1 Preface
2J. Frangois Gabriel
3 Henry Ford reputedly said that customers could have his automobiles in any color, as long as that color was black. A parallel can be drawn with the shape of our rooms, which could come in any shape but are essentially cubic. Very few rooms are perfect cubes, it is true; most are in the shape of flattened or elongated cubes, but the majority of our buildings are conceived as an assemblage of cubic forms, and that is what they look Eke: piles of shoe boxes.
4 We use the cube as if it were the only acceptable model for our living spaces and, in doing so, we ignore countless other forms that might lead to more efficient, more beautiful, more economical, and certainly less worn-out environments. Why do we do it? Mr. Ford told us we must drive his black cars, but who told us that we must dwell in square or rectangular spaces, bound by four vertical walls intersecting at right angles?
5 Would all the painters in the world agree to throw out all their colors and limit their palette to one color only? Would all the writers agree to limit their language to words of three syllables? Would all the composers…? Of course they would not. Yet, like it or not, most of us end up living in cubes, or nearcubes. This book makes a case for a family of shapes that often make more sense than the cube: polyhedra. Indeed, the cube itself is a polyhedron, and many of the forms used or described in this book have a direct, if not always obvious, relationship with the cube. My intention is to show, with the help of contributions from structural engineers, architects, historians, and others whose expertise spans several fields, that polyhedra provide all the elements for a formal language of extraordinary versatility that can satisfy the essential demands of buildings: solidity, beauty, and convenience.
6 Briefly, this book is organized as follows: We begin, logically enough, with a look at the past. The first chapter contains a historical survey of polyhedra. Then we discuss the attitudes of three great designers of the 20th century toward polyhedral forms. Chapters 5 to 7 look at a more recent past and show how space frames, formed by aggregates of polyhedra, shaped three important buildings, each using a space frame in an original way. In the next five chapters, we focus on the theoretical aspects of polyhedra, their formal bonds with the cube, the kinship that exists between one polyhedron and another, their symbolic meaning, their proportional relationships, their specific structure, and their representation. The subject of the three following chapters is the future: tensegrity, space labyrinths, and quasicrystals, all of which are in their experimental stage, but already suggesting architectural possibilities that may materialize soon.
7 The most important criterion of architecture is not about looks, but about the quality of the spaces within. An architecture of space frames and polyhedra will be viable if the spaces formed by it are at least as good (as convenient, as beautiful, etc.) as those found within conventional, cubic frameworks. This critical question is discussed in the last chapter of the book.
8 This is not the definitive book on architecture beyond the cube. Nor can it be, for the field of space frames and polyhedra is continually changing and expanding, enriched by the discovery of new configurations, new design methods, and new applications. However, a genuine effort is made to present a broad, accessible, and faithful picture of the state of the art.
9 Very few polyhedra are found in the natural world. Most of them are a creation of the mind. With experience in architecture and the history of structural design, Jos Tomlow looks in Chapter 1 at the discovery, the perception, and the use of polyhedra, from Pythagoras to Alexander Graham Bell. With few exceptions, polyhedra were not seen as structural or architectural objects. It is only in this century, and particularly in the last 50 years, that the connection was fully made. Interestingly enough, the representation of polyhedra emerges as one of the most intriguing aspects of their history. Everybody seriously involved with polyhedra knows the challenge of making their forms comprehensible to the onlooker, or even to oneself. The cube is the easiest polyhedron to draw, and I suspect this to be one of the explanations for its extravagant popularity. Of course, the price we pay for using it indiscriminately is a greatly impoverished spatial experience for all of us.
10 How did some of the most inventive minds of the first half of this century approach problems of structure and architectural modeling? Bruce Goff’s work is characterized by fantasy and ingenuity and he came close to a literal geometry, for he sometimes merged polyhedra and architecture into a single entity. He did not use simple geometric solids in isolation, which is easy to do but which might not be of great significance. He chose to shelter varied architectural functions in regular patterns of polyhedra, or infinite structures, as they are called. He gave polyhedral shapes to his rooms, and he also used the same polyhedra to create different room shapes in the same building, thus proving that it could be done without monotony. He also proved something else. Rollie Ristine, a student of Bruce Goff, shows in Chapter 2 that his teacher was not a geometer, and it is doubtful that he even knew by their names the polyhedra he used in his designs. However, he thought as an architect when giving order to space, and he came intuitively to geometry, showing that geometry is as good a way to organize space as any other. One might even wonder if there is any better way.
