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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: Popko, Edward S. Divided Spheres : Geodesics and the Orderly Subdivision of the Sphere. Hoboken : CRC Press, ©2012 |

Named Person: | R Buckminster Fuller; R Buckminster Fuller |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Edward S Popko |

ISBN: | 9781466504301 1466504307 |

OCLC Number: | 798535752 |

Description: | 1 online resource (525 pages) |

Contents: | Front Cover; Table of Contents; Preface; Acknowledgments; 1. Divided Spheres; 2. Bucky's Dome; 3. Putting Spheres toWork; 4. Circular Reasoning; 5. Distributing Points; 6. Polyhedral Frameworks; 7. Golf Ball Dimples; 8. Subdivision Schemas; 9. Comparing Results; 10. Computer-Aided Design; 11. Advanced CAD Techniques; A. Spherical Trigonometry; B. Stereographic Projection; C. Geodesic Math; D. Schema Coordinates; E. Coordinate Rotations; Glossary; Useful Websites; Bibliography; About the Author. |

### Abstract:

## Reviews

*Editorial reviews*

Publisher Synopsis

This well-illustrated book-in color throughout-presents 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 explains the principles of spherical design and the three main categories of subdivision based on geometric solids (polyhedra). He illustrates how basic and advanced CAD techniques apply to spherical subdivision and covers modern applications in product design, engineering, science, games, and sports balls. Divided SpheresWorking with SpheresMaking a PointAn Arbitrary NumberSymmetry and Polyhedral DesignsSpherical WorkbenchesDetailed DesignsOther Ways to Use PolyhedraSummaryAdditional ResourcesBucky's DomeSynergetic GeometryDymaxion ProjectionCahill and Waterman ProjectionsVector EquilibriumIcosa'sThe First DomeNC State and Skybreak CarolinaFord Rotunda DomeMarines in RaleighUniversity CircuitRadomesKaiser's DomesUnion Tank CarCovering Every AngleSummaryAdditional ResourcesPutting Spheres to WorkTammes ProblemSpherical VirusesCelestial CatalogsSudbury Neutrino ObservatoryClimate Models and Weather PredictionCartographyHoneycombs for SupercomputersFish FarmingVirtual RealityModeling SpheresDividing Golf BallsSpherical Throwable Panoramic CameraHoberman's MiniSphereRafiki's Code WorldArt and ExpressionAdditional ResourcesCircular ReasoningLesser and Great CirclesGeodesic SubdivisionCircle PolesArc and Chord FactorsWhere Are We?Altitude-Azimuth CoordinatesLatitude and Longitude CoordinatesSpherical TripsLoxodromesSeparation AngleLatitude SailingLongitudeSpherical CoordinatesCartesian CoordinatesÏ , Ï , Î» CoordinatesSpherical PolygonsExcess and DefectSummaryAdditional ResourcesDistributing PointsCoveringPackingVolumeSummaryAdditional ResourcesPolyhedral FrameworksWhat Is a Polyhedron?Platonic SolidsSymmetryArchimedean SolidsAdditional ResourcesGolf Ball DimplesIcosahedral BallsOctahedral BallsTetrahedral BallsBilateral SymmetrySubdivided AreasDimple GraphicsSummaryAdditional ResourcesSubdivision SchemasGeodesic NotationTriangulation NumberFrequency and HarmonicsGrid SymmetryClass I: Alternates and FordClass II: TriaconClass III: SkewCovering the Whole SphereAdditional ResourcesComparing ResultsKissing-TouchingSameness or Nearly SoTriangle AreaFace AcutenessEuler LinesParts and T . 257Convex HullSpherical CapsStereogramsFace OrientationKing IcosaSummaryAdditional ResourcesComputer-Aided DesignA Short HistoryCATIAOctet Truss ConnectorSpherical DesignThree Class II Triacon DesignsPanel SphereClass II Strut SphereClass II Parabolic StellationsClass I Ford Shell31 Great CirclesClass III SkewAdditional ResourcesAdvanced CAD TechniquesReference ModelsAn Architectural ExampleSpherical Reference ModelsPrepackaged Reference and Assembly ModelsLocal Axis SystemsAssembly ReviewDesign-in-ContextAssociative GeometryDesign-in-Context versus ConstraintsMirrored EnantiomorphsPower CopyPower Copy PrototypeMacrosPublicationData StructuresCAD Alternatives: Stella and AntiprismAntiprismSummaryAdditional ResourcesSpherical TrigonometryBasic Trigonometric FunctionsThe Core TheoremsLaw of CosinesLaw of SinesRight TrianglesNapier's RuleUsing Napier's Rule on Oblique TrianglesPolar TrianglesAdditional ResourcesStereographic ProjectionPoints on a SphereStereographic PropertiesA History of Diverse UsesThe AstrolabeCrystallography and GeologyCartographyProjection MethodsGreat CirclesLesser CirclesWulff NetPolyhedra StereographicsPolyhedra as CrystalsMetrics and InterpretationProjecting PolyhedraOctahedronTetrahedronGeodesic StereographicsSpherical IcosahedronSummaryAdditional ResourcesGeodesic MathClass I: Alternates and FordsClass II: TriaconClass III: SkewCharacteristics of TrianglesStoring Grid PointsAdditional ResourcesSchema CoordinatesCoordinates for Class I: Alternates and FordCoordinates for Class II: TriaconCoordinates for Class III: SkewCoordinate RotationsRotation