GEODESIC DOMES A Fuller Understanding of Applied Geometry

T he geodesic dome originates from one man’s effort to model universal patterns of force. Working in the mid-

20th century, Buckminster Fuller sensed that a coherent mathematical system could account for structural similarities among natural phenomena of diverse scale and material. He observed that interacting fields of force move toward equilibrium and that structures organize themselves according to the requirements of minimum energy. Fuller considered his approach “energetic geometry,” adopting “synergetic-energetic geometry” and later “synergetics” to emphasize synergy, that notion that a whole system is greater than the sum of its parts. As Fuller sought to apply synergetic principles to human constructions, geodesic domes arose from the nexus of mathematical theory and practical design. Understanding the mathematics of geodesic domes requires preliminary knowledge about polyhedra, three-dimensional figures whose vertices join an equivalent number of identical faces. To take advantage of the most rigid planar figure, Fuller used polyhedra with triangular faces as his basic building blocks. Models of struts and connectors demonstrate the strength of a triangle. When flexible connectors join struts of a square or a larger polygon, an applied force can distort the figure, but when flexible connectors join three struts in a triangle, no force can alter the structure without deforming or breaking the connectors. Triangles are central to Fuller’s concept of a system, a collective of interrelated elements that divides the universe into two parts: that which it contains and that which is external. Four non-coplanar points represent the minimum system, or the simplest way to enclose space. The Greeks called a symmetrical arrangement of four such points a tetrahedron, referring to its four triangular sides. Since energy automatically travels along the triangulating diagonals of any polygon to which force is applied, triangles also represent economical energy

GEODESIC

DOMES

A Fuller Understanding of Applied GeometryLAURA GARZON ʻ02

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DARTMOUTH UNDERGRADUATE JOURNAL OF SCIENCE4 SPRING 2003 5

configurations. From these observations, Fuller concluded that “omni triangulated, omni symmetric systems require the least energy effort to effect and regenerate their own structural stability” (Marks 43). Tetrahedron has too few faces to form a practical basis for spherical design, but octahedron and icosahedron are at the core of Fuller’s domes (Figure 1).

Figure 1. Symmetrical Polyhedra with Triangular Faces: Tetrahedron (4 faces, 4 vertices, 6 edges) Octahedron (8 faces, 6 vertices, 12 edges) and Icosahedron (20 faces, 12 vertices, 30 edges) (Edmondson 208).

The sphere that circumscribes octahedron and icosahedron guides the progression from polyhedra to domes. Since everything in the universe is in motion, Fuller emphasized the spin of systems. He identified three types of axes of rotational symmetry in polyhedra that “connect pairs of either polar-opposite vertices, mid-edge points, or face centers” (Edmondson 209). Spinning about one of these axes, a polyhedron generates a circle midway between the poles, like an equator. This “great circle” has a center that corresponds with the center of the sphere and a plane that divides the sphere in two equal parts (Pugh 56). For instance, an icosahedron spinning on axes that link 6 pairs of opposite vertices will thereby produce 6 great circles. It can also rotate on axes between 10 pairs of opposite faces to produce an additional 10 and on axes linking 15 pairs of opposite edges to make 15 great circles. The number of great circles for icosahedron thus totals 6+10+15, or 31 (Figure 2). Since their paths offer the most direct connection between any two points on a sphere, great circles incorporate the “mathematical phenomenon known as a geodesic,” or the shortest distance between two points across a surface (Fuller 1975: 373). To Fuller, these routes “represented the least possible expenditure of time and energy,” and he reasoned that an intersecting network of geodesic lines could provide the geometry for materials-efficient structures (Baldwin 117). Great circles also show potential to generate particularly sturdy structures, since they divide the circumscribing sphere of polyhedra into triangles.

Figure 2. The Great Circles of Icosahedron: Six great circles; 10 great circles ; 15 great circles (Kenner 50); 31 total great circles (Edmondson 214).