11 With Louis Kahn, in Chapter 3, we come to one of the most widely acclaimed architects of the postwar era. Unlike F. L. Wright, Le Corbusier, or Mies van der Rohe, Kahn had a formal architectural education and earned an architectural degree. The school he attended was traditional, even classical, since at the time architectural education in the United States was completely dominated by the Beaux-Arts system. Far from rebelling against his training, Kahn assimilated the fundamentals of classical architecture and reasserted its principles in his mature work. Simple geometric forms such as the square, the circle, and the equilateral triangle, bilateral symmetry, and clarity, which are typical of classical designs, are among the constants of Kahn’s work. The memorable impression made by his buildings is a direct outcome of classical principles. Geometry infuses Kahn’s designs, and it is no wonder that, under the influence of his collaborator Anne Griswold Tyng, he should have become interested in space frames. The building that first brought him to the attention of the architectural profession at large is the Yale University Art Gallery, where the floor structure is a space frame. The fascinating design for an office structure using a mega-space frame, although not built, gave him the aura of a prophet and made him famous worldwide. Irene Ayad shows how Kahn’s involvement with polyhedra fits in his architectural development.
12 Kahn, who was also a painter, was intensely visual. Buckminster Fuller, on the other hand, told me in the last year of his fife that he did not care how things looked. He seemed to think that, when things are done right, they end up looking right, for our aesthetic judgments are based on previous experience. Fuller’s geodesic domes, some very large and some rather small, are often as beautiful as pure polyhedra can be. And so is the space frame, which Fuller called the octettruss, and for which we owe him so much. However, those who believe that originality and personal expression are the first goals of architecture do not think much of structuralism. Fuller was neither an architect nor an engineer. He was essentially self-educated, and he has been variously called by others a poet, a philosopher, an inventor, an environmentalist, a scientist, an engineer, a maverick, a crackpot and, by himself, a comprehensivist. In 1952 he did receive an Award of Merit from the New York Chapter of the American Institute of Architects, and in 1960 the Gold Medal of the Philadelphia Chapter of the American Institute of Architects. It is fitting that Arthur Loeb, a maverick himself, should share with us personal recollections of a fruitful collaboration in Chapter 4, Buckminster Fuller and the Relevant Pattern.
13 Philip Johnson’s approach to design is radically different. Best described as a complex, sectioned, prismatic form, his Crystal Cathedral is a dramatic piece of abstract sculpture. It demonstrates in a masterful way that, far from hindering freedom of expression, space frames can lend themselves to the implementation of any building shape. It is not for ideological reasons that a space frame was selected by the architect, it is because nothing else could do the job, as Mr. Johnson explained to me in January of 1993. Familiar to millions of television worshippers who have seen it on Sunday mornings for 16 consecutive years, the building remains, on many levels, one of the most paradoxical of the second half of the century. In Chapter 5, Lawrence Davis gives us a fascinating account of these paradoxes.
14 The Javits Convention and Exhibition Center, by Pei, Cobb, and Freed, provides a neat contrast to the Crystal Cathedral. Other than its enormous size, the most impressive feature of the Javits Center is its designers’ sensitive recognition and acceptance of the geometry of a space frame. By geometry, I mean the shape of the ‘‘building blocks’’ and the pattern they form: octahedra juxtaposed to tetrahedra. As Matthys Levy, the engineer in charge of the project, shows us in Chapter 6, this pattern controls and, to a certain extent, determines the shape of the building. One admires, in particular, how smoothly space frames are made to ‘‘turn the corners’’ as well as the expressive effect obtained at the main entrance of this civic palace. Nothing extraneous was added to the space frame, and nothing was subtracted from it. Levy’s no-frills writing style seems perfectly suited to the restraint of the design.
15 Because of its location and because of its function, a certain austerity is expected in the facade of a building like the Javits Center. What was called for in the theme building of the exhibition ‘‘Portopia ‘81’’ in Kobe, Japan, to celebrate the completion of a large artificial island, is very different. The open and cheerful structure is achieved solely with space frames. The airy elegance of this building, all curves and smiles, makes its loss to demolition regrettable. Conceived to resist the effects of typhoons and seismic forces, it would have been interesting to see how it would have fared in the powerful 1995 earthquake. The chapter, entitled Double Curvature Space Frames, is written by the engineer in charge, the talented Masao Saitoh.
16 It cannot be said too often that a cubic frame with hinged joints is unstable, and that it requires some doctoring to be made indeformable. One treatment consists of adding one member to each square face, placed along one of its diagonals. If the shape one wishes to achieve is a cube, the presence of diagonal members on its faces changes nothing of the interior volume or its bounding surfaces. Because a tetrahedron can be the figure formed by the diagonals alone, it would have been more direct to ignore the cubic frame and adopt a tetrahedral frame to begin with. In other words, a cubic volume fits in a tetrahedral frame as well as it fits in a cube. This little bit of irreverent magic introduces Arthur Loeb’s chapter, Deconstruction of the Cube, where the coupling of tetrahedron to cube is shown to beget many other polyhedra. Names like Stella octangula and rhombic dodecahedron may sound a bit complicated, but they are descriptive and, once you know them, you find that they refer to interesting and friendly personalities. And know them one must if one is considering an expansion of design sources beyond the cube.
