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
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. 2.8. 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 Panʘnʘ ™ Panoramic Camera. 3.16. Termespheres. 3.17. Space Chip’s™. 3.18. Hoberman’s MiniSphere™. 3.19. V-Sphere™. 3.20. Gear Ball - Meffert's Rotation Brain Teaser. 3.21. Rhombic Tuttminx. 3.22. Rafiki’s Code World. 3.23. Japanese Temari Balls. 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 ɸ λ Coordinates. 4.16. Spherical Polygons. 4.17. Excess and Defect. 4.18 Summary. Additional Resources. 5. Distributing Points. 5.1. Covering. 5.2. Packing. 5.3. Volume. 5.4. Summary. Additional Resources. 6. Polyhedral Frameworks. 6.1. What Is a Polyhedron? 6.2. Platonic Solids. 6.3. Symmetry. 6.4. Archimedean Solids. 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. 8.6. Class II: Triacon. 8.7. Class III: Skew. 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. Subdivision Schemas. Geodesic Math. 10. Self-Organizing Grids. 10.1. Reduced Constraint Networks. 10.2. Symmetry. 10.3. Self-Organizing - Key Concepts. 10.4. Hexagonal Grids. 10.5. Rotegrities. 10.6. Future Directions. 10.7. Summary. Additional Resources. A. Stereographic Projection. B. Coordinate Rotations. C. Geodesic Math. Bibliography. Index.
"Divided Spheres presents a wealth of ideas and images for anyone interested in spherical constructions. The section on the history of geodesic dome design and the chapter with new material on rotegrities and nexorades are particularly notable.
Popko's book is sure to spark the creative juices of readers who love geometry and who think about making physical spherical structures."
– George Hart, Mathematician and Sculptor
"Edward Popko and Christopher Kitrick's Divided Spheres unfolds the Orderly Subdivision of the Sphere into a comprehensive and growing survey of the historical, spherical, polyhedral, mathematical and material patterns inherent in such a universal question... A fundamental resource for anyone interested in exploring the order, and part-to-whole relationships of any formal system in design."
– Daniel Lopez-Perez, Associate Professor of Architecture and a founding faculty member of the Architecture Program at the University of San Diego, and author of R. Buckminster Fuller, Pattern-Thinking
"I value Divided Spheres highly, as a pithy primer for anyone wishing a fresh approach to spherical geometry, and to geometry in general. As a curriculum developer, I include it in any syllabus that also includes R. Buckminster Fuller's Synergetics, as a complementary and enlightening investigation. The pattern language of spheres comes through loud and clear in this book."
– Kirby Urner, 4D Solutions
"Divided Spheres is simply a tour de force. The book’s comprehensive coverage and clear, elegant writing are remarkable—and enhanced by compelling figures throughout the text. Popko and Kitrick have managed to make three-dimensional geometry accessible without oversimplifying their material.
– Amy C. Edmondson, PhD., Author of A Fuller Explanation: The Synergetic Geometry of Buckminster Fuller, Professor, Harvard Business School
"Ed Popko & Christopher Kitrick’s newly revised and updated edition of Popko’s book, Divided Spheres is a marvellously detailed homage to the geometry of the sphere and spherical polyhedra. The field has grown significantly since 2012 when the original book was published. This new edition elevates Divided Spheres as the definitive resource for the history of 20th century breakthroughs in geodesics by two practitioners who are part of that history. Full color computer generated images and excellent writings by Kitrick introduce fresh and unique approaches. The eloquent foreword by geodesics expert Joseph Clinton clarifies how designers, students, scholars or practitioners of any discipline can take their own pathways through the voluminous material in the book to find value and insights along the way. It also serves as a comprehensive handbook featuring a range of techniques for tessellating the sphere."
– Bonnie DeVarco, FRSE, MediaX Fellow
"If the sphere is the sovereign of the kingdom of geometric shapes, then this book could rightly be said to be the complete chronicle of its many deeds. Its contents will undoubtedly be quoted and used by scientists and artists the world over (including Japanese researchers or creators of traditional spherical artifacts such as Temari balls or Origami polyhedra) thanks to the crystal-clear explanations and the wonderful accompanying graphics, which helps in overcoming even the steepest language barriers."
– Koji Miyazaki, Professor Emeritus at Kyoto University and author of The Japanese Encyclopedia of Polygons (2015), Polyhedra (2016) and Polytopes (2020).
"Divided Spheres is about spherical subdivision techniques and their applications. It provides an authentic description of the results of Buckminster Fuller and his circle in this field, but beyond geodesic domes, it discusses several old and new practical applications in cartography, astronomy, virus research, dimple patterns of golf balls, fish farming and in many more subjects. Important geometrical problems related to spherical subdivision such as spherical circle packings and coverings, as well as polyhedra with extremal properties are also treated. It contains a detailed mathematical description of effective subdivision methods according to which the reader can execute calculations and determine the data of his/her geodesic network. The book is well written and beautifully illustrated and can be read by specialists and general readers alike."
– Tibor Tarnai, Professor Emeritus at Budapest University of Technology and Economics
"This is an exceptionally well-produced book on spherical subdivisions with beautiful, colored illustrations, all digital. This book is on a topic of continuing interest despite its long history, and it always brings new surprises like that gem on ‘self-organizing grids’ included in this edition. Ideas, old and new, are collected in this careful methodical work which brings together the history and advancement of spherical structures and their implications across different fields. It is written by two expert practitioners from the early pioneering days of the field. Their treatise is a reminder that rigor, depth and beauty go hand in hand, and are a requirement for lasting works like this."
– Haresh Lalvani, Pratt Institute
This is a fascinating and beautiful book on a narrow topic (spherical geometry) of widespread appeal and application. Naturally, the book harks back to R. Buckminster Fuller’s geodesic domes and the 31 great circles of a spherical icosahedron; the main author is an architect. But spherical design occurs also in pollen grains, viruses, maps, fish pens, and implementations of puzzles similar to Rubik’s Cube. It turns out that it is not easy to distribute points evenly on a sphere! Applications include locations of cellphone towers, orbits of satellites, golf balls (there is a whole chapter on them), and pharmaceuticals. The book develops six schemas for subdividing a sphere and offers comparisons for different purposes. Other chapters present basic spherical geometry and grids on spheres. Appendices treat at length stereographic projection, coordinate rotations via matrices, and geodesic math. Almost every page contains a figure or illustration, many of them in color.
– Mathematics Magazine, MAA
"The treatment begins with a nice introduction to spherical polygons and the connection between choosing points on a sphere and constructing a subdivision grid. This is the main target of the book – to examine the coordinatizing process for spherical surfaces. This theme is returned to throughout the book and six basic means of dividing the surface are examined in detail.
[. . .] The strength of this book lies in its illustrations. They are beautiful and draw one into a deeper examination of the actual surface geometry of a sphere as well as that of the immersion of the sphere in three-dimensional space. This central dichotomy is present all the way through the book and constitutes a type of “picture explanation” of spherical trigonometry. [. . .] I would certainly recommend the book to anyone interested in Fuller’s work as it supplies a more stream-lined and less jargon-filled account. If the aim of the book is to excite students and present Synergetics in a more traditional context then by any measure this has been a successful endeavor."
– MAA Reviews, MAA
Praise for the previous edition
"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 a all-reference for illumination to one of nature's most perfect invention."
—Thomas T. K. Zung, Senior Partner, Buckminster Fuller, Sadao & Zung Architects
"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 an-tiquity. 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
"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