Two typos fixed. Plus "Arthur Goldberg" changed to "[Michael] Goldberg" (a mistake I fixed on Microarchitecture of the Virus as well -- thanks to Gregory J. Morgan for pointing out my error). Italics added. Original in archives.

Return to Memo to Ivars Peterson re 1998 Math Makeover

Date: Wed, 26 Nov 1997 12:01:48 -0800
From: Kirby Urner <>
Subject: 'Jungles of Randomness' re Fuller & viruses

                   November 26, 1997

TO: Ivars Peterson, Science News, science writer
FR: Kirby Urner, 4D Solutions, curriculum writer
RE: 'Jungles of Randomness' re Fuller & viruses

Dear Sir --

I've just been browsing your Jungles of Randomness, the 
section on viruses and R. Buckminster Fuller, and wanted to 
relay my own understanding of the events in question.

First, I congratulate you for bringing Fuller back into the 
narrative.  After his contribution got some front page press in 
the New York Herald Tribune, his name got dropped from 
subsequent accounts, ostensibly because Dr. Goldberg's 
mathematics had priority, but more accurately because Fuller 
was too much the maverick and his thinking too alien for the 
times, and therefore including him with tie-backs to 
Synergetics seemed more problematic than just wiring him 
out -- something of a relief that Coxeter helped uncover
that Goldberg stuff I'm sure.  

This decision to wire Fuller out was taken by Scientific American
which ran that article by Dr. Robert Horne, despite archived 
correspondence between Fuller and Horne in which Fuller takes 
great pains to show that his and Goldberg's thinking were not 
along the same lines at all i.e. both deserved mention in Horne's 
account (but by this time it was too late -- the damage had already 
been done).

So to see Fuller wired back into the narrative is heartening, 
as it shows we've overcome at least some of the initial 
prejudice.  This is an especially positive development in light 
of the hatchet job re Fuller's Synergetics we find in Hugh 
Aldersey-Williams' The Most Beautiful Molecule, wherein he 
implies in the conclusion of his chapter "A Fuller View" that 
we should feel no more compunctions about bleeping over 
synergetics in the fullerene chapter than we did in the virus 
chapter -- the fullerene chapter being a deja vu experience in 
a lot of ways, with science writers steadfastly ignoring the 
option to tie back to synergetics in any coherent manner, 
preferring to stick to safely superficial chatter about the 
geodesic domes.

What I found confusing in your account is you seem to suggest 
it was these 'magical' T numbers and the expression 10T + 2 
which Fuller was going on about in that quote of his you 
provide, whereas he was actually writing about 10F^2+2, which 
expression gives the number of spheres in successive shells of 
a cuboctahedron. If you dense-pack 12 spheres around a nuclear 
sphere, that's F=1.  A second layer retains the cuboctahedral 
conformation (as do all subsequent layers) and contains 42, 92, 
162 spheres (F=2,3,4...) and so on.  

What Fuller realized, and made use of when computing his domes, 
is that the cuboctahedron's shell count expression works 
equally well for the icosahedron, because of a transformation 
of the cuboctahedron into the icosahedron which does not alter 
the count, but merely the arrangement of the spheres in the 
outer layer (a transformation he calls the Jitterbug, though 
it was known to other mathematicians previously, at least in 
some aspects -- no others that in my awareness made it so 
central to their thinking as Fuller's synergetics did).

I think by keeping Fuller's precise contribution in view, while 
giving credit to others for their own respective inputs to the 
virus thing, we keep the link back to synergetics unobstructed 
by irrelevancies -- makes things easier from a pedagogical 
point of view.  Fuller was on his own track with the sphere 
packing investigations, doing something quite different from 
[Michael] Goldberg (what he was trying to communicate to Horne), 
and it makes sense that we tell the story with this in mind, 
neither giving Fuller more nor less than his due vis-a-vis the 
virus chapter.

In the case of fullerenes, my line is that the name is apt, 
even though Leonardo and others knew of the truncated icosa, 
because fullerenes come in multiple frequencies, just like the 
domes (and cuboctahedral sphere packings), and because we have 
correspondence on file showing Fuller was on the lookout for 
the fullerene hexa-pent pattern in chemistry in particular.[1]  
This is not to say that Fuller specifically anticipated 
fullerene (others before Kroto, Smalley et al did that, 
including in Japan), but that 'fullerene' makes more sense 
than 'Leonardoene' or 'Archimedesene' for example, for the 
reasons just given.

Thanks for helping to make our math curriculum a fun and 
interesting one for ourselves and those who come after us.  
I'm looking forward to studying more of your Jungles of 
Randomness plus other writings of yours, both past and 

Relevant citation:
(which now links to this memo).

[1] "...I am greatly intrigued by your discovery of the two
purines whose elementary components are hexagons and pentagons
and the pyramidines which have a hexagonal configuration only...
the pentagon occurs only as a consequence of its being a 
component of a polyhedral system."

Fuller to Petr Jandacek, Los Alamos, NM, 15 Dec 76 as included 
in the Synergetics Dictionary, compiled and edited by E.J. 
Applewhite (Garland Press, 1986), Vol 2., pg. 219, card 8.

Synergetics on the Web
maintained by Kirby Urner