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Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere [Kõva köide]

  • Formaat: Hardback, 532 pages, kõrgus x laius x paksus: 235x191x25 mm, kaal: 1252 g, Illustrations
  • Ilmumisaeg: 16-Jul-2012
  • Kirjastus: A K Peters
  • ISBN-10: 1466504293
  • ISBN-13: 9781466504295
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  • Formaat: Hardback, 532 pages, kõrgus x laius x paksus: 235x191x25 mm, kaal: 1252 g, Illustrations
  • Ilmumisaeg: 16-Jul-2012
  • Kirjastus: A K Peters
  • ISBN-10: 1466504293
  • ISBN-13: 9781466504295
Teised raamatud teemal:
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.

Arvustused

"... illustrations in the book, nearly all of them computer generated, are very good indeed. ... 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. ... remarkable book ... 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. ... 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 "... the ways in which spheres are modified that make them functional and more interesting ... [ are] the main point[ s] of the book. ... 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. ... 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. ... 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 ... 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

Preface xiii
Acknowledgments xix
1 Divided Spheres 1(12)
1.1 Working with Spheres
3(1)
1.2 Making a Point
3(1)
1.3 An Arbitrary Number
4(2)
1.4 Symmetry and Polyhedral Designs
6(2)
1.5 Spherical Workbenches
8(1)
1.6 Detailed Designs
9(1)
1.7 Other Ways to Use Polyhedra
10(1)
1.8 Summary
11(1)
Additional Resources
12(1)
2 Bucky's Dome 13(36)
2.1 Synergetic Geometry
15(2)
2.2 Dymaxion Projection
17(3)
2.3 Cahill and Waterman Projections
20(1)
2.4 Vector Equilibrium
21(1)
2.5 Icosa's 31
22(2)
2.6 The First Dome
24(2)
2.7 NC State and Skybreak Carolina
26(5)
2.8 Ford Rotunda Dome
31(2)
2.9 Marines in Raleigh
33(1)
2.10 University Circuit
34(1)
2.11 Radomes
35(3)
2.12 Kaiser's Domes
38(2)
2.13 Union Tank Car
40(1)
2.14 Covering Every Angle
41(6)
2.15 Summary
47(1)
Additional Resources
48(1)
3 Puffing Spheres to Work 49(30)
3.1 Tammes Problem
49(2)
3.2 Spherical Viruses
51(2)
3.3 Celestial Catalogs
53(1)
3.4 Sudbury Neutrino Observatory
54(2)
3.5 Climate Models and Weather Prediction
56(4)
3.6 Cartography
60(1)
3.7 Honeycombs for Supercomputers
61(2)
3.8 Fish Farming
63(2)
3.9 Virtual Reality
65(2)
3.10 Modeling Spheres
67(1)
3.11 Dividing Golf Balls
68(4)
3.12 Spherical Throwable Panoramic Camera
72(1)
3.13 Hoberman's MiniSphere
73(1)
3.14 Rafiki's Code World
74(1)
3.15 Art and Expression
75(2)
Additional Resources
77(2)
4 Circular Reasoning 79(48)
4.1 Lesser and Great Circles
81(2)
4.2 Geodesic Subdivision
83(2)
4.3 Circle Poles
85(1)
4.4 Arc and Chord Factors
86(1)
4.5 Where Are We?
87(1)
4.6 Altitude-Azimuth Coordinates
87(2)
4.7 Latitude and Longitude Coordinates
89(1)
4.8 Spherical Trips
90(1)
4.9 Loxodromes
91(2)
4.10 Separation Angle
93(1)
4.11 Latitude Sailing
94(1)
4.12 Longitude
94(1)
4.13 Spherical Coordinates
95(1)
4.14 Cartesian Coordinates
96(2)
4.15 ρ, ψ, λ Coordinates
98(1)
4.16 Spherical Polygons
99(15)
4.17 Excess and Defect
114(11)
4.18 Summary
125(1)
Additional Resources
126(1)
5 Distributing Points 127(10)
5.1 Covering
128(3)
5.2 Packing
131(2)
5.3 Volume
133(2)
5.4 Summary
135(1)
Additional Resources
136(1)
6 Polyhedral Frameworks 137(44)
6.1 What Is a Polyhedron?
138(1)
6.2 Platonic Solids
139(13)
6.3 Symmetry
152(10)
6.4 Archimedean Solids
162(17)
Additional Resources
179(2)
7 Golf Ball Dimples 181(10)
7.1 Icosahedral Balls
182(2)
7.2 Octahedral Balls
184(1)
7.3 Tetrahedral Balls
185(1)
7.4 Bilateral Symmetry
186(1)
7.5 Subdivided Areas
187(1)
7.6 Dimple Graphics
188(1)
7.7 Summary
189(1)
Additional Resources
190(1)
8 Subdivision Schemas 191(56)
8.1 Geodesic Notation
192(2)
8.2 Triangulation Number