ConceptsDirection and SequencesSimple RotationsReflectionsAntipodal PointsCompound RotationsRotation around an Arbitrary AxisPolyhedra and Class Rotation SequencesIcosahedron Classes I and IIIIcosahedron ClassOctahedron Classes I and IIIOctahedron ClassTetrahedron Classes I and IIITetrahedron ClassDodecahedron ClassCube ClassImplementing RotationsUsing MatricesRotation AlgorithmsAn ExampleSummaryAdditional Resources "â ¦ illustrations in the book, nearly all of them computer generated, are very good indeed. â ¦ The book contains an extremely detailed metrical treatment of all the regular and Archimedean polyhedra. An important construction is the space tessellating octahedron + tetrahedron which Fuller described as `simplest, most powerful structural system in the universe.' Taking tubes along the edges of the tessellation, he devised and patented a joint to which up to nine tubes could be connected, making a very rigid structure. This is called the `octet struss connector' and receives an entire, beautifully illustrated chapter in the book. â ¦ remarkable book â ¦ the sheer scale of the book, 509 pages on how to divide up the surface of a sphere, is amazing."-Peter Giblin, The Mathematical Gazette, March 2014"The text is written for designers, architects and people interesting in constructions of domes based on spherical subdivision. The book is illustrated with many figures and sketches and examples of real-life usage of the constructions developed during (roughly) the past 60 years. Overall, the book is written in a way accessible to a non-expert in mathematics and geometry. â ¦ The book could certainly be a good source for inspiration, with many applications, mostly in architecture and other related areas."-Pavel Chalmoviansky, Mathematical Reviews, May 2013"This well-illustrated book-in color throughout-presents 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 explains the principles of spherical design and the three main categories of subdivision based on geometric solids (polyhedra). He illustrates how basic and advanced CAD techniques apply to spherical subdivision and covers modem applications in product design, engineering, science, games, and sports balls."-L'ENSEIGNEMENT MATHEMATIQUE, 2013"â ¦ the ways in which spheres are modified that make them functional and more interesting â ¦ [are] the main point[s] of the book. â ¦ Implementations of tessellated spheres are used to describe real-world situations, from computer processor grids to fish farming to the surface of golf balls to global climate models. This is a very entertaining section, demonstrating once again how powerful and useful mathematics is. â ¦ this book is an existence proof of how complex, interesting and useful properly altered spheres can be."-Charles Ashbacher, MAA Reviews, December 2012 "In support of his primer, Popko provides a glossary of over 300 terms, a bibliography of 385 citations, reference to 28 useful websites, and an index of nine double columned pages. For some readers, these aids will be most useful in accessing and keeping track of the great diversity of ideas and concepts as well as practical and analytical procedures found in this complex and engaging volume. â ¦ a broad array of readers will find much of interest and value in this volume whether in terms of mathematics, conceptualization, application, or production."-Henry W. Castner, GEOMATICA, Vol. 66, No. 3, 2012"I have loved the beauty and symmetry of polyhedra and spherical divisions for many years. My own efforts have been concentrated on making both simple and complex spherical models using classical methods and simple tools. 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. Spherical subdivision is relevant today and useful for the future. 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"Edward Popko's Divided Spheres is the definitive source for the many varied ways a sphere can be divided and subdivided. From domes and pollen grains to golf balls, every category and type is elegantly described in these pages. The mathematics and the images together amount to a marvelous collection, one of those rare works that will be on the bookshelf of anyone with an interest in the wonders of geometry."-Kenneth Snelson, Sculptor and Photographer"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 invention."-Thomas T.K. Zung, Senior Partner, Buckminster Fuller, Sadao & Zung Architects"My own discovery, Waterman Polyhedra, was my way to see hidden patterns in ordered points in space. Ed's book Divided Spheres is about patterns and points too but on spheres. He shows you how to solve practical design problems based spherical polyhedra. Novices and experts will understand the challenges and classic techniques of spherical design just by looking at the many beautiful illustrations."