot of die-hard buckaneers better understand their great pirate "phantom captain" (insider joke).
Likewise Ed Applewhite, chief collaborator on Synergetics and former "Deputy Inspector General and Chief of the Inspection Staff" for the CIA, groomed a few proteges, me included, for more diplomatic cockfighting of this kind, plus he amassed lots of "collateral" (aka "ammo") some of which he sent off to the special collection at Stanford.
His chief concern: that academia would get away with bleeping over Fuller's work "with impunity". I'd say there's no chance of that now. These stories are just too good to keep under wraps, plus young scholars need to get on with their studies, can't afford to just dilly-dally. It pays to know your history and your heritage.
