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A Fuller Explanation
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Contents
Foreword
Foreword to the 1992 Edition
Preface
Preface to the 1992 Edition
Acknowledgements
Note to Readers
1 Introduction
2 Return to Modelability
From Geometry to Geodesics: A Personal Perspective
2.0.1 Operational Mathematics
Experimental Evidence
Nature’s Coordinate System
Universe
Generalized Principles
Return to Modelability
3 The Irrationality of Pi
"Nature Isn’t Using Pi"
Finite accounting system
Which Way Is "Up"?
Visual Literacy
Peaceful Coexistence
4 Systems and Synergy
Conceptual and Real Systems
Limits of Resolution as Part of the Whole-Systems Approach
4.0.1 Synergy
5 Tools of the Trade
5.0.1 Plato Presents
Triangles
Squares
Pentagons
A Limited Family
5.0.2 Euler’s Law
5.0.3 Duality
5.0.4 Truncation and Stellation
An Experiment
5.0.5 "Intertransformability"
5.0.6 Symmetry
6 Structure and “Pattern Integrity”
6.0.1 Pattern integrity
6.0.2 Structure
7 Angular Topology
7.0.1 Frequency and Size
Units of Measurement
Time and Repetition: Frequency versus Continuum
7.0.2 Topology and Vectors
Vector Polyhedra
7.0.3 Dimension
Size
Planes of Symmetry
Other Applications of Dimension
7.0.4 Angular topology
Angular Takeout: An Example
Angle Types
8 Vector Equilibrium
"Nature’s Own Geometry"
Spatial Considerations
Planar Equilibrium
Cuboctahedron as Vector Equilibrium
VE: Results
8.0.1 Degrees of freedom
9 Tales Told by the Spheres: Closest Packing
Equilibrium: Equalization of Distances
Symmetry versus Specificity of Form
Organization of Identical Units
New Level of Focus
9.0.1 Background: Closepacking
Planes of Symmetry
9.0.2 Fuller Observations
Tetrahedra
Vector equilibrium
Frequency
Icosahedron
9.0.3 Further Discoveries: Nests
"Interprecessing"
A Final Philosophical Note
10 Isotropic Vector Matrix
A Quick Comparison: "Synergetics Accounting"
Cells: "Inherent Complementarity"
10.0.1 A Complete Picture
Angles
Locating New Polyhedral Systems
10.0.2 Duality and the IVM
Angles
Polyhedra
Domain
10.0.3 Framework of Possibility
Invention: Octet Truss
11 Multiplication by Division: In Search of Cosmic Hierarchy
11.0.1 Volume
Results: Volume Ratios
Shape Comparisons: Qualities of Space
Volume: Direct Comparison
11.0.2 Multiplication by Division
Tetrahedron as Starting Point
Cube
Vector equilibrium
Rhombic Dodecahedron
Multiplication by Division
11.0.3 Cosmic Hierarchy (of Nuclear Event Patternings)
Volume Reconsidered
12 Jitterbug
12.0.1 Folding a Polyhedron
Volume and Phase Changes
12.0.2 Icosahedron
Single Layer versus IVM
"Trans-Universe" versus "Locally Operative"
12.0.3 Fives
"S modules"
Icosahedron and Rhombic Dodecahedron
Pentagonal Dodecahedron
12.0.4 Four Dimensions
12.0.5 Complex of Jitterbugs
12.0.6 Other Dynamic Models
Topology and Phase
13 “All-Space” Filling: New Types of Packing Crates
13.0.1 Plane Tessellations
13.0.2 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
14 The Heart of the Matter: A- and B-Quanta Modules
A-Quanta Modules
B-Quanta Modules
Energy Characteristics
14.0.1 Mite
Mirrors
Cubes into Mites
Rhombic Dodecahedra
14.0.2 Coupler
Volume and Energy
14.0.3 Review: All-Space Fillers
15 Cosmic Railroad Tracks: Great Circles
Why Are We Talking About Spheres?
New Classification System
15.0.1 Great-Circle Patterns
Least Common Denominator
LCD: "Intertransformability"
LCD of 31 Great Circles
VE’s 25 Great Circles
15.0.2 Operational Mathematics
Conservation of Angle
Foldable Systems
15.0.3 Energy Paths
Gas Molecules
Great-Circle Railroad Tracks of Energy
Icosahedron as Local Shunting Circuit
15.0.4 Inventory: Seven Unique Cosmic Axes of Symmetry
Excess of One
16 From Geodesic to Tensegrity: The Invisible Made Visible
Theory Behind Geodesic Structures: Summary
Geodesic Design in Nature
Geodesic Domes: Design Variables
16.0.1 Tensegrity
Nature’s Example
New Concept of Construction
Modeling the Invisible
Tensegrity Polyhedra
Pneumatics
Case in Point: Donald Ingber
17 “Design Science”
"Comprehensive Anticipatory Design Science"
"Comprehensive"
"Anticipatory"
Dymaxion Map 52
Suspended Storage Systems
More with Less
Appendix A: Trigonometric Calculations
Appendix B: Volume Calculations for Three Prime Structural Systems
Appendix C: Special Properties of the Tetrahedron
Appendix D: Lists of Illustrations, Photos, Tables
Glossary
Bibliography
Other Resources
Index of Synergetics Sections Cited
Index
About the Author
Appendix A: Trigonometric Calculations
Appendix A: Trigonometric Calculations
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