-Steve Waterman, Mathematician"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"Any math collection concerned with spherical modeling will find this offers a basic yet complex introduction â ¦ blends art with scientific inquiry, providing a college-level coverage of geometry that will bring math alive for any who want a discussion of sphere science."-Midwest Book Review Divided Spheres Working with Spheres Making a Point An Arbitrary Number Symmetry and Polyhedral Designs Spherical Workbenches Detailed Designs Other Ways to Use Polyhedra Summary Additional Resources Bucky's Dome Synergetic Geometry Dymaxion Projection Cahill and Waterman Projections Vector Equilibrium Icosa's The First Dome NC State and Skybreak Carolina Ford Rotunda Dome Marines in Raleigh University Circuit Radomes Kaiser's Domes Union Tank Car Covering Every Angle Summary Additional Resources Putting Spheres to Work Tammes Problem Spherical Viruses Celestial Catalogs Sudbury Neutrino Observatory Climate Models and Weather Prediction Cartography Honeycombs for Supercomputers Fish Farming Virtual Reality Modeling Spheres Dividing Golf Balls Spherical Throwable Panoramic Camera Hoberman's MiniSphere Rafiki's Code World Art and Expression Additional Resources Circular Reasoning Lesser and Great Circles Geodesic Subdivision Circle Poles Arc and Chord Factors Where Are We? Altitude-Azimuth Coordinates Latitude and Longitude Coordinates Spherical Trips Loxodromes Separation Angle Latitude Sailing Longitude Spherical Coordinates Cartesian Coordinates rho, phi, lambda Coordinates Spherical Polygons Excess and Defect Summary Additional Resources Distributing Points Covering Packing Volume Summary Additional Resources Polyhedral Frameworks What Is a Polyhedron? Platonic Solids Symmetry Archimedean Solids Additional Resources Golf Ball Dimples Icosahedral Balls Octahedral Balls Tetrahedral Balls Bilateral Symmetry Subdivided Areas Dimple Graphics Summary Additional Resources Subdivision Schemas Geodesic Notation Triangulation Number Frequency and Harmonics Grid Symmetry Class I: Alternates and Ford Class II: Triacon Class III: Skew Covering the Whole Sphere Additional Resources Comparing Results Kissing-Touching Sameness or Nearly So Triangle Area Face Acuteness Euler Lines Parts and T . 257 Convex Hull Spherical Caps Stereograms Face Orientation King Icosa Summary Additional Resources Computer-Aided Design A Short History CATIA Octet Truss Connector Spherical Design Three Class II Triacon Designs Panel Sphere Class II Strut Sphere Class II Parabolic Stellations Class I Ford Shell 31 Great Circles Class III Skew Additional Resources Advanced CAD Techniques Reference Models An Architectural Example Spherical Reference Models Prepackaged Reference and Assembly Models Local Axis Systems Assembly Review Design-in-Context Associative Geometry Design-in-Context versus Constraints Mirrored Enantiomorphs Power Copy Power Copy Prototype Macros Publication Data Structures CAD Alternatives: Stella and Antiprism Antiprism Summary Additional Resources Spherical Trigonometry Basic Trigonometric Functions The Core Theorems Law of Cosines Law of Sines Right Triangles Napier's Rule Using Napier's Rule on Oblique Triangles Polar Triangles Additional Resources Stereographic Projection Points on a Sphere Stereographic Properties A History of Diverse Uses The Astrolabe Crystallography and Geology Cartography Projection Methods Great Circles Lesser Circles Wulff Net Polyhedra Stereographics Polyhedra as Crystals Metrics and Interpretation Projecting Polyhedra Octahedron Tetrahedron Geodesic Stereographics Spherical Icosahedron Summary Additional Resources Geodesic Math Class I: Alternates and Fords Class II: Triacon Class III: Skew Characteristics of Triangles Storing Grid Points Additional Resources Schema Coordinates Coordinates for Class I: Alternates and Ford Coordinates for Class II: Triacon Coordinates for Class III: Skew Coordinate Rotations Rotation Concepts Direction and Sequences Simple Rotations Reflections Antipodal Points Compound Rotations Rotation around an Arbitrary Axis Polyhedra and Class Rotation Sequences Icosahedron Classes I and III Icosahedron Class Octahedron Classes I and III Octahedron Class Tetrahedron Classes I and III Tetrahedron Class Dodecahedron Class Cube Class Implementing Rotations Using Matrices Rotation Algorithms An Example Summary Additional Resources This well-illustrated book-in color throughout-presents 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 explains the principles of spherical design and the three main categories of subdivision based on geometric solids (polyhedra). He illustrates how basic and advanced CAD techniques apply to spherical subdivision and covers modern applications in product design, engineering, science, games, and sports balls. Read more...

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