Fuller attempted to construct a dome directly from the great circle patterns of primary polyhedra in 1949 while teaching at a college in North Carolina. His class produced a 15m-diameter “hemisphere out of old Venetian blinds, bolted together where they met at the intersections of 31 great circles,” but the blinds proved too flexible to stand (Pawley 116). At an architecture school in South Africa in 1958, he used the pattern of icosahedral great circles to construct a 5m dome with strips of corrugated aluminum. This structure withstood testing, and the students suggested that it could serve as a “truly economic Native housing unit” (Indlu 85). Fuller agreed that geodesic principles can produce cost-effective shelters, but by then he understood the drawbacks of any great circle design, whether it use recycled blinds or metal sheets. The diverse size and shape of the triangles that form from the intersecting circles can be disadvantageous, since “load distribution and resulting strength” is a function of symmetry (Edmondson 234). Also, limited arrangements of great circles do not “present a logical course for further subdivision” toward “progressively larger models with sufficient strength” (Edmondson 234). In fact, Fuller had spent much of the 1950’s working outside academia to produce practical structures for residential and industrial

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use since he believed that his “geometrical analogy” could help “defeat the old economy of scarcity” (Pawley 123). Observing icosahedral symmetry in natural enclosures like the eyes of flies and the protein shell of many viruses, Fuller was confident that synergetics described universal laws of efficient shape that could apply at any scale (Edmondson 238). These lofty visions took Fuller back to the basic polyhedra searching for more effective ways to develop geodesic domes. To make use of conventional flat and straight material, he replaced curved sections of great circles between vertices with chords. The change actually creates “a structural system of maximum economy because chords are shorter than arcs” (Marks 43). Preserving spherical shape through this transformation requires a method for shifting planar octahedron and icosahedron toward their circumscribing spheres. First, Fuller subdivided their faces with series of lines. He referred to the number of subdivisions as frequency, the synergetic term for “the number of times a repeating phenomena occurs within a specified interval” of space (Edmondson 66). Dividing a face with one line parallel to each original edge creates a 2-frequency (2v) polyhedron with 4 smaller triangles. Likewise, dividing a face with two lines parallel to each original edge creates a 3-frequency (3v) polyhedron with 9 triangles. Increasing frequency increases the number of resulting triangles by powers of 2, since number of triangles resulting in a polyhedra of frequency n is n^2. So, 2v has 4 triangles, 3v has 9, 4v has 16, 5v has 25, 6 has 36, and so on (Figure 3).

Figure 3: Subdivision Frequencies in a Triangular Polyhedron Face (Pugh 57).

To approximate a sphere, Fuller then projected all the resulting vertices to an equal distance from the figure’s center. The move alters chord lengths to create some scalene and isosceles triangles, as in 4v octahedron, or 4v octa (Figure 4). Since icosahedron’s 20 faces are closer to the imagined sphere than octahedron’s 8, the edges of 4v icosa remain “more nearly alike and the small triangles more nearly equilateral” than those of 4v octa (Kenner 38). Whether they belong to octa or icosa, triangles differ less at higher frequencies. Also, since increasing frequency multiplies the numbers of chords along with faces and vertices, higher frequency produces shorter, more stress-resistant chords. Each new vertex joins a greater number of struts to distribute forces in many directions. Furthermore, polyhedra with more vertices closer together and more chords closer to arcs become increasingly spherical.

Figure 4: a) Octahedron Projects to 4v octa. b) Icosahedron Projects to 4v icosa (Kenner 38)

The spherical polyhedra that arose from this process of subdivision provide the framework for the domes Fuller produced commercially. In the Cold War years of the early 1950’s, US Defense Department sought out Fuller’s designs for air-transportable, weather-hardy radar installations along the Arctic Circle. The 17m-diameter “radomes” he produced represented 75% of a sphere (Baldwin 110). A few years later, the Marine Corps commissioned pre-assembled shelters transportable by helicopter. The resulting 15m hemispheres were “only 3% of the weight of traditional tents” and required “6% of the packing volume” and “14% of the cost “ (Pawley 132). Fuller began working for the business sector in 1953, building a rotunda for the Ford Motor Company’s main showcase(Figure 5). His geodesic domes began appearing before a wider public when he designed a 30m dome for the US pavilion at the 1957 International Trade Fair. Increasingly large domes covered American exhibits at shows years afterwards, from the 60m dome at 1959 World’s Fair in Moscow to the 80m dome at Expo ‘67 in Montreal. To reinforce large, flatter domes that represent less than half a sphere, Fuller devised a surface truss of tetrahedral struts (Figure 5).

