III. Energetic Architecture - Buckminster Fuller’s Geometry of the Sphere

Fuller’s "Building Blocks"

Four hundred years after Durer and Kepler, Buckminster Fuller continued a similar process of experimental observation of structure in three dimensions. Fuller's approach to design was influenced by his Navy experience. During a long introduction to the design of ships on the sea, Fuller paid attention to designs which contained new angles and curves in order to navigate through a continually shifting, fluid medium. In his earliest writing, a 1928 document titled "Lightful Housing," he introduced a "Theory of the Spheres." In this paper he contended, "all matter in unforced state is spheroidal not cubistic, and these spheres are expanding for the life of their existence at a fixed rate."15 A very different version of this essay appeared in Fuller's self-published 1928 book titled "4-D Timelock." For the rest of his life he unraveled this way of looking at the earth from a spherical perspective. Finally in 1975 and 1979 respectively, Fuller released Synergetics and Synergetics 2 presenting the complete system of dynamic geometry from which he derived the geodesic dome, his icosahedral map and his octet truss building system.

For Fuller there were no flat planes. Everything was curved, from space to shape. Everything was in motion and was continually shifting. But the classical shapes remain as guideposts to form found throughout the Universe. His approach the same problems that earlier geometricians and artists tackled by "tiling the plane" or building polyhedra was through the closest packing of spheres. Delineating vector lines within closest packed spheres, Fuller defined the basic polyhedra that could be used as dynamic building blocks on a larger scale.

Fuller proposed that the tetrahedron, octahedron and icosahedron were the most important building blocks of nature. Fuller was the first to describe the tetrahedron as the simplest structural system with insideness and outsideness, and it was his most important building block, the form on which the rest of synergetic geometry hinged. The tetrahedron, with its four faces and four vertexes, was the three-dimensional form that could contain the least volume. It was the simplest "system" containing a set of relationships. Regardless of the earlier references to the family of regular polyhedra and their significance in life's architecture on a moving, spherical earth, humans had latched onto the cube as the main building block of mathematics. For Fuller, the 90 degree angles of the cube were a side effect or "precessional effect" of various processes in a universe of angles, curves and arcs. His cube was inscribed by the duotet, two interpenetrating tetrahedra whose eight outer points met cube's eight vertices and gave it an inherent stability.

Closest Packed Spheres and Space-Filling

 Like Kepler before him, Fuller was enamored with sphere packing. He generated his vector systems from the closest packing of spheres. Vector lines could be drawn from the center of one sphere to the center of the other. All of the vertices, then, in this closest packed group of unit radius spheres would be found in the center of each sphere at which each vector meets the others. A simple way to visualize this packing formation is to put three spheres close together on the table so that they all touch each other. Where they each touch each other Fuller called "the kissing point of spheres."

 Drawing a line from the center point of each sphere to the center points of the others, you will inscribe a vector triangle. But that "system" of vector relationships does not have an inside and an outside. If you place a sphere on top, centered on the three spheres and connect the centers of each of the three to the center of the top one, you will have four vertices -- a tetrahedron -- the simplest "system" with insideness and outsideness. Fuller started with the topology of closest packed spheres to map the vectorial relationships between them. He always worked in three dimensions, never two. Unlike the geometers before him, Fuller did not start with a point, then a line, then a plane to which he then added dimension. He started in the center of the sphere and out in all directions. The lines or vectors represented energy, direction and time. 

As space-fillers, tetrahedra cannot pack together. They will instead insist on twisting into a "tetrahelix" -- a helical structure made of tetrahedrons which expands in either direction. The same could be said for the octahedron if you tried to stack one atop another. In addition, if you truncate each of the four points of a tetrahedron, you will have a beautiful octahedra laying on its side in the middle. But if you pack tetrahedrons and octahedrons together, you can fill space with ease. The presence of 60-degree angles within a lattice of tetrahedrons and octahedrons gives the final structure an inherent cross-bracing in all directions.

