What This Book Is
1Geodesics is a technique for making shell-like structures that hold themselves up without supporting columns, by exploiting a three-way grid of tensile forces. They are very strong and can be very large: the geodesic bubble erected to house the United States exhibits at Expo ‘67 in Montréal encloses some 6 million cubic feet; it is approximately three-fourths of a sphere 250 feet in diameter. They are also very light for what they do: the Montréal bubble weighs about 600 tons, Plexiglas skin and all. It has been calculated that a geodesic sphere approximately one-half mile in diameter would float away like a soap bubble if the air inside it were one degree warmer than the air outside. They have been used as homes, as offices, as fair pavilions, as locomotive round-houses, as gymnasiums, as auditoriums, as banks, as playground structures for children to climb on, as housings for radar installations on the DEW line, and for observatories buried under snow at the South Pole. Yet, considering their apparent potential, in the quarter-century since Buckminster Fuller introduced them they haven’t been used very widely.
2 That is partly because they are mathematically derived structures, and the mathematics hasn’t been easily available. Parts for the self-supporting frame must be fabricated to close specifications. The fabrication, with today’s technology, is no problem; the problem is learning what the specifications should be. If we know them, we can achieve extraordinary savings of material, weight, and effort. If we don’t, we have no resource save one of the conventional methods of building, which by geodesic standards means gross overbuilding.
3 My assumption is that if architects, designers, engineers knew how to get past the first step, which is calculating the pertinent details of a geodesic structure’s geometry, they would explore geodesic potentials more than they have. This book shows how to do that—and without access to a large computer. Hobbyists and math buffs should find it interesting, too.
4 It would help, moreover, if geodesic design could be disentangled from its historical reliance on spheres. More than once a geodesic approach has been shunned because the designer, for one good reason or for several, didn’t want his structure to resemble a slice from a sphere. But anyone who masters the design procedure in this book will find that, once he has mastered geodesic spheres, geodesic eggs, whether tall or flat, are only one step more complicated, and free-form contours are no more difficult than eggs. You can work from an equation if you have one; or you can even draw a contour freehand and calculate without much trouble an equivalent geodesic structure.
5 Finally, the great strength of geodesic structures needs demystifying. They are always much lighter and tougher than people expect, partly because we are used to thinking that walls bear weight the way posts do, and are unprepared for the resources of a hidden tensile system. In fact, Fuller’s geodesic domes constitute a special case of a larger class of Fuller constructs called Tensegrities, and the way to an intuitive understanding of domes is to understand Tensegrity first. So the book begins by developing Tensegrity mathematics, something hitherto unexplored. (That may be one reason no useful structures exploiting pure Tensegrity—tension wholly separated from compression—have been built at all.)
6 As to what you can do with the calculations: you can immediately build fascinating models, out of sticks and string and soda straws and paper fasteners. If you want to build something useful to get inside of, you will be well advised to build a model first.
7 After that you will encounter other problems for which I have no help to offer. I can show you how to calculate, to any degree of accuracy you need, the length of every member in the framing system, and every angle governing their junctions. You yourself will have to decide what to make the members from, how to join them, how to apply a skin made out of what, how to waterproof and fenestrate and ventilate the structure, what to do with the space inside. All this entails a whole new building technology which, despite some spectacular achievements, may be estimated to be still in the log-cabin stage. If inability to make the first calculations has helped ensure that people who might be solving such problems are employing their time otherwise, then this book may help unblock development.
8 There is no use in anyone’s pretending that there are no problems.
9 There are problems in keeping an ordinary flat roof from leaking, problems generations of building experience have pretty well solved. When someone nails plywood triangles onto a wooden geodesic frame and discovers that cold weather pulls the joints apart, he has no more discredited geodesics than a leaky outhouse discredits the procedures that will ensure a snug bungalow. He has merely proved that one kind of geodesic technology won’t work. (Fiberglassing the whole exterior of a wooden dome apparently will work, and may be the cheapest way to get a homogenous shell. Or there may be better ways.)
10 Strength calculations remain another problem area. Many geodesic structures—including Fuller’s own first commercial building, the dome over the Ford rotunda in Dearborn, Michigan—have been overbuilt because no one really knew what the minimum framing system would be. That hidden tensile net defeats post-and-beam calculations. Shell analysis—which in effect postulates an infinity of components of zero dimension—is said to yield the most reliable results, but one would think something less distant from structural reality could be devised. Possibly this book’s approach via Tensegrity may suggest to some imaginative engineer the way to go.
