Richard Buckminster Fuller (1895-1983, Bucky, as he has become to be known) was one of the brilliant group of 20th-century engineers that followed on Thomas Edison and Alexander Bell's legacy of exploration and discovery. Like many in that group, which included Edwin Armstrong (inventor of AM and FM radio technology) and many of the physicists of the Manhattan project, Fuller was self-taught in engineering, and this disestablishment background may well have been instrumental in his maverick approach to everything he did. He developed ideas for everything from toys to cosmography, causing change and consternation everywhere he turned his attention.
Fuller did not invent geodesic domes for architectural use; that honor belongs to Dr. Walter Bauersfeld, the director of the planetarium project at the Carl Zeiss optical works in Germany, and his co-workers. Zeiss wanted to build a huge projector sphere that would both hold a large number of people and show the motions of the planets as well as the stars, what we today call a planetarium. After years of thought Dr. Bauersfeld developed the idea of creating a dome and placing the projectors in the center inside the dome rather than outside the screen. He developed and built such a dome on the top of the Zeiss building in Jena, Germany in 1919; before turning it over to a building contractor to complete, he designed and built the skeleton for it from 3,480 struts accurate in length to .002 inch, a remarkable feat of computation, design and fabrication. Many Bucky advocates dismiss Bauersfeld's design in giving Bucky credit (Bucky himself used some of his usual double-talk to denigrate it) but that is just not tenable.
What Fuller did, though, was to point out that the dome had a lot more in its favor than its roundness. He developed a theory of the dome from the principles of tensegrity (one of the unending stream of new words that Bucky was ever fond of generating), and showed that the dome was completely self supporting. Ultimately, he patented the design. This is important, and I'll tell you why.
The first dome structures were developed in prehistoric antiquity, as a way in which common stone, despite it's lack of stretching strength, could be used without continuous support, by channeling the stress due to gravity on the stone and its load into places where stones high compressive strength could be used. The Roman's famous arched aqueducts and the medieval flying buttresses used in the medieval cathedrals are examples of arch support. Arches depend on being able to channel downward forces sideways, where they are balanced by similar forces on both sides. That makes the a single, standing arch or dome (like the dome of St. Peter's cathedral in Rome) difficult to build, because it has no brothers beside it to balance the sideways forces. Normally a dome like St. Peter's would tend to bulge outwards from the forces created only by the dome's own weight, and be squashed like an eggshell pressed down by a book. The medieval answer to that was to create an band around the dome to hold it together; in St. Peter's case, a huge chain that encircles the dome bottom to keep it from expanding outwards.
But it turns out that the geodesic dome is different. It does not need such a band to keep it from squashing - the tension needed to keep it from flying apart is distributed throughout the structure, as long as it is complete (a whole sphere, or equivalently supported). This ability allows for extremes of structural lightness. It is estimated that a strutted geodesic dome 1/2 mile in diameter could be built that would support itself and a Canadian snow-load, and which would float free if the air inside it were 2 degrees higher than that of it's environment; such buoyancy cannot be achieved by any other dome technology. (This is not to say, however, that a loaded dome, one with, say, a really heavy snow load and wind load, could not collapse - simply that the dome can easily support itself against gravitic forces. A loaded dome has to be strengthened for that load, though to much less an extent than other dome designs would have to be.)
So, how are geodesic domes constructed? The basic idea is to start with a regular geometrical solid whose vertexes (Bucky called them "vertexia", probably out of pique at the arguments between the supporters of traditional, irregular "verticies" and the regular form that I used) describe a sphere. All regular geometrical solids do so, and even irregular solids can be used, though the resulting domes are not smoothly regular, so there are many possible starting points. Only three are usually considered: the Tetrahedron (four equilaterally-triangular faces), the Octahedron (eight faces) and the Icosahedron (twenty faces). Each of these solids has the property that from the center of the solid all their vertexes lie on the same sphere (they are all equidistant from the center). These solids can, of course, be realized just as they are, and are referred to as domes of frequency one. The ancient Egyptian pyramids and the new Louvre pyramid in Paris represent frequency 1 domes of half of an octahedron. Hereafter I will call the equilaterally-triangular faces of the frequency 1 solids "superfaces", for reasons that will soon become apparent.