All builders know that while cubes fill space perfectly, any cube or square with 90-degree angles must either be made of strong materials or have cross bracing to ensure the building's structural strength. Fuller's octet truss system makes the 60-degree angles of cross bracing primary. Any 90-degree angle in Fuller's system appears only as a side effect, inscribing various vector relationships throughout the isotropic vector matrix (IVM). He called this new, inherently strong space-filling matrix an "octet truss." Although the octet truss had already been built by Alexander Graham Bell, Fuller’s early investigations left him unaware of Bell’s architectures. He did not know about Bell’s strong, lightweight and beautiful octet truss kites out of wood with metal braces He was not familiar with Bell’s octet truss tetrahedral tower braced together with a custom-designed hub system of metal spheres. Fuller became aware of Bell’s structures well after he had discovered and demonstrated the strength of the octet truss sytem both in theory and in practice. 16a

Fuller found also that if an octet space filling truss was built outwardly in successive shellings it would result in a successive series of nested polyhedra, the tetrahedron, octahedron and the cuboctahedron, or "vector equilibrium" (with six square faces and eight triangular faces on its outer surface) replicating the structures of its components on a larger scale. He called this larger trusswork an "isotropic vector matrix" because the angles and strut lengths were everywhere the same.

Isotropic vector matrixes were known and described well before Fuller named them such. Coxeter published a picture of a tetrahedral octet matrix he called "solid tessellation" in a mathematics publication as early as 1939.16b Matila Ghyka, in his interdisciplinary masterwork of 1946, The Geometry of Art and Life showed a similar image called an "isotropic partition of space by four sets of planes" in a section on closest packing and the cuboctahedron.16c Although Fuller was not aware of Coxeter's solid tesselation, there is evidence that he was familiar with Ghyka's. So what was so different about Fuller's isotropic vector matrix? Through it, he introduced this matrix of 60-degree angles as a model of primary coordinate system from which all other systems could be generated. In addition, Fuller attempted to effectively replace the platonic and Archimedean "solids" as vectored "energy systems" and built them with struts with flexible connectors in order to demonstrate how they transform from one shape to another in time.

In a closest packed group of 12 spheres around one, a vector equilibrium can be inscribed. Fuller called his simple vector equilibrium, when made out of flexible hubs and struts, a "jitterbug" because it twisted to exhibit, while in continuous motion, a series of shapes which accommodate and transform into one another. In its most open stage, it is the cuboctahedron. If it is twisted and contracted, it will become an open icosahedron with six struts missing and with one more contraction it will become the octahedron. It can then be folded down further into a tetrahedron and finally to a simple triangle. Then, simply unfold, untwist and the jitterbug pops back to its original shape, the cuboctahedron or, in Fuller’s dynamic system, the Vector Equilibrium.

The key to Fuller's jitterbug was its ability to embody and demonstrate the "motion" and transformation of polyhedral forms. Fuller's jitterbug could be considered the first polyhedral model in more than two thousand years of mathematical and structural exploration that can demonstrate the energy characteristics of expansion and contraction. Because of the natural twist, like the spiral of a nautilus shell which draws its form from the mathematical rules of golden proportion, the jitterbug in motion can move through symmetrical forms which, if omnitriangulated internally, will span the oscillating continuum of symmetrical and asymmetrical form.

Fuller insisted that since there were no straight lines, the lines that emerge from the 90 degree angles of the "ghostly cube" are wave forms resembling straight lines. His cube was merely the outer "case" of a negative and one positive tetrahedron -- the duotet -- which was meant to be seen in motion. An octahedron (the dual of the cube) can be inscribed in the center of the two interpenetrating tetrahedra. Only in this duotet configuration is the cube completely stable with a diagonal edge of one of the tetrahedra at each face. The duotet was known and depicted graphically well before Fuller, however. Pacioli deemed it "raised octahedron" and pictured it in his 1509 treatise, "De Divina Proportione." Leonardo da Vinci called it the "star tetrahedron" and in the 1600s Kepler gave it its more popular title, "stella octangula." In all of its manifestations, two interpenetrating tetrahedra have represented the most important fact about the tetrahedron -- that it is the only regular polyhedron that is its own self-dual with exactly the same number of vertices as faces. Fuller placed his duotet as the energetic corollary to his vector equilibrium and allowed it to replace the cube in his synergetic geometry.