11 The Tensegrities themselves are tantalizing. Everyone who sees a model of one—resilient, nearly indestructible despite its local fragilities, recovering like a rubber ball from gross deformations—thinks at once that a building framed like that would be virtually invulnerable to earthquakes. So far as I know, no building has ever been framed like that: the skin, for one thing, would need to be as flexible as the frame. Fuller at one time intended that the Montreal bubble should be a giant Tensegrity, but time and budget inhibited the necessary research. If Tensegrity has a practical use, other than yielding models and pieces of sculpture and helping us understand geodesics, the first principles of that usefulness remain to be investigated. By making a rigorous design procedure available, this book may get some visionary genius started. I trust tests will be run before any contracts are let. I have absolutely no idea how any of these designs would behave on a large scale under load. I make no claims whatever in that domain and decline all responsibility for misfortune.
12 I am neither an engineer nor a mathematician. I’ve checked most things out with small models, but that has been the extent of my hardware implementation. I’ve thought it best to assume that the reader’s mathematics is about where mine was when Bucky Fuller’s proximity got me interested in this subject: algebra and high-school trig, inoperative from decades of disuse. I have also had the great advantage of not having access to a large digital computer. Computers in the past have helped obfuscate the subject by making unsuitable methods workable. It was in trying to avoid the sheer donkey work a computer shoulders uncomplainingly that I stumbled on the applicability of the spherical coordinate system, which I came across in Professor George E. Owen’s Fundamentals of Scientific Mathematics (Baltimore, Johns Hopkins Press: 1961) [Owe61]. To avoid mind-numbing weariness, you do need a good pocket calculator with trig and power functions. I used the Hewlett-Packard HP-35, which can handle any procedure in this book without one’s having to write down intermediate results. For the more cumbersome equations, I give HP-35 and HP-21 routines here. Users of other calculators will have to devise their own.
13 For repetitive work—and much work on large-scale geodesic structures is repetitive—the programmable HP-65 is of course still better. All you do is enter data; it repeats routines indefinitely without chance of error. I got one when I started compiling the tables. But the simpler machine works fine.
14 A word for mathematicians. Conceptually this book is very simple. Part 1 offers an essentially Newtonian analysis of Tensegrities, treating them as diagrams of their own system of equilibrated forces, and shows how they can be designed by using trig to predict their equilibrium states. There is also a brief account of an approach using calculus, which depends on the fact that the tension system seeks a minimum length. Part 2 simply applies in detail the elementary insight that spherical structures are best described by spherical coordinates. Books explain this coordinate system as a sort of conceptual metaphor but don’t seem to explain how to use it, so I’ve had to devise most of the algorithms myself.
15 They may not be the most elegant possible, but they all work.
16 At three points where my own resources failed me, I received invaluable help from Professor Paul Kelly, University of California, Santa Barbara; Mr. Roland O. Davis of Goleta, California; and Professors George Owen and Rufus Isaacs at The Johns Hopkins University.
17 A letter from Professor H.S.M. Coxeter, explaining a tantalizing regularity I’d noticed, gave the key to a simple method of finding the coordinates themselves; Mrs. Cindy Engers first glimpsed the way of putting it into workable form.
18 Much of the book’s terminology is that of Joe Clinton, whose Advanced Structural Design Concepts for Future Space Missions (NASA Contract NGR-14-008-002, 1970) [Lau+68] remains the pioneer job of systematizing geodesic geometry. I am grateful to him for a copy. When the definitive treatment of geodesics is compiled, he will be at the center of the enterprise. Mine is meanwhile an interim report, to go on from. Peter Calthorpe, Lloyd Kahn, and Bob Easton were generous with help when I was getting started. All future workers will be indebted to Kahn’s Domebook 1 (Los Gatos, Ca., Pacific Domes: 1970) [Kha70] and Domebook 2 (Bolinas, Ca., Pacific Domes: 1971) [Kha71] for first making information accessible. And historians will some day disentangle from the early history of the art the contributions of Donald Richter and Duncan Stuart, neither of whom I’ve met. Much of the time I’ve been aware of handling ideas—I don’t know which ones—that they and others first formulated; just as much of my analysis of Tensegrity was incited by structures Tony Pugh first showed me, which in turn embodied concepts Kenneth Snelson—unknown to Tony—had once intuited. Everyone in this list is indebted—as am I—to Buckminster Fuller, instigator nonpareil.