Higher frequency domes are executed by taking each of the superfaces of the solid and dividing it up. If the evenly-spaced dividing lines parallel one of the triangle's edges the division is called Class I (or "alternate", though I haven't found out what it is alternative to). If the divisions are through a superface's vertex and the center of the opposite edge, it is Class II (or "triacon"). Finally, Class III is any other division. The number of dividing lines (plus 1) defines the frequency of the resulting solid. Class II divisions only exist in even frequencies. The small triangles that result from the division of a superface are called simply "faces", they will be the actual faces of the dome when we are through.
By playing with GeoDome you can look at the various combinations of domes that are created by these rules. The most regular domes originate from the icosahedron (because it has the most superfaces to start with), and most people seem to prefer them. Octahedral domes are good for creating 1/4 and 1/8 sphere sections because they are symmetric on all three axes. Tetrahedron spheres have a great variation in triangle sizes, and look a bit anarchic, but interesting for all that. Class II frequency divisions can give rise to what are called "kite-shaped" facets (by combining pairs of triangles together), and have excellent construction properties.
Now that we have the superfaces all divided up into the final faces, there is one final, crucial step to perform. Draw a line from the center of the sphere through each of the small face corners, and extend the line outwards until it meets the sphere that is defined by the verticies of the initial solids. That is the final resting place of each corner. This makes every corner of every triangle, super or small faced, rest on the sphere; they all define the same sphere, given the center point. Stretching the triangles this way makes them slightly different than they were in repose on the primary solid faces; they are all a bit longer, and some longer than others. This stretching is also what gives a dome it's strength - since the triangles have been expanded, forcing them back into the plane of the superface would involve breaking and buckling the struts that form the faces, which is mechanically much more difficult than simply depressing a flat surface.
Since all the superfaces we started with on the solid were identical, the development of each face is likewise identical, so we only need to consider one such superface to have considered them all. The diagram on the left depicts what happens with a superface from any of the basic solids, after the triangle is divided to frequency 3, class I, and then developed into a segment of the sphere; the basic pattern can be inferred into all frequencies. On the right is the same face at frequency 4, Class II. The faces close to the triangle's corner stretch less then those close to the center, so the edge lengths are a bit shorter. In the class II depiction, "kites" may be built out of pairs of faces , making them single faces. The spectacular point about that simple transformation is that all the kites thus developed, while a bit more difficult to physically construct than triangles, are all identically sized, making mass-production efforts easier and cheaper.
The final construction choice we face is the truncation of the sphere. A complete sphere is a handsome object, but generally one wants only a part of the sphere - say, 1/2 of the sphere or some other amount, with the rest cut off by a plane at which all the struts become embedded in a support structure and the dome is thereby opened. This truncation plane usually sits atop a cylindrical wall that affords access into the dome from below. Domes generally have truncations from 25% to 75% of the diameter of the sphere; truncation outside those limits yields a rather unsteady balance, or too little overhead for the dome to be useful. Note that to keep the strength of the sphere while slicing it one has to firmly embed the struts in an rigid support medium.
There are as many construction techniques for building domes as there are dome types. The most famous spheres in the world, as well as those created for temporary recreational purposes, use the popular strut techniques with relatively fragile or filmy cover materials. For permanent personal structures more conventional wood covering material may be used; the struts may be metal or wood. Some examples are created using prefabricated wood triangles, which are self-strutted. I've read that Fuller recommended construction using, essentially, 4'x8' sheets of plywood as the struts, with additional layers added to close the gaps; the result was very functional (the strength of a slightly bowed 1/4" plywood sheet can be amazing).
One final word. Building a dome is not all beer and skittles; there are costs involved which may make them uneconomical. Most of these objections are beginning to be addressed by manufacturers as more structures are built, but there is a long way to go before domes become the common man's home as Bucky predicted in the 1950's.
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