Divided Spheres : Geodesics and the Orderly Subdivision of the Sphere

**Praise for the previous edition** [. . .] Dr. Popko’s elegant new book extends both the science and the art of spherical modeling to include Computer-Aided Design and applications, which I would never have imagined when I started down this fascinating and rewarding path. His lovely illustrations bring the subject to life for all readers, including those who are not drawn to the mathematics. This book demonstrates the scope, beauty, and utility of an art and science with roots in antiquity. [. . .] Anyone with an interest in the geometry of spheres, whether a professional engineer, an architect or product designer, a student, a teacher, or simply someone curious about the spectrum of topics to be found in this book, will find it helpful and rewarding. **__– Magnus Wenninger, Benedictine Monk and Polyhedral Modeler__** Ed Popko's comprehensive survey of the history, literature, geometric, and mathematical properties of the sphere is the definitive work on the subject. His masterful and thorough investigation of every aspect is covered with sensitivity and intelligence. This book should be in the library of anyone interested in the orderly subdivision of the sphere. **__– Shoji Sadao, Architect, Cartographer and lifelong business partner of Buckminster Fuller__** Edward Popko's __Divided Spheres__ is a "thesaurus" must to those whose academic interest in the world of geometry looks to greater coverage of synonyms and antonyms of this beautiful shape we call a sphere. The late Buckminster Fuller might well place this manuscript as an all-reference for illumination to one of nature's most perfect inventions. **__– Thomas T. K. Zung, Senior Partner, Buckminster Fuller, Sadao, & Zung Architects.__** This first edition of this well-illustrated book presented a thorough introduction to the mathematics of Buckminster Fuller’s invention of the geodesic dome, which paved the way for a flood of practical applications as diverse as weather forecasting and fish farms. The author explained the principles of spherical design and the three classic methods of subdivision based on geometric solids (polyhedra). This thoroughly edited new edition does all that, while also introducing new techniques that extend the class concept by relaxing the triangulation constraint to develop two new forms of optimized hexagonal tessellations. The objective is to generate spherical grids where all edge (or arc) lengths or overlap ratios are equal. New to the Second Edition * New Foreword by Joseph Clinton, lifelong Buckminster Fuller collaborator * A new chapter by Chris Kitrick on the mathematical techniques for developing optimal single-edge hexagonal tessellations, of varying density, with the smallest edge possible for a particular topology, suggesting ways of comparing their levels of optimization * An expanded history of the evolution of spherical subdivision * New applications of spherical design in science, product design, architecture, and entertainment * New geodesic algorithms for grid optimization * New full-color spherical illustrations created using __DisplaySphere__ to aid readers in visualizing and comparing the various tessellations presented in the book * Updated Bibliography with references to the most recent advancements in spherical subdivision methods Cover Half Title Title Page Copyright Page Dedication Table of Contents Foreword Preface Divided Spheres Graphic Conventions Acknowledgments 1. Divided Spheres 1.1. Working with Spheres 1.2. Making a Point 1.3. An Arbitrary Number 1.4. Symmetry