17 As indicated by the title of his chapter, The Polyhedral World, Pieter Huybers reaches away from the cube and sets out to understand the geometric laws that govern these shapes. This understanding is necessary if we are to adapt polyhedra to our architectural and structural needs. Archimedean solids, prisms and antiprisms, domes, and folded-plate structures are included along with space frames in the discussion, which contains a minimum of mathematics.
18 With Rene Motro in Chapter 10, we look for meaning in space frames and polyhedra. For as long as they have been known, polyhedra have aroused the interest of brilliant minds who, from Plato to Buckminster Fuller, have tried to understand the world as a synergy. The five ‘‘perfect’’ polyhedra are often referred to as Platonic solids, not because Plato discovered them but because each one symbolized one of the five elements in Pythagoras’s cosmology, in which he, Plato, was deeply interested. Thus the cube stands for the earth, whereas the octahedron stands for the air. However, symbolic meaning can be found in everything, and symbolism is not a science. According to Motro, however, one rule applies, and it stipulates that a symbol cannot be defined without suffering mutilation, distortion, or total elimination. As for proportion, it can be more than an attempt to please the eye. Proportion is related to symbolism when it is understood as an expression of divine harmony, that is, perfection. In reading Motro’s chapter, we will see that he is an engineer whose thinking is as clear as it is rigorous.
19 We return to earth, so to speak, with Ture Wester’s chapter, The Structural Morphology of Basic Polyhedra. Wester is also a structural engineer, and an imaginative one at that. He does not confine his thinking to post-and-beam structures stabilized by triangulation or other means. This would not do with polyhedra, the structural problems of which have little in common with those of their brother the cube. Three of the five regular polyhedra make perfect rigid lattices. They are the tetrahedron, the octahedron, and the icosahedron, which all have triangular faces. Three make perfect plate structures, and they are the cube, the dodecahedron and, again, the tetrahedron. These three have three-branched vertices. This observation forms the basis of Wester’s elegant general theory, structural duality, which leads to the formulation of simple rules for analyzing the rigidity of any arbitrary polyhedra, simple design methods for geometrically complicated but highly efficient plate structures, and other interesting possibilities. Some structures of the natural world are included in his demonstration.
20 Hoshyar Nooshin has developed a mathematical tool called forniex algebra for the processing of all kinds of configurations. The advent of the computer made the structural analysis of space frames easier and faster. In so doing it became the indirect cause of the profiferation of space frames from the 1950s on. CAD now plays an increasingly useful role in visualization and formal transformations, which are routine in all architectural and structural design, and are even more critical with noncubic forms, where the use of the traditional T-square may be too slow or considered old-fashioned. Nooshin’s chapter, written in collaboration with P. L. Disney and O. C. Champion, lays down the foundations of a comprehensive approach for computer-aided processing of polyhedral configurations.
21 Chapters 13 to 15 open new horizons for polyhedra in three directions. Tensegrity, a term coined by Buckminster Fuller, represents structures with discontinued compression. All structures include parts that are under compression. In tensegrity structures, these elements are not in direct contact with one another: They are held together by intermediate cables. Tensegrity structures are a marvel to behold. Ariel Hanaor, in Chapter 13, suggests that practical applications may be found in combining tensegrity with deployable, or retractable structures, which are brought on the site in bundles and erected rapidly with a variety of mechanical devices.
22 Space labyrinths are structures made of a continuous surface dividing space into two parts, one being the inside and the other one being the outside. In some cases, inside and outside are interchangeable and some labyrinths can be constructed with one single module. Space labyrinths are open-ended, meaning that they could theoretically go on forever. One can see from this rough description how space labyrinths could revolutionize our concept of architectural space. Haresh Lalvani conceived his chapter as a pictorial essay that could be part of a visual encyclopedia of form and structure. Higherdimensional diagrams show space labyrinths, some already known, others presented here for the first time, grouped in families. Included are hyperbolic and nonperiodic, that is, quasicrystalline, space labyrinths.
23 Tony Robbin defines quasicrystals as three-dimensional manifestations of higher-dimensional cubes. Essentially, quasicrystals are assemblies of two different polyhedra with similar topological properties, both derived from the cube, capable of clustering in compact arrangements and capable of forming nonrepeating patterns. This means that the pattern may duplicate itself, but not necessarily at predictable intervals. In his realizations, Robbin aims at making works of art, but he is aware of the structural and architectural capabilities of quasicrystals, which might be realized at full scale in the years ahead.
