In 1976 literary critic Hugh Kenner published this fully-illustrated practical manual for the construction of geodesic domes, which had been invented 25 years previously by R. Buckminster Fuller. Now returned to print for the first time since 1990.

The aim of this book is to present the fundamental concepts and properties of the geodesic flow of a closed Riemannian manifold. The topics covered are close to my research interests. An important goal here is to describe properties of the geodesic flow which do not require curvature assumptions. A typical example of such a property and a central result in this work is Mane's formula that relates the topological entropy of the geodesic flow with the exponential growth rate of the average numbers of geodesic arcs between two points in the manifold. The material here can be reasonably covered in a one-semester course. I have in mind an audience with prior exposure to the fundamentals of Riemannian geometry and dynamical systems. I am very grateful for the assistance and criticism of several people in preparing the text. In particular, I wish to thank Leonardo Macarini and Nelson Moller who helped me with the writing of the first two chapters and the figures. Gonzalo Tomaria caught several errors and contributed with helpful suggestions. Pablo Spallanzani wrote solutions to several of the exercises. I have used his solutions to write many of the hints and answers. I also wish to thank the referee for a very careful reading of the manuscript and for a large number of comments with corrections and suggestions for improvement.

It was 1976-twenty-five years after R. Buckminster Fuller introduced geodesic domes when literary critic Hugh Kenner published this fully-illustrated practical manual for their construction. Now, some twenty-five years later, Geodesic Math and How to Use It again presents a systematic method of design and provides a step-by-step method for producing mathematical specifications for orthodox geodesic domes, as well as for a variety of elliptical, super-elliptical, and other nonspherical contours. Out of print since 1990, Geodesic Math and How To Use It is California's most requested backlist title. This edition is fully illustrated with complete original appendices.

The book provides an introduction to Differential Geometry of Curves and Surfaces. The theory of curves starts with a discussion of possible definitions of the concept of curve, proving in particular the classification of 1-dimensional manifolds. We then present the classical local theory of parametrized plane and space curves (curves in n-dimensional space are discussed in the complementary material): curvature, torsion, Frenet’s formulas and the fundamental theorem of the local theory of curves. Then, after a self-contained presentation of degree theory for continuous self-maps of the circumference, we study the global theory of plane curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of class C2, and Hopf theorem on the rotation number of closed simple curves. The local theory of surfaces begins with a comparison of the concept of parametrized (i.e., immersed) surface with the concept of regular (i.e., embedded) surface. We then develop the basic differential geometry of surfaces in R3: definitions, examples, differentiable maps and functions, tangent vectors (presented both as vectors tangent to curves in the surface and as derivations on germs of differentiable functions; we shall consistently use both approaches in the whole book) and orientation. Next we study the several notions of curvature on a surface, stressing both the geometrical meaning of the objects introduced and the algebraic/analytical methods needed to study them via the Gauss map, up to the proof of Gauss’ Teorema Egregium. Then we introduce vector fields on a surface (flow, first integrals, integral curves) and geodesics (definition, basic properties, geodesic curvature, and, in the complementary material, a full proof of minimizing properties of geodesics and of the Hopf-Rinow theorem for surfaces). Then we shall present a proof of the celebrated Gauss-Bonnet theorem, both in its local and in its global form, using basic properties (fully proved in the complementary material) of triangulations of surfaces. As an application, we shall prove the Poincaré-Hopf theorem on zeroes of vector fields. Finally, the last chapter will be devoted to several important results on the global theory of surfaces, like for instance the characterization of surfaces with constant Gaussian curvature, and the orientability of compact surfaces in R3.

Each of the models in this book is both beautiful and interesting to make. Each also plays its part as a hands-on introduction to geodesic domes. With the aid of its models, this book explains the underlying theory for designing geodesic domes and shows how a sphere can be divided and subdivided symmetrically in order to create dramatic buildings which are light and strong and also have no need of internal support.

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

X 1 O S R Cher lecteur, J'entre bien tard dans la sphere etroite des ecrivains au double alphabet, moi qui, il y a plus de quarante ans deja, avais accueilli sur mes terres un general epris de mathematiques. JI m'avait parle de ses projets grandioses en promettant d'ailleurs de m'envoyer ses ouvrages de geometrie. Je suis entiche de geometrie et c'est d'elle dontje voudrais vous parler, oh! certes pas de toute la geometrie, mais de celle que fait l'artisan qui taille, burine, amene, gauchit, peaufine les formes. Mon interet pour le probleme dont je veux vous entretenir ici, je le dois a un ami ebeniste. En effet comme je rendais un jour visite il cet ami, je le trouvai dans son atelier affaire a un tour. Il se retourna bientot, puis, rayonnant, me tendit une sorte de toupie et me dit: {laquo}Monsieur Besse, vous qui calculez les formes avec vos grimoires, que pensez-vous de ceci?)) Je le regardai interloque. Il poursuivit: {laquo}Regardez! Si vous prenez ce collier de laine et si vous le maintenez fermement avec un doigt place n'importe ou sur la toupie, eh bien! la toupie passera toujours juste en son interieur, sans laisser le moindre espace.)) Je rentrai chez moi, fort etonne, car sa toupie etait loin d'etre une boule. Je me mis alors au travail ...

This volume presents lectures given at the Summer School Wisła 18: Nonlinear PDEs, Their Geometry, and Applications, which took place from August 20 - 30th, 2018 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures in the first part of this volume were delivered by experts in nonlinear differential equations and their applications to physics. Original research articles from members of the school comprise the second part of this volume. Much of the latter half of the volume complements the methods expounded in the first half by illustrating additional applications of geometric theory of differential equations. Various subjects are covered, providing readers a glimpse of current research. Other topics covered include thermodynamics, meteorology, and the Monge–Ampère equations. Researchers interested in the applications of nonlinear differential equations to physics will find this volume particularly useful. A knowledge of differential geometry is recommended for the first portion of the book, as well as a familiarity with basic concepts in physics.

The first edition of Connections was chosen by the National Association of Publishers (USA) as the best book in OC Mathematics, Chemistry, and Astronomy OCo Professional and ReferenceOCO in 1991. It has been a comprehensive reference in design science, bringing together in a single volume material from the areas of proportion in architecture and design, tilings and patterns, polyhedra, and symmetry. The book presents both theory and practice and has more than 750 illustrations. It is suitable for research in a variety of fields and as an aid to teaching a course in the mathematics of design. It has been influential in stimulating the burgeoning interest in the relationship between mathematics and design. In the second edition there are five new sections, supplementary, as well as a new preface describing the advances in design science since the publication of the first edition. Contents: Proportion in Architecture; Similarity; The Golden Mean; Graphs; Tilings with Polygons; Two-Dimensional Networks and Lattices; Polyhedra: Platonic Solids; Transformation of the Platonic Solids I; Transformation of the Platonic Solids II; Polyhedra: Space Filling; Isometries and Mirrors; Symmetry of the Plane. Readership: Polytechnic students, architects, designers, mathematicians and general readers."