and Polyhedral Designs 1.5. Spherical Workbenches 1.6. Detailed Designs 1.7. Other Ways to Use Polyhedra 1.8. Summary Additional Resources 2. Bucky’s Dome 2.1. Synergetic Geometry 2.2. Dymaxion Projection 2.3. Cahill and Waterman Projections 2.4. Vector Equilibrium 2.5. Icosa’s 31 2.6. The First Dome 2.7. Dome Development 2.7.1. Tensegrity 2.7.2. Autonomous Dwellings and Fly’s Eye 2.7.3. A Full-Scale Project 2.7.4. NC State and Skybreak Carolina 2.7.5. Ford Rotunda Dome 2.7.6. Marines in Raleigh 2.7.7. Plydome 2.7.8. University Circuit 2.7.9. Radomes 2.7.10. Kaiser’s Domes 2.7.11. Union Tank Car 2.7.12. Spaceship Earth 2.8. Covering Every Angle 2.9. Summary Additional Resources 3. Putting Spheres to Work 3.1. The Tammes Problem 3.2. Spherical Viruses 3.3. Celestial Catalogs 3.4. Sudbury Neutrino Observatory 3.5. Cartography 3.6. Climate Models and Weather Prediction 3.7. H3 Uber’s Hexagonal Hierarchical Geospatial Indexing System 3.8. Honeycombs for Supercomputers 3.9. Fish Farming 3.10. Virtual Reality 3.11. Modeling Spheres 3.12. Computer Aided Design 3.13. Octet Truss Connector 3.14. Dividing Golf Balls 3.15. Spherical, Throwable PanonoTM Panoramic Camera 3.16. Termespheres 3.17. Space ChipsTM 3.18. Hoberman’s MiniSphereTM 3.19. Rafiki’s Code World 3.20. V-SphereTM 3.21. Gear Ball—Meffert’s Rotation Brain Teaser 3.22. Rhombic Tuttminx 66 3.23. Japanese Temari Balls 3.23.1. Basic Ball and Design Layouts 3.23.2. Platonic Layouts 3.24. Art and Expression Additional Resources 4. Circular Reasoning 4.1. Lesser and Great Circles 4.2. Geodesic Subdivision 4.3. Circle Poles 4.4. Arc and Chord Factors 4.5. Where Are We? 4.6. Altitude-Azimuth Coordinates 4.7. Latitude and Longitude Coordinates 4.8. Spherical Trips 4.9. Loxodromes 4.10. Separation Angle 4.11. Latitude Sailing 4.12. Longitude 4.13. Spherical Coordinates 4.14. Cartesian Coordinates 4.15. ρ,φ,y Coordinates 4.16. Spherical PolygonsSpherical 4.16.1. LunesLunes, 4.16.2. Quadrilaterals 4.16.3. Other Polygons 4.16.4. Caps and Zones 4.16.5. Gores 4.16.6. Spherical Triangles 4.16.7. Congruent and Symmetrical Triangles 4.16.8. Nothing Similar 4.16.9. Schwarz Triangles 4.16.10. Area and Excess 4.16.11. Steradians 4.16.12. Solid AnglesThe 4.16.13. Spherical Degrees and Square Degrees 4.17. Excess and Defect 4.17.1. Visualizing Excess 4.17.2. Centering in on Triangles 4.17.3. Euler Line 4.17.4. Surface Normals 4.18. Summary Additional Resources 5. Distributing Points 5.1. Covering 5.2. Packing 5.2.1. 200-Year-Old Kissing Puzzle 5.3. Volume 5.4. Summary Additional Resources 6. Polyhedral Frameworks 6.1. What Is a Polyhedron? 6.2. Platonic Solids 6.2.1. Platonic Duals 6.2.2. Shorthand for the Unpronounceable 6.2.3. Circumsphere and Insphere 6.2.4. Vertex-Face-Edge Relationships 6.2.5. The Golden Section 6.2.6. Precise Platonics 6.2.7. Platonic Summary 6.3. Symmetry 6.3.1. Symmetry Groups 6.3.2. Icosahedral Symmetry 6.3.3. Octahedral Symmetry 6.3.4. Tetrahedral Symmetry 6.3.5. Schwarz Triangles and Symmetry 6.3.6. Deltahedra 6.4. Archimedean Solids 6.4.1. Cundy-Rollett Symbols 6.4.2. Truncation 6.4.3. Archimedean Solids with Icosahedral Symmetry 6.4.4. Archimedean Solids with Octahedral Symmetry 6.4.5. Archimedean Solids with Tetrahedral Symmetry 6.4.6. Chiral Polyhedra 6.4.7. Quasi-Regular Polyhedra and Natural Great Circles 6.4.8. Waterman Polyhedra 6.5. Circlespheres and Atomic Models 6.6. Atomic Models Additional Resources 7. Golf Ball Dimples 7.1. Icosahedral Balls 7.2. Octahedral Balls 7.3. Tetrahedral Balls 7.4. Bilateral Symmetry 7.5. Subdivided Areas 7.6. Dimple Graphics 7.7. Summary Additional