24 I have been interested for many years in the shape of spaces generated by space frames. I am even more interested in the architectural spaces that can be found within space frames. What are they like? How can they be connected with one another? How can they be accessed? To what use do they lend themselves? How do they compare with square rooms? How can they be built? Some of my investigations are reviewed in the last chapter of the book. The conclusive ones are given names, such as hexmods and star bea?ns. More are in progress, and many more remain to be discovered. I think we are in the prehistoric phases of discovery in an immense and promising field, which we have only begun to probe.
25 In general, there is a regrettable shortage of actual polyhedral buildings. This is difficult to understand for those of us who, having explored and experimented for years with these forms, marvel at their inexhaustible richness and have a vision of the poetry that emanates from some of them. This book presents a small but significant portion of the work done around the world by a number of architects, engineers, and others. Some speculate, and others build. Some do both. All our efforts are experiments, and many are successful enough to sustain our enthusiasm. Space frames and polyhedra will change our ways of building. Eventually, they will bring about a gentle revolution in the way we design architecture. I hope readers find pleasure in this book, as the material presented here should stimulate their imagination and encourage them to satisfy their curiosity.
27 The origin of this book can be traced to my teens, when a Monsieur Prevot taught me descriptive geometry for three years with wonderful toughness. He instilled in me a lasting desire to better understand complex relationships between form and space. On this foundation, the training I received at the Ecole des Beaux-Arts developed in me a great respect for the pure, basic, and beautiful forms on which classical architecture is based.
28 I owe my first insights into space structures to Robert Le Ricolais and Stephane Du Chateau, who both advised me on my thesis. Others who kindly encouraged my independent pursuits are Felix Candela, Keith Critchlow, Buckminster Fuller, Gulzar Haider, Zygmunt S. Makowski, Stefan Medwad- owski, Max Mengeringhausen, Peter Jon Pearce, Duncan R. Stuart, Moshe Safdie, and Yona Friedman.
29 The students I was privileged to teach, many of them American, played a large part in my continued interest. Their curiosity about spatial relationships, their desire to explore new forms, to understand structural action, and to integrate architecture and structure helped me learn more. John Ray Hoke, Jr., now a Fellow of the American Institute of Architects, was one of these students, and he convinced me to prepare this book.
30 I am particularly grateful to the fourteen contributors who put aside their own important activities to contribute their talent and their expertise to make this book a remarkable compendium, bringing together aspects of many fields bearing upon architecture: aesthetics, social history, structural engineering, and other fields that do not accept well-defined boundaries. Several contributors also advised me on important matters related to the book.
31 I am also grateful to the following, who generously gave of their time and wisdom to read and comment on the contents: Edward J. Applewhite, writer, Washington, D.C.; Professor Thomas F. Banchoff, Mathematics Department, Brown University; Dr. John Chilton, Department of Architecture and Planning, University of Nottingham; Dr. H. Martyn Cundy, Kendal, Cumbria, U.K.; Larry Wayne Grantham, architect, Foley, Alabama; Professor Yasuhiko Hangai, Institute of Industrial Science, University of Tokyo; Datuk Lim Chong Keat, Emmanuel College, University of Cambridge; Matthys Levy, Weidlinger Associates, Inc., Consulting Engineers, New York; Dr. Rowland J. Mainstone, D. Eng., Hon. R.I.B.A.; Dr. Robert C. Meurant, Institute of Traditional Studies, Auckland; Professeur Rene Motro, Laboratoire de Mecanique et Genie Civil, Universite Montpellier II; Professor John O’Brien, School of Architecture, University of Tennessee, Knoxville; Peter Jon Pearce, Pearce Research and Design, Studio City, California; Professor Theoman Pekdz, Department of Civil Engineering, Cornell University; Professor Luis Sanchez-Cuenca, Departament d’Arquitectura i Enginyeria de la Construccio, Universitat de Girona; Professor Franz Schulze, Department of Art, Lake Forest College; Professor Ronald Shaeffer, School of Architecture, Florida A & M University; Dr. Charles H. Thornton, Thornton-Tomasetti Engineers, New York; Dr. Anne Griswold Tyng, F.A.I.A.; Professor Patricia Waddy, President of the Society of Architectural Historians; and Bruno Zevi, Director of Architettura.
32 A number of people helped in several ways. Bruce Abbey, Dean of the School of Architecture at Syracuse University, was instrumental in obtaining for me the leave of absence needed to complete the project. Dr. Gershon Vin-cow, Vice-Chancellor for Academic Affairs, approved the leave. Dr. Ben Ware, Vice-President for Research and Graduate Affairs, believed in the value of my work. Amanda L. Miller, the editor at John Wiley and Sons, Inc., with her assistants Mary Masi and MaryAlice Yates, gave, in addition to technical guidance, the support of her never-flagging enthusiasm for the book project during the entire process. P.G. Wodehouse helped me keep things in perspective.
33 This book is dedicated to Laura A. Martin, my wife.
34 Beyond the Cube