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Geodesic Domes: Design Variables
The study of great-circle patterns seems at first like a completely abstract endeavor. However, the construction of models demonstrates that spheres with a greater number of shorter arc-segments have a significant structural advantage over simpler structures, and suggests potential applicability. Fuller's early models, based directly on great-circle patterns showed considerable strength but did not go far enough. A new method of generating geodesic structures was needed to produce higher-frequency structureswith less variation among chord lengthsthan the 31-circle pattern. This progression led Fuller to concentrate years of attention on icosahedral geodesic designs. He discovered that the inherently self-stabilizing geodesic polyhedron could be truncated as the basis of a stable dome (Fig. 15-6). These domes, which could be any fraction of a geodesic sphere, must sit on the ground to complete a "system." Depending on the situation, a half-sphere truncation, 3/4, 1/8, or any number of other fractions might be desirable. Fuller saw that the possibilities were endless; geodesic domes could be designed to be built out of almost any material at any frequency.
The geodesic pattern is also a variable. In addition to the above "checkerboard," Fuller developed a number of other "geodesic breakdowns." An icosahedral face can be subdivided by a variety of patternseach one yielding a different design for potential domes. The most common breakdown subdivides each triangular face with lines parallel to its three edges as explained above (Fig. 15-7a). The number of segments along each icosahedral edge specifies the frequency of the resulting geodesic dome, and the number of triangles per icosa face will always be f ² for an "f-frequency" structure. A geodesic dome might be 2- or 32-frequency depending on size and material; there's no inherent upper limit, but the exponentially increasing numbers of different strut types become prohibitively complicated at very high frequencies.
The second breakdown subdivides icosahedral faces into triangles with lines perpendicular to icosahedral edges, producing slightly fewer triangles on the overall structure than the parallel version (12 per face for a 4-frequency breakdown, as compared to 16) and is somewhat less symmetrical (Fig. 15-7b). Fuller's diamond pattern is yet another choice (Fig. 15-7c). This design is unstable unless constructed out of panels rather than struts. If the situation allows mass production of panels, the diamond pattern will have an advantage over the first breakdown because of its fewer different types of faces. A number of aluminum domes have been built according to this design.
Finally, the deliberate omission of certain chords in Figure 15-7a produces Fuller's "basket-weave" design, characterized by hexagons and pentagons entirely framed by triangles (Fig. 15-7d). This pattern lends itself to building domes out of bamboo for example: struts in a three-way weave are simply tied together at crossings.
There are many other potential designs. The field is as wide open as the number of ways to symmetrically subdivide an icosahedron with great-circle chords. The fundamental characteristics of the resulting enclosures are the same. Geometric principles are exploited to gain unprecedented structural efficiency, just as nature, in response to the interplay of physical forces and the constraints of space, produces icosahedral geodesic patterns for many enclosures.
Geodesic domes of virtually unlimited size can be built by increasing the frequency as needed. The mathematics behind this undertaking is cumbersome but not conceptually difficult. Calculations are simplified (or at least kept under control) by an understanding of symmetry: complete information for an entire dome of any frequency is contained within its LCD triangle. Using the formulae of spherical trigonometry, we can manipulate the central and surface angles of spherical triangles to derive values with which to obtain strut lengths. Struts are great-circle chords, and each one is subtended by a specific central angle (Fig. 15-8a).
Depending on construction methods and materials, we might strive to keep struts as close to the same length as possible, or instead try to develop triangles with maximally similar shapes, or work toward any number of other preferred solutions. Frequency is itself an important variable; greater numbers of shorter struts may be more efficient in terms of supporting loads, but construction is correspondingly more difficult. Such tradeoffs must be carefully weighed, and fortunately trigonometry allows enormous flexibility in terms of potential solutions. Taking advantage of symmetry, we are able to experiment with different strut lengths in the LCD triangle, thereby only manipulating angles and lengths for a very small portion of the whole system, while developing the mathematical specifications for an entire geodesic dome (Fig. 15-8b). Finally, geodesic domes can be elongated or pear-shaped, or (theoretically) even shaped like elephants. The constant curvature of a sphere produces the greatest strength, but these other options do exist and can be developed with no more than a pocket calculator and a lot of paper.
As a final note, it is important to realize that while the theory behind geodesic domes is strikingly simpleand previously contemplated by others before Fuller was granted U.S. Patent 2,682,235 in 1954the actual translation from theory to practical structures involved fantastically intricate mathematical development. As with most invention and design, the initial insight was not enough to produce a 250-foot-diameter clear-spanning structure overnight. The subsequent calculations required enormous aptitude and perseverance. Consider the precision necessary to have six strutS meeting at the same point, at thousands of different vertices; minute errors in strut lengths at only a few points will accumulate and produce vast discrepancies elsewhere on the dome. Dealing with tolerances similar to that of the aircraft industry rather than the relatively crude building world, Fuller had to develop absolutely reliable trigonometric data to enable the construction of extremely large domes.
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