Resources 8. Subdivision Schemas 8.1. Geodesic Notation 8.2. Triangulation Number 8.3. Frequency and Harmonics 8.4. Grid Symmetry 8.5. Class I: Alternates and Ford 8.5.1. Defining the Principal Triangle 8.5.2. Edge Reference Points 8.5.3. Intersecting Great Circles 8.5.4. Four Class I Schemas 8.5.5. Equal-Chords 8.5.6. Equal-Arcs (Two Great Circles) 8.5.7. Equal-Arcs (Three Great Circles) 8.5.8. Mid-Arcs 8.5.9. Subdividing Other Deltahedra 8.5.10. Summary 8.6. Class II: Triacon 8.6.1. Schwarz LCD Triangles 8.6.2. How Frequent 8.6.3. A Quick Overview 8.6.4. Establish Your Rights 8.6.5. Subdividing the LCD 8.6.6. Grid Points 8.6.7. Completed Triacon 8.6.8. Subdividing Other Polyhedra 8.6.9. Summary 8.7. Class III: Skew 8.7.1. What Is Class III 8.7.2. Snubbed Relatives 8.7.3. Enantiomorphs 8.7.4. Harmonics 8.7.5. Developing Grids 8.7.6. The BC Grid 8.7.7. From Two to Three Dimensions 8.7.8. Scale and Translate 8.7.9. PPT Standard Position 8.7.10. Projection 8.7.11. Class III PPT 8.7.12. Other Polyhedra and Classes 8.7.13. Summary 8.8. Covering the Whole Additional Resources 9. Comparing Results 9.1. Kissing-Touching 9.2. Sameness or Nearly So 9.3. Triangle Area 9.4. Face Acuteness 9.5. Euler Lines 9.6. Parts and T 9.7. Convex Hull 9.8. Spherical Caps 9.9. Stereograms 9.10. Face OrientationThus 9.11. King IcosaBy 9.12. Summary Additional Resources 10. Self-Organizing Grids 10.1. Reduced Constraint Networks 10.1.1. Hexagonal Grids 10.1.2. Rotegrities, Nexorades, and Reciprocal Frames 10.1.3. Organizing Targets 10.2. Symmetry 10.2.1. Uniqueness 10.2.2. The BC Grid 10.3. Self-Organizing—Key Concepts 10.3.1. Initial State 10.3.2. Neighborhoods and Local Optimization 10.3.3. Global Propagation 10.3.4. Target Goal 10.3.5. Computation Sequence Algorithm 10.3.6. Summary 10.4. Hexagonal Grids 10.4.1. Initial Condition 10.4.2. Hexagonal Neighborhood and Local Optimization 10.4.3. Global Propagation 10.4.4. Hexagonal Grid Example One—Icosahedron 10.4.5. Hexagonal Grid Example Two—Octahedron 10.4.6. Hexagonal Grid Example Three—Tetrahedron 10.4.7. Results 10.4.8. Summary 10.5. Rotegrities 10.5.1. Initial Condition 10.5.2. Rotegrity Neighborhood and Local Optimization 10.5.3. Global Propagation 10.5.4. Icosahedral Rotegrity—Example One 10.5.5. Octahedral Rotegrity—Example Two 10.5.6. Tetrahedral Rotegrity—Example Three 10.5.7. Summary 10.6. Future Directions 10.6.1. Additional Reduced Constraint Networks 10.6.2. Non-Symmetrical Reduced Constraint Configurations 10.6.3. Reduced Constraint Configurations on Non-Spherical Surfaces 10.7. Summary Additional Resources A. Stereographic Projection A.1. Points on a Sphere A.2. Stereographic Properties A.3. A History of Diverse Uses A.4. The Astrolabe A.5. Crystallography and Geology A.6. Cartography A.7. Projection Methods A.8. Great Circles A.9. Lesser Circles A.10. Wulff Net A.11. Polyhedra Stereographics A.12. Polyhedra as Crystals A.13. Metrics and Interpretation A.14. Projecting Polyhedra A.15. Octahedron A.16. Tetrahedron A.17. Geodesic Stereographics A.18. Spherical Icosahedron A.19. Summary Additional Resources B. Coordinate Rotations B.1. Rotation Concepts B.2. Direction and Sequences B.3. Simple Rotations B.4. Reflections B.5. Antipodal Points B.6. Compound Rotations B.7. Rotation Around an Arbitrary Axis B.8. Polyhedra and Class Rotation Sequences B.9. Icosahedron Classes I and III B.10. Icosahedron Class II B.11. Octahedron Classes I and III B.12. Octahedron Class II B.13. Tetrahedron Classes I and III B.14. Tetrahedron Class II B.15. Dodecahedron Class II B.16. Cube Class II B.17. Implementing Rotations B.18. Using