194(1)
8.3 Frequency and Harmonics
195(2)
8.4 Grid Symmetry
197(2)
8.5 Class I: Alternates and Ford
199(20)
8.6 Class II: Triacon
219(12)
8.7 Class III: Skew
231(13)
8.8 Covering the Whole Sphere
244(1)
Additional Resources
245(2)
9 Comparing-Results 247(28)
9.1 Kissing-Touching
248(3)
9.2 Sameness or Nearly So
251(2)
9.3 Triangle Area
253(2)
9.4 Face Acuteness
255(1)
9.5 Euler Lines
255(2)
9.6 Parts and T
257(3)
9.7 Convex Hull
260(2)
9.8 Spherical Caps
262(1)
9.9 Stereograms
263(4)
9.10 Face Orientation
267(5)
9.11 King Icosa
272(1)
9.12 Summary
273(1)
Additional Resources
273(2)
10 Computer-Aided Design 275(56)
10.1 A Short History
276(2)
10.2 CATIA
278(1)
10.3 Octet Truss Connector
278(12)
10.4 Spherical Design
290(4)
10.5 Three Class II Triacon Designs
294(1)
10.6 Panel Sphere
295(7)
10.7 Class II Strut Sphere
302(5)
10.8 Class II Parabolic Stellations
307(8)
10.9 Class I Ford Shell
315(7)
10.10 31 Great Circles
322(4)
10.11 Class III Skew
326(3)
Additional Resources
329(2)
11 Advanced CAD Techniques 331(32)
11.1 Reference Models
331(1)
11.2 An Architectural Example
332(2)
11.3 Spherical Reference Models
334(1)
11.4 Prepackaged Reference and Assembly Models
335(1)
11.5 Local Axis Systems
336(1)
11.6 Assembly Review
337(2)
11.7 Design-in-Context
339(1)
11.8 Associative Geometry
339(1)
11.9 Design-in-Context versus Constraints
340(2)
11.10 Mirrored Enantiomorphs
342(1)
11.11 Power Copy
343(1)
11.12 Power Copy Prototype
344(2)
11.13 Macros
346(3)
11.14 Publication
349(1)
11.15 Data Structures
350(1)
11.16 CAD Alternatives: Stella and Antiprism
350(5)
11.17 Antiprism
355(6)
11.18 Summary
361(1)
Additional Resources
362(1)
A Spherical Trigonometry 363(10)
A.1 Basic Trigonometric Functions
363(2)
A.2 The Core Theorems
365(1)
A.3 Law of Cosines
366(1)
A.4 Law of Sines
366(1)
A.5 Right Triangles
367(1)
A.6 Napier's Rule
367(1)
A.7 Using Napier's Rule on Oblique Triangles
368(1)
A.8 Polar Triangles
369(2)
Additional Resources
371(2)
B Stereographic Projection 373(22)
B.1 Points on a Sphere
374(1)
B.2 Stereographic Properties
374(1)
B.3 A History of Diverse Uses
375(1)
B.4 The Astrolabe
375(2)
B.5 Crystallography and Geology
377(1)
B.6 Cartography
378(1)
B.7 Projection Methods
379(2)
B.8 Great Circles
381(1)
B.9 Lesser Circles
382(3)
B.10 Wulff Net
385(1)
B.11 Polyhedra Stereographics
386(1)
B.12 Polyhedra as Crystals
386(1)
B.13 Metrics and Interpretation
387(1)
B.14 Projecting Polyhedra
388(2)
B.15 Octahedron
390(1)
B.16 Tetrahedron
390(1)
B.17 Geodesic Stereographics
391(1)
B.18 Spherical Icosahedron
392(1)
B.19 Summary
393(1)
Additional Resources
394(1)
C Geodesic Math 395(22)
C.1 Class I: Alternates and Fords
397(2)
C.2 Class II: Triacon
399(6)
C.3 Class III: Skew
405(2)
C.4 Characteristics of Triangles
407(1)
C.5 Storing Grid Points
408(7)
Additional Resources
415(2)
D Schema Coordinates 417(6)
D.1 Coordinates for Class I: Alternates and Ford
418(2)
D.2 Coordinates for Class II: Triacon
420(2)
D.3 Coordinates for Class III: Skew
422(1)
E Coordinate Rotations 423(29)
E.1 Rotation Concepts
424(1)
E.2 Direction and Sequences
424(1)
E.3 Simple Rotations
425(1)
E.4 Reflections
426(2)
E.5 Antipodal Points
428(1)
E.6 Compound Rotations
429(1)
E.7 Rotation around an Arbitrary Axis
430(1)
E.8 Polyhedra and Class Rotation Sequences
431(1)
E.9 Icosahedron Classes I and III
432(2)
E.10 Icosahedron Class 11
434(2)
E.11 Octahedron Classes I and Ill
436(1)
E.12 Octahedron Class II
436(3)
E.13 Tetrahedron Classes I and III
439(1)
E.14 Tetrahedron Class II
439(3)
E.15 Dodecahedron Class 11
442(1)
E.16 Cube Class II
443(2)
E.17 Implementing Rotations
445(1)
E.18 Using Matrices
446(1)
E.19 Rotation Algorithms
446(5)
E.20 An Example
451(1)
E.21 Summary
451(1)
Additional Resources 452(1)
Glossary 453(26)
Useful Websites 479(2)
Bibliography 481(20)
Index 501(10)
About the Author 511

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