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(1) Buckminster Fuller, Synergetics: The Geometry of Thinking (New York: Macmillan Publishing, 1976), p.30, section 216.03: "Comprehension of conceptual mathematics and the return to modelability are among the most critical factors governing humanity's epochal transition from bumblebee-like self's honey-seeking preoccupation into the realistic prospect of a spontaneously coordinate planetary society." (My italicsA.E.) As stated in the introductory "Note to Readers," quotations from Synergetics will hereafter simply be followed by a numerical section reference in parenthesis. Quotation from Synergetics 2: Further Explorations in the Geometry of Thinking will be followed by the section number and the letter "b."
(2) These statements were included in many of Fuller's lectures. I am quoting from memory and from personal notes made over the years. Hereafter, assume this to be the source if not otherwise referenced.
(5) Sir Arthur Eddington, (1882-1944), English astronomer and physicist.
(6) Quoted from epic videotaped lecture session, recorded in January 1975, in Philadelphia, over a ten-day period. The total length of these lectures, which are part of Fuller's archives, now located in the Buckminster Fuller Institute in Los Angeles, is 43 hours. (Hereafter referred to as "43-hour videotape.")
(7) See e.g. Powers of Ten, by Philip and Phyllis Morrison and The Office of Charles and Ray Eames (New York: Freeman, Scientific American Books, 1982).
(8) See Appendix C for biographical references.
(9) Arthur L. Loeb, Space Structures (Reading, MA: Addison-Wesley, Advanced Book Program, 1976), p. xvii.
(10) F = GMm/r²: The attractive force due to gravity between two objects is equal to the "gravitational constant" (G) times the product of their masses and divided by the separation-distance raised to the second power.
(1) See Glossary, "whole number."
(2) See Appendix C.
(3) It must be noted that tetrahedra cannot fit together face to face to form the larger tetrahedra shown, but rather must alternate with octahedra, as will be explored at length in later chapters (8, 9, 10, 12, 13). The values given for the volume of each tetrahedron are based on using a unit-length tetrahedron as one unit of volume, in exactly the same manner that a unit-length cube is conventionally employed. Despite the tetrahedron's inability to fill space, the relative volumes of tetrahedra of increasing size are identical to those exhibited by cubes of increasing size.
(1) See Loeb's preface to Fuller's Synergetics: "In rejecting the predigested, Buckminster Fuller has had to discover the world by himself" (p. xv).
(2) Buckminster Fuller, Intuition (Garden City, N.Y.: Anchor Press/ Doubleday, 1973), p.39.
(4) 43-hour videotape.
(5) Quoted several times by Fuller, in 43-hour videotape.
(1) "Contribution to Synergetics" is a 52-page supplement by Arthur Loeb included in Fuller's Synergetics, pp. 821-876.
(2) See Glossary, "-valent".
(3) Arthur Loeb, Space Structures, Chapter 6.
(4) Loeb, p.11.
(5) Chapter 5 explains why "structural system" implies triangulated system.
(6) Loeb, p.40.
(7) Loeb, p.63.
(1) The best renditions of the precession sequence are found in videotaped lectures, because Fuller's gestures are as important as his words; the 43-hour videotape contains an especially good version. Original written document was published in Fortune, May, 1940. Fuller wrote the two-page piece in response to a request (or challenge!) from the Sperry Gyroscope company.
(2) See Glossary, "Greater Intellectual Integrity of Universe."
(3) Loeb, pp.29-30. The derivation of the equation 3V - E = 6, which is thought to have originated with Maxwell, can be derived from degree-of-freedom analysis of molecular spectra, and was applied by Loeb to polyhedral systems.
(1) See Glossary, "special case."
(2) 43-hour videotape.
(3) Kenner, pp.129-131.
(4) Appendix A, "chord factors."
(1) William Morris, Editor, The American Heritage Dictionary (New York: American Heritage Publishing Co., 1970), p.442.
(2) 43-hour videotape.
(3) It must be noted however that if the cuboctahedron is sliced in half and one "hemisphere" is rotated 60 degrees with respect to the other, the resulting "twist cuboctahedron" (Fig. 7-6b) maintains the radial-circumferential equivalence. With its asymmetrical arrangement of faces, however, this shape is not similarly suited to model equilibrium. The desired balance of vectors is therefore achieved through the straight cuboctahedron (Fig. 7-6a).
(4) Amy Edmondson, "The Minimal Tensegrity Wheel," Thesis for completion of B.A. degree requirements, Harvard University, 1980. Refer to VES Teaching Collection, Carpenter Center, Harvard University.
(5)Fuller's use of the term "degrees of freedom" must be distinguished from the conventional treatment of the subject which specifies that a rigid body has six degrees of freedom (three translational and three rotational) which have two directions each, thus requiring twelve unidirectional constraints.
(1) N. J. A. Sloane, "The Packing of Spheres," Scientific American (January 1984), p.116.
