Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere

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 Panono™ Panoramic Camera
3.16. Termespheres
3.17. Space Chips™
3.18. Hoberman’s MiniSphere™
3.19. Rafiki’s Code World
3.20. V-Sphere™
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