Polyhedra as Energy Systems

Fuller’s deep understanding of spherical and polyhedral forms as energy systems was a view both profound and unprecedented in the history of mathematics and geometry. He sought to prove how the same design elements were reflected in these dynamic forms on every scale from micro to the macro. In the same way that Fuller looked at new "qualities" such as weight, functionality and flexibility in his approach to shelter design, he imbued mathematical forms with energetic, behavioral qualities that chemists, biologists and physicists use to describe the effects of pressure and heat kinetics in a system or chemical bonding such as Avogadro's law and Willard Gibbs' phase rule.17

Fuller proposed that the behaviors, qualities and design elements of energy systems could be understood and utilized to build stronger, more functional structures using less and less material. Fuller created an economical approach to shelter design by applying the dynamics of these "energized" forms acting in concert with one another to structure on a larger scale. Through the application of "synergetic-energetic geometry" Fuller offered two new building systems. One was the strong and stable octet truss, the network of tetrahedra and octahedra, and the more spherical system of geodesics in which he omni-triangulated the sphere to create strong, yet flexible triangulated spheres or portions of spheres of any size. Each of these structures used the tetrahedron as its base. Although a lightweight geodesic structure had already been designed and built by Walter Bauersfeld in 1922, it was Fuller who, unaware of the Bauersfeld’s Zeiss planetarium in Jena, systematized a mathematical system of omni-triangulating the sphere. This system called geodesics which Fuller patented in 1954 laid the foundation on which geodesics based on the icosahedron could be built in any size and frequency. It also ushered in a whole field of lightweight geodesic architecture.

In the same way that Fuller produced a form of narrative geometry of the sphere based on three-dimensional modeling, Durer created a descriptive geometry of curves based on two-dimensional geometry. Although he made some polyhedral models, most of Durer's geometric studies were in two dimensions. This great contribution was also overlooked though history, yet modern crystallographer Donald Caspar has recently returned to Durer's early tessellations as well as Kepler's and superimposed them with the most recently discovered quasi-periodic tilings of Roger Penrose in order to better understand the growth patterns of quasi-crystals in 1996.18 Fuller's energetic geometry of the sphere is now being overlooked similarly, yet may still have much to contribute to the newest findings in chemistry and physics and to the newest challenges and opportunities of bio- and nanotechnology.

Where the proto geometry of Plato, Pythagoras and Archimedes dealt with structure as if it were frozen, solid and symmetrical, Fuller's geometry was concerned with mapping the intersection between the worlds of the physical and metaphysical, the oscillating continuum of symmetry and asymmetry. His focus was on the dynamics of bodies in motion before they are crystallized into form. While geometry more than twenty centuries ago emerged through the explorations of "philomorphs" or lovers of form, Fuller himself, working with the same problems of design and structure, could be called a "philovigeo" or lover of energy. The early vector systems Fuller used to model the Platonic solids in a new way helped him visualize energy by "mapping" the invisible latticework of forces within structure. The fluid processes of transformation helped him to describe these energy systems and latticeworks of forces. His geodesic domes, especially the largest almost full-sphere domes such as the Montreal Expo '67 dome Fuller designed and built with architect Shoji Sadao, and tensegrity spheres demonstrated a dynamic architecture based on nature's principles.

Energizing the Sphere

At the beginning of his life Fuller's focus was shelter. His first architectural applications were toward finding the best design for a "machine for living." While most architectural approaches used materials to introduce structural strength, his shelter system relied on the strongest designs -- designs which could be made out of flexible materials of the lightest weight. Fuller observed dynamics which could make such lightweight shelter systems possible in the undulating curves and spherical qualities of nature rather than the hard cubic designs of the industrial world. In his earliest unpublished writings on this system he called "Lightful Housing," Fuller articulated an entirely new approach to design based on the geometry of the sphere.

