1. Pure buckminsterfullerene forms a more disorderly hexagonal closest packing of molecules (hcp). Hexagonal packing, like face centered cubic packing, consists of layers of spheres sandwiched together, each layer appearing like pool balls in a triangular rack. In the face-centered arrangement, spheres are consistently located above the voids two levels below. Hexagonal packings do not follow this rule and are therefore more random. They have the same density, however.

2. Assuming a sought-for polynomial of the third degree N = AF3+ BF2+ CF + D, one may substitute corresponding values of F and N, e.g. the (F,N) pairs (1, 13), (2, 55), (3, 147) and (4, 309). The resulting set of four linear equations may then be solved to obtain the coefficients (A,B,C,D).

3. S. Bjornholm, "Clusters, condensed matter in embryonic form," (Contemporary Physics, 1990, Vol 31. No. 5), pg. 313.

4. An isomatrix of higher frequency will accommodate both the bcp and XYZ lattices.

5. Fuller sets his first nucleated cube at 364 around one in Synergetics (415.12) but remembers to include the 63 sphere cube in Synergetics 2. Both are developed by adding 1/8 octahedra (made of spheres) to the cuboctahedron's triangular faces, i.e. 63 = (1+12 +42) + 8 and 364 =(12+42+92+162) + 8 x 7 (7 spheres = a 1/8 octahedron, added to each triangular facet of the cuboctahedron).

6. H.S.M. Coxeter (University of Toronto), "Virus Macromolecules and Geodesic Domes," reprinted from A Spectrum of Mathematics, essays presented to H.G. Forder, edited by J.C. Butcher (Aukland University Press, 1967) pg. 98 -107.

7. This one-sided treatment is evident in the following invented dialog: "'Buckminster Fuller? Wasn't he an architect?' 'That's right. But he took a lot of inspiration from mathematics. He designed geodesic domes - spheres built up from triangular patterns - on the same principles as the Goldberg-Casper-Klug virus shapes." From Ian Stewart's, "Build your own virus " in Game Set and Math (Basil Blackwell).

8. The author wishes to express his thanks to Dr. Robert Whetten and his associates for sharing about their discovery of this structure, and other information about fullerenes, with those of us meeting in his UCLA office on 8/19/91 after the Synergetics Workshop with Yasushi Kajikawa. Dr. Whetten has also worked with embryonic clusters.

9. Synergetics is not designed to prohibit or preclude our use of XYZ reference frames, which may be modeled with their origins at the centers of 4D octahedra (octahedra may be developed from tetrahedra by truncation). We may also employ linear transformations for converting Cartesian cubes into rhombohedra and vice versa, rhombohedra being the "fundamental parallelotopes" of the fcc lattice, thereby allowing all integral coordinates (a,b,c) to attach to all the centers of closest packed spheres. Six prime vectors, and their six 180 degree counterparts, pointing from a sphere center to all 12 surrounding sphere centers, may then be defined using combinations of 1s and 0s for (a,b,c). Andrew Frank, a former BFI volunteer, supplied some of the insights leading to this construction.

10. The other Voronoi cell that appears in disorderly hexagonal closest packings is a 12-sided polyhedron with trapezoidal faces. It has the same volume. Keith Critchlow calls this shape a "twist rhombic dodecahedron" in his Order and Space (New York: Thames and Hudson, 1969), Appendix 1.

11. The synergetics constant for volumes (S3) turns cubic units into tetrahedral ones. It is derived from the cube with a diagonal of 2, which in the XYZ system has a volume of 23, but a volume of precisely 3 in synergetics (isomatrix edges have a Cartesian length of 2, by convention). S3 = root(9/8) = 1.0606602 (approx.)

12. Barry Cipra, "Music of the Spheres," Science, Vol 251 pg. 1028.

13. The vector equilibrium (VE) is actually formed by 8 edge-bonded tetrahedra gathered around a common central vertex, which gives 24 radial vectors in addition to the 24 circumferential vectors.

14. John Polkinghorne, Science and Creation (Boston: New Science Library, 1989), pp. 35-6.

15. Rolf Landaur, "Wanted: a physically possible theory of physics," IEEE Spectrum, September, 1967, pg. 105.

16. Elizabeth Corcoran, "Computing Reality," Scientific American, January, 1991, p. 107.

17. The reason NASA would fund such a study is to better understand how structures might be folded into a payload and then unfolded in space - see 2:795.05.

18. For a more comprehensive inventory of jitterbug-style transformations and treatment of their computerization see: H.F. Verheyen, "The Complete Set of Jitterbug Transformations and the Analysis of their Motion," Computers Math. Applic. (Vol 17, No. 1-3, pp. 203-250, 1989), reprints from Pergamon Press.

19. E.J. Applewhite, Synergetics Dictionary (New York: Garland Publishing, Inc., 1986); Vol. 1, p. 326, cards 1-2 (Computer as Antibody).

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