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Appendix A: Trigonometric Calculations Appendix A: Trigonometric Calculations Finding Interior Angles of Polygons Pentagon 360 0 -;- 5 = f3 72 0 = f3 f3 + 2(ta) = 1800 a = 1800 - f3 a = 108 0 Hexagon 360 0 -;- 6 = f3 = 60 0 f3 + a = 180 0 a = 120 0 ...... 4 ...... I /' """"f3 ....,I I I Heptagon 360 0 -;- 7 = f3 = 51.43 0 a = 1800 - 51.43 0 a = 128.570 272 A Fuller Explanation Chord Factors ~ Polyhedral ...--------' edge Radius Central angle (corresponds to a polyhedral edge) 1./ sin 1. a = 2.... 2 R R sin ~a = ~/ 2R sin ~a = / R [2 sin ~a 1 = / R X [chord factor] = / For every central angle a, there is a numerical value called a "chord factor," generated by the equation [2 sin ~a], which can be multiplied by the desired radius of a polyhedral system to calculate the exact length of the chord (edge) subtended by a. Appendix B: Volume Calculations for Three Prime Structural Systems Tetrahedron Using traditional formula: v = t AbH, where A b = area of base and H = height. (1) Area of base: Bh 1 x sin 60° Ab= 2 = 2 1(0.8660) 2 = 0.4330. (2) Height of tetrahedron: H 2 + (0.5774)2 = 12, H~ H = 0.8165. ~ 0.5774 (3) Volume: Ab X H V = = 0.11785. 3 ~74 ~ 0.5 274 A Fuller Explanation Octahedron Two square-based pyramids: AbH ] [-3- X 2 = V. (1) Area of base: " // " / ""x / = 1 / "- / "- / " / " H 1110~. 8~6~60 \.'-- -v.---' o.s (2) Height of pyramid: H2 + (0.5)2 = (0.8660f, H = 0.7071. (3) Volume of half octahedron: (0 .7071)(1) ---- = 0.2357. 3 (4) Volume of octahedron: 2 X 0.2357 = 0.47140. Icosahedron Twenty pyramids with equilateral bases and side-edge length equal to the icosahedron radius. Appendix B: Volume Calculations for Three Prime Structural Systems 275 (1) Pentagonal cross-section: x = 2[sin 54 0 X 1] = 1.61803 / (2) Radius R: (2R)2 = (1.61803)2 + (1)2, R = 0.95106. 0.5774 ~7 4 0.5 (3) Pyramid height H: (0.95106)2 = (.5774)2 + H2 , H = 0.7557. 276 A Fuller Explanation (4) Area of pyramid base is equal to area of base of regular tetrahedron: 1(0.866) Ab = = 0.4330. 2 (5) Pyramid volume: t(AbH) = 0.1091. (6) 1Volurne of icosahedron: 20 X (0.1091) = 2.1817. Appendix C: Sources of Additional Information All books, unless noted by an asterisk, are available from the Buckminster Fuller Institute, 1743 S. La Cienega Boulevard, Los Angeles, CA 90035. A number of versions of the Dymaxion Map can also be obtained; write the Institute for a complete list. General Some of the other books by R. Buckminster Fuller (in addition to those listed in the bibliography) recommended for further reading: Operating Manual for Spaceship Earth (Carbondale, Ill.: Southern Illinois University Press, 1969). No More Secondhand God (Carbondale, Ill.: Southern Illinois University Press, 1969). Poems and essays. Biographical Material Robert Snyder, Buckminster Fuller: An Autographical Monologue Scenario, (New York: St. Martin's Press, 1980). A great introduction, written by Fuller's son-in-law using Fuller's own words, with hundreds of photographs from the Fuller archives. E. J. Applewhite, Cosmic Fishing (New York: Macmillan, 1977). Especially good for its description of the collaboration between Bucky and his long-time friend E. J. Applewhite in the process of writing Synergetics. Hugh Kenner, BUCKY: A Guided Tour of Buckminster Fuller (New York: William Morrow, 1973). Geodesic Mathematics Hugh Kenner, Geodesic Math: And How to Use It (Berkeley, University of California Press, 1976). 278 A Fuller Explanation World Game "50 Years of Design Science Revolution and the World Game," historical documentation (articles, clippings) with commentary by Fuller, 1969. Available only through the Buckminster Fuller Institute. Medard Gabel, Earth, Energy and Everyone (Garden City, N.Y.: Doubleday, 1975). Medard Gabel, HO-ping: Food For Everyone (Garden City, N.Y.: Doubleday, 1976). Write for newsletter, information about seminars and workshops, and list of available material to The World Game, University City Science Center, 3508 Market Street #214, Philadelphia, PA 19104. Appendix D: Special Properties of the Tetrahedron (1) Minimum system: the tetrahedron is the first case of insideness and outsideness. (2) The regular tetrahedron fits inside the cube, with its edges providing the diagonals across the cube's six faces, and thereby supplying the six supporting struts needed to stabilize the otherwise unstable cube. Furthermore, two intersecting regular tetrahedra outline all eight vertices of the cube. (3) The tetrahedron is unique in being its own dual. (4) The six edges of the regular tetrahedron are parallel to the six intersecting vectors that define the vector equilibrium. (5) Similarly, the four faces of the regular tetrahedron are the same four planes of symmetry inherent in the vector equilibrium and in cubic closepacking of spheres. The tetrahedron is thus at the root of an ornnisymmetrical space-filling vector matrix, or isotropic vector matrix. (6) When the volume of a tetrahedron is specified as one unit, other ordered polyhedra are found to have precise