Spherical Models

Photo A. Icosidodecahedron with pentagrams.

Photo B. An 8-frequency geodesic dome.

Spherical Models

Magnus J. Wenninger

DOVER PUBLICATIONS, INC.

Mineola, New York

Copyright

Copyright © 1979 by Cambridge University Press

New Appendix copyright © 1999 by Magnus J. Wenninger

All rights reserved.

Bibliographical Note

This Dover edition, first published in 1999, is an unabridged and corrected republication of the work originally published by Cambridge University Press, Cambridge, England, in 1979. The original color frontispiece has been reproduced here on the inside front cover. A new Appendix has been added and the References have been updated in the present edition.

Library of Congress Cataloging-in-Publication Data

Wenninger, Magnus J.

Spherical models / Magnus J. Wenninger, p. cm.

Originally published: Cambridge [Eng.]; New York : Cambridge University

Press, 1979.

Includes bibliographical references.

eISBN 13: 978-0-486-14365-1

1. Sphere—Models. 2. Geodesic domes—Models. I. Title.

QA491. W43 1999

516'. 15—dc21                                                         99-045829

Manufactured in the United States by Courier Corporation

40921X03

www.doverpublications.com

to the memory of my father and my mother

and

to Professor H. S. M. Coxeter without whose inspiration this book could never have been written

Certe in Dei Creatoris mente consistit Deo coaetemo figurarum harum veritas.

Beyond doubt the true form of all these shapes exists eternally in the mind of God the Creator.

Johann Kepler, 1611

Contents

Foreword by Arthur L. Loeb

Preface

Introduction: Basic properties of

the sphere

I.The regular spherical models

The spherical hexahedron or cube

General instructions for

making models

The spherical octahedron

The spherical tetrahedron

The spherical icosahedron

and dodecahedron

The polyhedral kaleidoscope

Summary

II.The semiregular spherical models

The spherical cuboctahedron

The spherical icosidodecahedron

Spherical triangles as characteristic

triangles

The five truncated regular

spherical models

The rhombic spherical models

The rhombitruncated spherical models

The snub forms as spherical models

The spherical duals

Summary

III.Variations

Regular and semiregular variations

Star-faced spherical models

IV.Geodesic domes

The simplest geodesic domes

Geodesic domes derived from

the icosahedron

General instructions for making

geodesic models

An alternative method of approaching

geodesic segmentation

Introduction to geodesic symbolism

and classification

Geodesic models derived from the

dodecahedron

An alternative for geodesic

segmentation of the dodecahedron

A second alternative for geodesic

segmentation of the icosahedron

An alternative for geodesic

segmentation of the snub

dodecahedron

A third alternative for geodesic

segmentation of the icosahedron

Final comments

V.Miscellaneous models

Honeycomb models, edge models,

and nolids

An introduction to the notion of

polyhedral density

Edge models of stellated forms

Some final comments about

geodesic domes

Epilogue

Appendix

References

List of models

Foreword

In order to provide the reader with an indication of his standards of judgment, a book reviewer must carefully weigh and clearly state the pros and cons of any book. Unqualified praise is suspect, for even the most excellent of books will be graced by some features that will displease some reviewer.

It is very gratifying to find that a minor criticism in my review of Magnus Wenninger’s superb Polyhedron models may in some small measure have provided an impetus for his present Spherical models. I am delighted to find my role changed from reviewer of the earlier volume to contributor of a foreword to the latter.

The projection of polyhedrons onto a spherical surface has distinct conceptual advantages. Although not flat, the surfaces of such polyhedrons are twodimensional, allowing two degrees of freedom of travel on them. The spherical surface provides these polyhedrons with a standard frame of reference, having two degrees of freedom, on which to represent, transform, and interrelate these forms without change in the radial coordinates of their vertices, edges, and faces. Although the rational relationships between the volumes of some of these solids are obscured by spherical projection, edge lengths and distances obtain simple relations, expressible as arc lengths of great and small circles on the surface of the sphere. Furthermore, spherical projection relates polyhedrons to dome structures, since edges of these polyhedrons constitute or relate to geodesics, that is, to paths of minimal length between two points on the surface of a sphere.

The tessellation, or tiling, of a surface has fascinated artists and designers for many centuries. The tiling of a plane is easily related to that of a cylinder, because the plane may be rolled up into a cylinder. Certain constraints need to be specified along the seam, but basically the problem of tiling the cylinder is not very different from that of tiling the plane. In turn, a cylinder may be bent so that its ends join; the result is a toroid. Aside from another seam constraint, and some distortion resulting from the bending, the problem of tiling a toroid does not essentially differ from that of tiling the plane. Sphere tessellations, on the other hand, follow entirely different rules, which are essentially those determining polyhedral configurations. The present book, then, is a book concerned in a unified manner with polyhedrons, sphere tilings, and dome structures.

The beauty of Magnus Wenninger’s models is beyond doubt. It would have been easy for him to display these live and through photographs, withholding the secrets of their manufacture, or to veil their mathematics in obscure formulas or symbolism, restricting their accessibility to a cultural elite. Instead the author has devised an optimum method of construction, using minimal, and principally planar, trigonometry. Teachers of design science can now provide their students with construction materials and this book and feel confident that successful models will emerge. And once these attractive models become prevalent, they are bound to influence both environmental art and architecture and, thus, have a very positive effect on our visual environment. We owe Magnus Wenninger a debt of gratitude for making his jewels, or three-dimensional mandalas, so accessible!

Arthur L. Loeb

April, 1978

Cambridge, Massachusetts

Preface

Since the publication of my book Polyhedron models (Cambridge University Press, 1971), one of my ambitions has been to clarify for the general reader the section entitled Mathematical classification, pp. 4-10, of that book. Even so highly qualified a reader as Arthur L. Loeb of the Department of Visual and Environmental Studies at Harvard University commented in his review of the book: This section suffers from trying at once too much and too little: in its conciseness it is very difficult to follow and not really extensive enough to provide a sufficient background to the relationships between the various polyhedra (Leonardo, vol. 7, p. 73).

It is my hope that the present book will come to the aid of any interested reader and supply what is needed for a better understanding of polyhedral relationships. On p. 4 of Polyhedron models I said: The ideas are easier to visualize with the aid of models, and the same is true here. Fortunately, the methods used for making the three spherical models illustrated on p. 7 of Polyhedron models can be generalized. Therefore the present book is simply entitled Spherical models and may be considered a companion volume to Polyhedron models, though by no means as comprehensive in scope.

The spherical models of this book are closely related to geodesic domes. Many people are vaguely aware of the relationship of polyhedral forms to geodesic domes. Polyhedron models generally are found to be very attractive mainly because of their symmetries. This remains equally true for the spherical models given here. It will, therefore, also be the aim of this book to show explicitly the relationship between polyhedrons and geodesic domes and to show how models of such domes can be made in paper.

Even though all the models illustrated in this book are truly spherical, it may be in place here to say immediately that the mathematics involved remains elementary throughout. The book will be directed mainly