(2) Cubic packing thus corresponds to the VE (cuboctahedron), while hexagonal packing corresponds to the "twist" VE. See Footnote 3 in Chapter 7 for the difference.
(3) I. Rayment, T. S. Baker, D. L. D. Caspar, and W. T. Murakami, "Polyoma Virus Structure at 22.5 Å Resolution," Nature, 295 (14 January 1982), p.110-115.
(4) Donald Caspar and Aaron Klug, "Physical Principles in the Construction of Regular Viruses," Cold Spring Harbor Symposia on Quantitative Biology, XXVII (1963), pp.1-3.
(1) See Glossary, "supplementary angles."
(2) A. L. Loeb, "Contribution to Synergetics," in Synergetics: The Geometry of Thinking (New York: Macmillan, 1976), pp. 860-875; "A Systematic Survey of Cubic Crystal Structure,"J. Solid State Chemistry, 1 (1970), pp. 237-267.
(3) 43-hour videotape.
(1) A. L. Loeb, "Contribution to Synergetics," pp.821-876. Volume ratios are derived by comparing geometrically similar polyhedra so that "shape constants" cancel out of the formulae. Credit for this approach belongs to Loeb and Pearsall, and it must be noted that this section (H) of Contribution to Synergetics is adapted from an article by Loeb in The Mathematics Teacher.
(2) Loeb, p.836.
(3) Loeb, p.832.
(1) Loeb, p.829.
(2) A. L. Loeb, "Addendum to Contribution to Synergetics," published in Synergetics 2: Further Explorations in the Geometry of Thinking, (New York: Macmillan Publishing, 1979), pp.473-476.
(3) Dennis Dreher of Bethel, Maine designed an onmidirectional hinging joint, which allows the necessary twisting of adjacent VE triangles in the jitterbug. This joint can be seen in Photograph 11-1.
(1) " Bohr, Niels" Encyclopaedia Britannica, 1971.
(2) lsaac Asimov, Asimov's Guide to Science (New York: Basic Books,1972), pp.334-5.
(3) Loeb, "Contribution to Synergetics," pp.836-847.
(4) Loeb, "Coda," Space Structures, pp.147-162.
(1) Discovered independently by Loeb and called "moduledra."
(2) Loeb, "Contribution to Synergetics" pp.847-855;
(3) Space Structures, "Coda," pp.147-162.
(1) See Appendix E, "right isosceles triangles."
(2) Fuller, Synergetics, Figure 455.20, p.178.
(3) Fuller, Synergetics, Figure 458.12, p.189.
(1) Buckminster Fuller, Critical Path (New York: St. Martins Press, 1981), p.13.
(2) This fact was demonstrated in the "tensegrity" bicycle wheel experiment mentioned in Chapter 7. In tests to determine the minimum number of tension (Dacron string) "spokes" required to stabilize its hub, the wheel's eventual structural failure originated with buckling of its rim. Long arc spans (which were a consequence of the low number of radial spokes) were too thin to withstand the compression force created by loading the hub and transmitted to the rim through the tension spokes. The usual arrangement, which consists of a fairly large number of spokes (36 or more), therefore turns out to be advantageous; despite the fact that there are many more spokes than necessary to restrain the hub, this large number does serve to subdivide the otherwise vulnerable arc segments of the compression-element rim. See also Edmondson, "The Minimal Tensegrity Wheel."
(3) Recalling the principle of angular topology in Chapter 6, we also know that the "angular takeout" is 720 degrees. That is, if we subtract the sum of the surface angles at each vertex of a convex polyhedron from 360 degrees, the sum of all these differences will be exactly 720 degrees.
(4) Ernst Haeckel, Art Forms in Nature (New York: Dover, 1974), plate 1 (Radiolaria) and plate 5 (calcareous sponges).
(6) Note that the engineering term "pure axial force" is thus a convenient simplification, which is effective in terms of structural analysis, rather than an accurate scientific description. A strut which is to carry either axial compression or tension can be significantly lighter than one which is subject to bending or torque.
(7) Buckminster Fuller, "Tensegrity," Creative Science and Technology, IV, No.3 (January-February, 1981), p.11.
(8) Synergetics, p.354.
(10) This very general summary is based on Dr. Donald Ingber's paper, "Cells as Tensegrity Structures: Architectural Regulation of Histodifferentiation by Physical Forces Transduced over Basement Membrane," published as a chapter in Gene Expression During Normal and Malignant Differentiation, (L. C. Anderson, C. G. Gahrnberg, and P. Ekblom, eds.; Orlando, Fla.: Academic Press, 1985), pp.13-22.
(11) Taken from a letter from Ingber to Buckminster Fuller, April 5, 1983.
(1) This approach characterizes Fuller's "World Game" studies. See Appendix C for sources of information about this research.
(2) Buckminster Fuller and Shoji Sadao, Cartographers; Copyright R. Buckminster Fuller, 1954. "Dymaxion Map" is a trademark of the Buckminster Fuller Institute.
(3) Fuller, Critical Path, p. xxiii.
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