Praise for the previous edition [. ] Dr. Popkoâs 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. â 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. â 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. â 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 Fullerâs 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 . Language: English