A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller

In a broad sense Design Science is the grammar of a language of images rather than of words. Modern communication techniques enable us to transmit and reconstitute images without the need of knowing a specific verbal sequential language such as the Morse code or Hungarian. International traffic signs use international image symbols which are not specific to any particular verbal language. An image language differs from a verbal one in that the latter uses a linear string of symbols, whereas the former is multidimensional. Architectural renderings commonly show projections onto three mutually perpendicular planes, or consist of cross sections at differ­ ent altitudes representing a stack of floor plans. Such renderings make it difficult to imagine buildings containing ramps and other features which disguise the separation between floors; consequently, they limit the creativity of the architect. Analogously, we tend to analyze natural structures as if nature had used similar stacked renderings, rather than, for instance, a system of packed spheres, with the result that we fail to perceive the system of organization determining the form of such structures.

1. Return to Modelability.- From Geometry to Geodesics: A Personal Perspective.- Operational Mathematics.- Experimental Evidence.- Nature’s Coordinate System.- Universe.- Generalized Principles.- Return to Modelability.- 2. The Irrationality of Pi.- “Nature Isn’t Using Pi”.- Finite Accounting System.- Which Way Is “Up”?.- Visual Literacy.- Peaceful Coexistence.- 3. Systems and Synergy.- Conceptual and Real Systems.- Limits of Resolution as Part of the Whole-Systems Approach.- Synergy.- 4. Tools of the Trade.- Plato’s Discovery.- Triangles.- Squares.- Pentagons.- A Limited Family.- Euler’s Law.- Duality.- Truncation and Stellation.- An Experiment.- “Intertransformability”.- Symmetry.- 5. Structure and “Pattern Integrity”.- Pattern Integrity.- Structure.- 6. Angular Topology.- Frequency and Size.- Units of Measurement.- Time and Repetition: Frequency versus Continuum.- Topology and Vectors.- Vector Polyhedra.- Dimension.- Size.- Planes of Symmetry.- Other Applications of Dimension.- Angular Topology.- Angular Takeout: An Example.- Angle Types.- 7. Vector Equilibrium.- “Nature’s Own Geometry”.- Spatial Considerations.- Planar Equilibrium.- Cuboctahedron as Vector Equilibrium.- VE: Results.- Degrees of Freedom.- 8. Tales Told by the Spheres: Closest Packing.- Equilibrium: Equalization of Distances.- Symmetry versus Specificity of Form.- Organization of Identical Units.- New Level of Focus.- Background: Closepacking.- Planes of Symmetry.- Fuller Observations.- Tetrahedra.- Vector Equilibrium.- Frequency.- Icosahedron.- Further Discoveries: Nests.- “Interprecessing”.- A Final Philosophical Note.- 9. Isotropic Vector Matrix.- A Quick Comparison: “Synergetics Accounting”.- Cells: “Inherent Complementarity”.- A Complete Picture.- Angles.- Locating New Polyhedral Systems.- Duality and the IVM.- Angles.- Polyhedra.- Domain.- Framework of Possibility.- Invention: Octet Truss.- 10. Multiplication by Division: In Search of Cosmic Hierarchy.- Volume.- Results: Volume Ratios.- Shape Comparisons: Qualities of Space.- Volume: Direct Comparison.- Multiplication by Division.- Tetrahedron as Starting Point.- Cube.- Vector Equilibrium.- Rhombic Dodecahedron.- Multiplication by Division.- Cosmic Hierarchy (of Nuclear Event Patternings).- Volume Reconsidered.- 11. Jitterbug.- Folding a Polyhedron.- Volume and Phase Changes.- Icosahedron.- Single Layer versus IVM.- “Trans-Universe” versus “Locally Operative”.- Fives.- “S-Modules”.- Icosahedron and Rhombic Dodecahedron.- Pentagonal Dodecahedron.- Four Dimensions.- Complex of Jitterbugs.- Other Dynamic Models.- Topology and Phase.- 12. “All-Space” Filling: New Types of Packing Crates.- Plane Tessellations.- Filling Space.- Complementarity.- Other Space Fillers.- The Search Continues.- The Dual Perspective.- Duality and Domain in Sphere Packing.- Truncated Octahedron.- Two to One: A Review.- 13. The Heart of the Matter: A- and B-Quanta Modules.- A-Quanta Modules.- B-Quanta Modules.- Energy Characteristics.- Mite.- Mirrors.- Cubes into Mites.- Rhombic Dodecahedra.- Coupler.- Volume and Energy.- Review: All-Space Fillers.- 14. Cosmic Railroad Tracks: Great Circles.- Why Are We Talking About Spheres?.- New Classification System.- Great-Circle Patterns.- Least Common Denominator.- LCD: “Intertransformability”.- LCD of 31 Great Circles.- VE’s 25 Great Circles.- Operational Mathematics.- Conservation of Angle.- Foldable Systems.- Energy Paths.- Gas Molecules.- Great-Circle Railroad Tracks of Energy.- Icosahedron as Local Shunting Circuit.- Inventory: Seven Unique Cosmic Axes of Symmetry.- Excess of One.- 15. From Geodesic to Tensegrity: The Invisible Made Visible.- Theory Behind Geodesic Structures: Summary.- Geodesic Design in Nature.- Geodesic Domes: Design Variables.- Tensegrity.- Nature’s Example.- New Concept of Construction.- Modeling the Invisible.- Tensegrity Polyhedra.- Pneumatics.- Case in Point: Donald Ingber.- 16. “Design Science”.- “Comprehensive Anticipatory Design Science”.- “Comprehensive... ”.- “... Anticipatory... ”.- Dymaxion Map.- Suspended Storage Systems.- More with Less.- Appendices.- A. Trigonometric Calculations.- B. Volume Calculations for Three Prime Structural Systems.- C. Sources of Additional Information.- D. Special Properties of the Tetrahedron.- E. Glossary.- Notes.