Divided Spheres

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

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

Ben-Dor (law, U. of Southampton) developed this book concerning critical constitutionalism from his doctoral thesis at University College London. In it, he interprets unpublished and recently published texts by Jeremy Bentham (1748-1832), first reconstructing the most general argument about Bentham's legal and political thought as a founder of utilitarianism, and then analyzing Bentham's work within the context of contemporary debates in legal and political philosophy. He concludes that the technical and reductionist methodology associated with utilitarianism don't do justice to the theory, which identifies the maximization of pleasure as the most fundamental self-interest guiding people. Distributed by ISBS. c. Book News Inc.

This book constitutes the refereed proceedings of the International Symposium on Surgery Simulation and Soft Tissue Modeling, IS4TM 2003, held in Juan-Les-Pins, France in June 2003. The 33 revised full papers presented together with 3 invited papers were carefully reviewed and selected from 45 submissions. The papers are organized in topical sections on soft tissue models, haptic rendering, cardiac modeling, and patient specific simulators.

Friendship has always been a universal category of human relationships and an influential motif in literature, but it is rarely discussed as a theme in its own right. In her study of how friendship gives direction and shape to new ideas and novel strategies of plot, character formation, and style in the British novel from the 1760s to the 1830s, Katrin Berndt argues that friendship functions as a literary expression of philosophical values in a genre that explores the psychology and the interactions of the individual in modern society. In the literary historical period in which the novel became established as a modern genre, friend characters were omnipresent, reflecting enlightenment philosophy’s definition of friendship as a bond that civilized public and private interactions and was considered essential for the attainment of happiness. Berndt’s analyses of genre-defining novels by Frances Brooke, Mary Shelley, Sarah Scott, Helen Maria Williams, Charlotte Lennox, Walter Scott, Jane Austen, and Maria Edgeworth show that the significance of friendship and the increasing variety of novelistic forms and topics represent an overlooked dynamic in the novel’s literary history. Contributing to our understanding of the complex interplay of philosophical, socio-cultural and literary discourses that shaped British fiction in the later Hanoverian decades, Berndt’s book demonstrates that novels have conceived the modern individual not in opposition to, but in interaction with society, continuing Enlightenment debates about how to share the lives and the experiences of others.