whole-number volume ratios, as opposed to the cumbersome and often irrational quantities generated by employing the cube as the unit of volume. Furthermore, the tetrahedron has the most surface area per unit of volume. (7) Of all polyhedra, the tetrahedron has the greatest resistance to an applied load. It is the only system that cannot "dimple"; reacting to an external force, a tetrahedron must either remain unchanged or tum completely "inside out." (8) The surface angles of the tetrahedron add up to 720 degrees, which is the" angular takeout" inherent in all closed systems. (9) The tetrahedron is the starting point, or "whole system," in Fuller's "Cosmic Hierarchy," and as such contains the axes of symmetry that characterize all the polyhedra of the isotropic vector matrix, or face-centered cubic symmetry in crystallography. (10) Packing spheres together requires a minimum of four balls, to produce a stable arrangment, automatically forming a regular tetrahedron. The centers of the four spheres define the tetrahedral vertices. In Fuller's words. "four balls lock." 280 A Fuller Explanation (11) It has been demonstrated that many unstable polyhedra can be folded into tetrahedra, as in the jitterbug transformation. (12) Fuller refers to the six edges of a tetrahedron as one "quantum" of structure, because the number of edges in regular, semi regular, and high­ frequency geodesic polyhedra is always a multiple of six. Appendix E: Glossary Includes specific terminology coined and/or used in an unusual way by Buckminster Fuller (in italics) as well as terms from conventional mathematics and geometry that may be unfamiliar. A-module: Asymmetrical tetrahedron, which encompasses one twenty-fourth of the regular tetrahedron. acute angle: Angle less than 90 degrees. angle: Formed by two lines (or vectors) diverging from a common crossing. An angle is necessarily independent of size. angular topology: Fuller's term, intended to describe a principal function of synergetics, that is, description of all structure and pattern by variation of only two variables: angle and frequency. (See "frequency.") arc: A segment of a curve. B-module: Asymmetrical tetrahedron, which is the result of subtracting an A module from one forty-eighth of the regular octahedron. complementarity, or inherent complementarity of Uniwrse: Necessary coexistence, inseparable pairs. Fuller emphasizes that Universe consists of complementary teams, such as concave-convex, tension-compression, positive-negative, and male-female, and concludes that "unity is inherently plural." concave: Curved toward the observer, such as the surface of a sphere, or other enclosure, as seen from the inside. convex: Curved away from the observer, such as the surface of a sphere as seen from the outside. coupler: Semisymmetrical octahedron, which consists of sixteen "A­ Modules" and eight "B-Modules," or eight "Mites." degrees of freedom: This term is used by Fuller to mean the number of independent forces necessary to completely restrain a body in space, and by Loeb in reference to the overall stability of systems. design: Deliberate ordering of components. design science: See Chapter 16. dimension: See Chapter 6. Used by Fuller to include spatial extent, orders of complexity, and distinct facets of symmetry. dimpling: Yielding inwardly to produce local indent in structural system, caving-in. 282 A Fuller Explanation Dymaxion: Fuller's trademark word, created by Marshall Fields Department Store in the late 1920s, for use in promotion of exhibit featuring Fuller's revolutionary house design. Also used by Fuller to mean "doing more with less." edge: Geometry term: connection between points (or vertices) and boundary between two faces, line. Fuller prefers to substitute" vector" or else refer to the specific material of construction, such as toothpick or straw. energy event: Fuller's substitute for geometric term" point." Also descriptive term for natural phenomena, discrete constituents of Universe. Replaces "obsolete" vocabulary such as "solid," "point," "thing," etc. ephemeralization: Doing more with less, via design science and technological invention. face: Geometric term: polyhedral window, polygon, area. finite accounting system: Describes concept of physical reality consisting, on some level, of discrete indivisible particles, as opposed to continuous surfaces or masses. See Chapter 2. frequency: Used by Fuller to specify length and size in general. The intention is to employ a more precisely descriptive term for both geometric systems and events in nature, than provided by specific units of measurement. Fuller points out that frequency never relates to the quantity "one," for it necessarily involves a plurality of experiences. high-frequency energy event: Describes most tangible structures, which might be popularly thought of as "solids." A good example to illustrate the concept of "high-frequency energy event" through a visible image is found in the white foam of breaking ocean waves.
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