... Now here is the real statement of things as they are visible or invisible, soundful or silent, tangible or intangible, etc. ALL MATTER visible or invisible is naturally a perfect sphere or spheroidal in shape. Due to dynamic forces and contacts of other spheres of greater kinetic energy none of our spheres are perfect in shape. In as much as they cannot exist without time they all have a time dimension. ALL THE SPHERES OF MATERIAL THINGS BE THEY OF OUR BODY OR OF STONE (organic or inorganic, so improperly called) ARE, like radio waves, EXPANDING SPHERES AND THEY KEEP ON EXPANDING ON A FIXED WAVE LENGTH FOR THE TIME LIMIT OF THEIR EXISTENCE. As a cross sectional demonstration of the fourth dimension, let us go to the stone dropped in the pool of water it creates waves which radiate out until they are overpowered with some dynamic force, when they subside. Measuring from the center where the stone was dropped to the final extreme outer circle reached we have the time limit or life of existence. "Now returning to mathematics that is the abstract one that of arithmetic and algebra we find that they, knowing no material bounds were correct, and that inasmuch as we can have as many right angle lines addressing the surface of a sphere as we may wish to infinity, we can therefore keep raising any figure, indicative of a mathematical division of matter, to any power we wish, merely indicating the size of that much greater sphere by the distance from the center of the sphere by that much greater a time unit. In as much as time then is space (not a line) between the center and the surface which would be indicated by the radii of the sphere we realize that time alone is the shortest distance between two points, for starting with matter actually as it is and taking the roots of it or dividing it up we can do so to any degree we wish for we can never eliminate matter by dividing it.19a

Fuller's earliest approaches to geometry and the harmony of the sphere drew unconsciously from those who came before, yet introduced the metaphysical concept of ephemeralization, an expansion into the weightlessness of space. He treated the sphere from the center outward as a moving, expanding structure, not as a solid finished structure.

Fuller's primary building block for subdividing a sphere was not a cube but a tetrahedron. The tetrahedron is the simplest structural system with insideness and outsideness, the simplest system which can contain complexity. Thus, the tetrahedron contains all the other polyhedra and can be contained within the other polyhedra. Through omnitriangulating the sphere, Fuller designed the geodesic dome and the icosahedral projection world map (he called it a transformation rather than a projection because rather than projecting from the outside it was generated from the central point in the interior of the sphere). This map is very different than standard world maps which subdivide the spherical surface with a grid of squares, causing large discrepancies in the depiction of the earth's surface. Like Durer's tesselations which introduced new angles to the basic 90-degree grid, Fuller's newer triangular grid system accomodated the curves of the sphere with different angles. His map with its 20 triangular faces could be used folded into an icosahedron or laid out flat. Its surface contains the least amount of distortion of the relative sizes and shapes of the continents than any world maps that came before it.

Fuller's icosahedral map and his geodesic domes demonstrate the same approach to the subdivision of the sphere, both based on the tetrahedron. All full geodesic spheres based on the icosahedron will have 12 pentagons surrounded by different numbers of hexagons. Through an elaborate system of chord factors, calculated by the measurement from the center of the geodesic sphere to various different points on its surface, each vector's length and its angular relationship to the others could be drawn. Fuller called this system "geodesic math." [During the early years it took many man hours to mathematically calculate the chord factors for each geodesic dome using slide rules and tables; the same can be done today in a heartbeat through an advanced geodesics computer program].

Fuller's geometry does not identify a linear hierarchy. His hierarchy went outward in all directions and inward as well -- movement in his polyhedral hierarchy was always omnidirectional. Fuller used the term, "frequency" to denote the various levels of complexity of systems. The higher the frequency, the higher number of internal relationships (a simple example would be to compare his 18-frequency Expo '67 dome with a small three frequency dome that a lot of builders made into homes). There is no up and down in synergetics, only in, out and around. Space is curved. The only point is the dynamic center. The dynamics are everywhere and every thing interconnected and in continual motion.

In synergetics, the basis for Fuller's geodesic math, the relationships between components rather than the components themselves were the focus; the associations were primary forces and parts were secondary elements. Hence, his understanding of frequency growth was the omnidirectional amplification of relationships. This use of the term, frequency to describe the omnidirectional growth of systems is an elegant approach to understanding both simplicity and complexity.

Fuller's models exhibit a quality of universality. Like those before him, Fuller used a language as ancient as speech and dance -- the language of numbers for an architecture based on the geometry of the sphere. In the making of things, in the doing of things, in experience and real events, Fuller attempted to learn and demonstrate what was relevant and what was not in this form of dynamic architecture. By looking at nature, he learned what worked and what didn't. Fuller looked for the rules that worked in all cases -- generalized principles rather than special-case.

Table of Contents

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References

copyright 1997, Bonnie DeVarco