Matrices B.18.1. Identity B.18.2. Specifying Angles B.18.3. Matrix Multiplication B.19. Rotation Algorithms B.19.1. Identity B.19.2. Rotation Around the X-axis B.19.3. Rotation Around the Y-axis B.19.4. Rotation Around the Z-axis B.19.5. Translation B.19.6. Matrix Multiplication (Concatenation) B.19.7. Transform Points B.20. An Example B.21. Summary Additional Resources C. Geodesic Math C.1. Class I: Alternates and Fords C.1.1. Step 1: Define the PPT Apex Coordinates C.1.2. Step 2: Define PPT Edge Reference Points C.1.3. Step 3: Subdivide the PPT C.1.4. Class I Summary C.2. Class II: Triacon C.2.1. Step 1: Position and Define the Triacon LCD C.2.2. Step 2: Subdivide the Triacon LCD C.2.3. Step 3: Define Grid Points C.2.4. Class II Summary C.3. Class III: Skew C.3.1. Step 1: Define the PPT Grid C.3.2. Step 2: Position PPT for Projection C.3.3. Step 3: Project Trigrid Points C.3.4. Class III Summary C.4. Characteristics of Triangles C.5. Storing Grid Points C.5.1. gcsect(): Intersection Points of Two Great Circles C.5.2. gdihdrl(): Dihedral Angle between Two Planes C.5.3. gtricent(): Centroid of a Triangle C.5.4. stABC(): Surface Angles of a Spherical Triangle C.5.5. vabs(): Length of a Vector C.5.6. vadd(): Add Two Vectors C.5.7. varcv(): Locate a Point on a Geodesic Arc between Two Other Points C.5.8. vcos(): Cosine of Angle between Two Vectors C.5.9. vcrs(): Cross Product of Two Vectors C.5.10. vdir(): Point on a Parametric Line C.5.11. vdis(): Distance between Two Points C.5.12. vdot(): Dot Product C.5.13. vnor(): Normal Vector to a Plane C.5.14. vrevs(): Reverse a Vector’s Sense C.5.15. vscl(): Scale a Vector or Point C.5.16. vsub(): Subtract a Vector or Point C.5.17. vuni(): Normalize a Vector C.5.18. vzero(): Initialize a Vector C.6. Symmetrical Uniqueness C.6.1. Storing BC Grid Points C.6.2. hspace(): Which Side of a 2D Line Does Point Lie C.6.3. section0(): Determines If Point in BC Grid Is in Section Zero (S0) for Class III Additional Resources Bibliography Index About the Authors "This first edition of this well-illustrated book presented a thorough introduction to the mathematics of Buckminster Fuller's invention of the geodesic dome, which paved the way for a flood of practical applications as diverse as weather forecasting and fish farms. The author explained the principles of spherical design and the three classic methods of subdivision based on geometric solids (polyhedra). This thoroughly edited new edition does all that, while also introducing new techniques that extend the class concept by relaxing the triangulation constraint to develop two new forms of optimized hexagonal tessellations. The objective is to generate spherical grids where all edge (or arc) lengths or overlap ratios are equal. New to the Second Edition New Foreword by Joseph Clinton, life-long Buckminster Fuller collaborator A new chapter by Chris Kitrick on the mathematical techniques for developing optimal single-edge hexagonal tessellations, of varying density, with the smallest edge possible for a particular topology, suggesting ways of comparing their levels of optimization An expanded history of the evolution of spherical subdivision New applications of spherical design in science, product design, architecture and entertainment New geodesic algorithms for grid optimization New full-color spherical illustrations created using DisplaySphere to aid readers in visualizing and comparing the various tessellations presented in the book. Updated Bibliography with references to the most recent advancements in spherical subdivision methods"-- Provided by publisher