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Sphere Packing and Capsomere Counts

Sometime in the 1960s virologists first approached Fuller with 
a puzzle: new x-ray diffraction patterns revealed that many 
viruses were similar in structure to geodesic domes. The 
protein sheaths housing the RNA were 5-fold symmetric, 
icosahedral structures, and the number of capsomeres numbered 
12, 42, 92... Virologists wondered whether Fuller recognized 
these numbers from his explorations with geodesic domes. He 
answered with his sphere-packing formula for the number of 
spheres in the outer layer of an expanding cuboctahedral 
packing: 10F^2+2, where F=1,2,3.. and is the number of 
intervals between sphere centers along an outermost edge. That 
Fuller's investigations were proving relevant, and that 10F^2+2 
in particular was matching empirical data, made the front page 
of the New York Herald Tribune.

Given that the jitterbug transformation turns a cuboctahedral 
layer into an icosahedral one, the geometric basis for the 
number of viral capsomeres was clear. However, other common 
capsomere counts were not accounted for by the cuboctahedral 
numbers, as when the capsomeres 'wrap' around an icosahedron in 
a skewed pattern. Virologists Casper and Klug derived a formula 
that fit even more of the data, which they subsequently 
connected back to the geometric investigations of Arthur 
Goldberg, a mathematician. 

When Dr. Robert Horne wrote a groundbreaking article for 
Scientific American about these matters, he chose not to credit 
Fuller for having discovered anything new, although students of 
synergetics (including Arthur Goldberg) realized at the time 
that Fuller's pioneering investigations, like Goldberg's, also 
derived from more generalized principles -- in Fuller's case 
these were generalizations about sphere packing in a variety of 
conformations, not just the cuboctahedral. 

Armed with Goldberg's 1922 paper, virologists felt they were 
off the hook vis-a-vis hard-wiring anything from Fuller's 
synergetic geometry into their discipline. Many felt relieved 
because Fuller, something of a maverick (regarded as such even 
by architects), and without the academic credentials expected 
of a scientist, might only serve to undermine the status quo if 
admitted into the literature as a recognized contributor in 
such professional circles. The idea that virology might also 
serve as a useful access point leading to a more generic 
synergetics- informed curriculum, linking in turn to many other 
disciplines, at the time seemed quasi-irrelevant, if not 
downright alien, to these rather narrowly focused specialists. 

With the subsequent discovery of a new allotrope of carbon in 
the 1980s, again showing five-fold symmetric icosaspherical 
geometry (variably frequenced hexa-pents), the opportunity to 
provide a useful curriculum access point to synergetics again 
arose. This time, by naming the newly discovered C60 molecule 
buckminsterfullerene (and other frequencies fullerenes) its 
discoverers helped forge a chemistry-synergetics link (see E.J. 
Applewhite's The Naming of Buckminsterfullerene for more on 
this story). 

The Scientific American has subsequently published an article 
about synergetics and five-fold symmetry in its Japanese 
edition (dinosaurs on the cover), focusing on Yasushi 
Kajikawa's modular decompositions of well-known five-fold 
symmetric shapes (e.g. the icosahedron). It's cover story in 
January, 1998 was also devoted to a central synergetics theme, 
tensegrity (as a principle for understanding cell structure), 
which likewise belongs to Ken Snelson. 

More relevant to the this virus thread than tensegrity, 
however, was a story in the previous issue (December, 1997) 
about metal atoms clustering in icosahedral and cuboctahedral 
conformations -- no link to Fuller's groundbreaking 
sphere-packing studies was included, even in the notes section. 

For further reading: 

A Case of Deja Vu in The Invention Behind the Inventions: 
Synergetics in the 1990s 

Memo to Ivars Peterson re Fuller & viruses in his Jungles of 
Randomness 

Review of Hugh Aldersey-Williams' chapter on Fuller in his book 
The Most Beautiful Molecule about buckminsterfullerene 

Some further discussion including w/ Hugh Aldersey-Williams 

Kroto on Fuller and the naming of buckminsterfullerene 

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