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Divided Spheres Geodesics and the Orderly Subdivision of the Sphere 2nd Edition by Edward S Popko, Christopher J Kitrick ISBN 0367680033 9780367680039

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Divided Spheres Geodesics and the Orderly Subdivision of the Sphere 2nd Edition Edward S. Popko Digital Instant Download

Author(s): Edward S. Popko, Christopher J. Kitrick
ISBN(s): 9780367680039, 0367680033
Edition: 2
File Details: PDF, 26.01 MB
Year: 2021
Language: english
SKU: EB-33350364 Category: Tags: ,

Divided Spheres Geodesics and the Orderly Subdivision of the Sphere 2nd Edition by Edward S Popko, Christopher J Kitrick – Ebook PDF Instant Download/Delivery: 0367680033, 9780367680039
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ISBN 10: 0367680033
ISBN 13: 9780367680039
Author: Edward S Popko, Christopher J Kitrick

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

Divided Spheres Geodesics and the Orderly Subdivision of the Sphere 2nd Table of contents:

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 Coordinates
4.16 Spherical Polygons
 4.16.1 Lunes
 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 Angles
 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 Sphere
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 Orientation
9.11 King Icosa
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

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Tags: Edward S Popko, Christopher J Kitrick, Divided Spheres, Orderly Subdivision