Magnus Wenninger – Benedictine Monk and Polyhedral Modeler

Magnus Wenninger, Monk 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, Divided Spheres, 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.


Kenneth Snelson – Tensegrity Sculptor and Photographer

Ken Snelson, Sculptor 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.


Shoji Sadao, Architect, Cartographer and lifelong business partner of Buckminster Fuller

Shoji Sadao, ArchitectEd 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.


Thomas T. K. Zung, Senior Partner, Buckminster Fuller, Sadao & Zung Architects.
Thomas Zung, ArchitectEdward 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.


Steve Waterman – Mathematician

Steve WatermanMy 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 on spherical polyhedra. Novices and experts will understand the challenges and classic techniques of spherical design just by looking at the many beautiful illustrations.