Amy C. Edmondson

A Fuller Explanation

Contents Chapters 1 through 8 [ Chapters 9 through 16 ] Series Editor Foreward ........................................................ xv Preface ....................................................................... xix Acknowledgments ............................................................... xxi Note to Readers ............................................................... xxiii Introduction .................................................................. xxv 1. Return to Modelability ................................................... 1 From Geometry to Geodesics: A Personal Perspective ....... 3 Operational Mathematics ...................................................... 6 Experimental Evidence ........................................................ 7 Nature's Coordinate System ................................................... 9 Universe ..................................................................... 11 Generalized Principles ....................................................... 12 Return to Modelability ....................................................... 14 2. The Irrationality of Pi ................................................ 15 "Nature Isn't Using Pi" .......................................................... 16 Finite Acounting System .......................................................... 18 Which way is "Up"? ............................................................... 19 Visual Literacy .................................................................. 22 Peaceful Coexistence ............................................................. 23 3. Systems and Synergy ...................................................... 25 Conceptual and Real Systems ......................................................... 28 Limits of Resolution as Part of the Whole-System Approach ........ 30 Synergy ............................................................................. 33 4. Tools of the Trade ....................................................... 36 Plato's Discovery ................................................................... 37 Triangles ........................................................................... 38 Squares ............................................................................. 40 Pentagons ........................................................................... 41 A limited Family .................................................................... 41 Euler's Law ......................................................................... 43 Duality ............................................................................. 45 Truncation and Stellation ........................................................... 46 An Experiment ..................................................................... . 48 "Intertransformability" ............................................................. 51 Symmetry ............................................................................ 52 5. Structure and "Pattern Integrity" ......................................... 54 Pattern Integrity ................................................................ 56 Structure ........................................................................ 60 6. Angular Topology ................................................................. 65 Frequency and Size ...................................................................... 66 Units of Measurement ................................................................. 66 Time and Repetition: Frequency versus Continuum ............ 67 Topology and Vectors .................................................................... 68 Vector Polyhedra ..................................................................... 68 Dimension ............................................................................... 70 Size ................................................................................. 72 Planes of Symmetry ................................................................... 73 Other Applications of Dimension ................................. 74 Angular Topology ........................................................................ 75 Angular Takeout: An Example ................................... 78 Angle Types .......................................................................... 79 7. Vector Equilibrium ............................................................ 82 "Nature's Own Geometry" ............................................................ 84 Spatial Considerations ............................................................. 85 Planar Equilibrium ................................................................. 87 Cuboctahedron as Vector Equilibrium ................................ 90 VE: Results ........................................................................ 91 Degrees of Freedom ................................................................. 93 8. Tales Told by the Spheres: Closest Packing ............... 100 Equilibrium: Equalization of Distances ............................................. 100 Symmetry versus Specificity of Form ............................................. 101 Organization of Identical Units ................................................. 101 New Level of Focus .............................................................. 102 Background: Closepacking ........................................................ 102 Planes of Symmetry .............................................................. 106 Fuller Observations ............................................................. 109 Tetrahedra ...................................................................... 110 Vector Equilibrium .............................................................. 114 Frequency ....................................................................... 114 Icosahedron ..................................................................... 117 Further Discoveries: Nests ...................................................... 120 "Interprecessing .................................................................. 121 A Final Philosophical Note ........................................................ 125 Appendices .......................................... 271 A. Trigonometric Calculations .......................... 271 B. Volume Calculations for three Prime Sructural Systems 273 C. Sources of Additional Information ......... 277 D. Special Properties of the Tetrahedron ........................ 279 E. Glossary .............................................. 281 Bibliography ..................................... 287 Notes ....................................... 289 Index ....................................... A-H Index ....................................... I-R Index ....................................... S-T Index ....................................... U-Z