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Smaller and more hemispherical designs do not require these trusses, nor do they demand the high frequency subdivision evident in structures like the Ford rotunda. Domes of 15m or less, like those that serve as homes or greenhouses, usually derive from 3-frequency polyhedra (Baldwin 77). Throughout the 1960’s, eager amateurs with utopian visions attempted to construct their own geodesic domes using whatever materials they had and the mathematics they could muster. Seeing a potential for profit, commercial builders began offering more professional, precise designs. Domes became popular during the energy crisis of the 1970s and the “staggering cost of energy” in the early 1990s created renewed interest (Knauer 1992: 29). In 1996, 200,000 geodesic domes worldwide enclosed “far more space than the work of any other architect” (Baldwin 119). Sources on the Internet today list almost forty dome manufacturers in the United States, and many more around the world (Rader 2001). With their spherical design, geodesic domes in fact prove highly resource-efficient. A sphere contains the greatest amount of volume with the least amount of surface of any figure, and domes, as truncated reflections of spherical polyhedra, inherit the benefits of this ratio. For instance, a cube-shaped house 20m wide will contain 8,000m3 within four walls and a flat roof, totaling 2,000 m2 of surface area. Using the same 2,000 m2 of surface, a hemisphere contains approximately 1.5 times the volume. As with other three-dimensional figures, larger domes have smaller surface-to-volume ratios. The surface area of a sphere equals 4πr2 and its volume equals 4/3πr3, so surface increases by powers of 2 while its volume increases by powers of 3. Bigger domes also conserve more energy as well, since a smaller percentage of the air they contain touches the surface where most heat escapes or enters. Meanwhile, the aerodynamics of domes promotes passive

heating and cooling, making them “30-40% more energy efficient than traditional houses” (Knauer 1992: 31). Wind slides more smoothly over the curved exterior of domes than over conventional buildings, creating fewer disturbances to the outer layer of air that insulates against heat loss from inside. With tactful ventilation, their concave interiors can also facilitate rolling currents that pull cool air from a hole in the apex and release warm air through openings along the side (Figure 6).

Figure 6: Passive Cooling with Rolling Currents (Baldwin 115).

The “Fuller” explanation of a dome’s efficiency involves the concept of tensional integrity, or tensegrity in synergetic terminology. Tension and compression function simultaneously in any system, but one usually dominates over the other. Ever since ancient architects began piling up stones, constructions have depended mostly on compression. Historically, domes have also relied on bulk to sustain their load. The world’s largest domes before the mid-20th century, Rome’s St. Peters and the Pantheon, each span about 50 meters and weigh 15,000 tons; Fuller’s first geodesic dome of the same diameter weighed one-thousand times less (Pawley 115). Geodesic domes exploit tensional behavior rather than

Figure 5: Dome at Expo ʻ67 in Montreal (left), Ford Rotunda Dome (right)

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exclusively local compressional behavior (Fuller 1975: 372). A “system of equilibrated omni directional stresses” results, and local pressure transmits uniformly throughout the structure (Kenner 5). Since each part of the structure need not receive loads unassisted, valuable economies in material are possible. Also, while compression pushes to eventually bend stressed parts, tension pulls to reinforce their shape. Assuming increasingly strong materials, there is no “geometrical limit to the length of a tension component” (Edmondson 246). In fact, Fuller provided the calculations for a dome large enough to cover fifty blocks of Manhattan that could protect the downtown area from rain and snow while controlling sunlight and air quality. Embracing the natural order of structure, synergetics forms the “discipline behind Fuller’s fantastic visions of a sustainable future” (Edmondson 1). More than half a century later, synergetic principles continue to

influence construction. Domes enclose contemporary industrial facilities and concert halls. City planners use geodesic domes as low-cost shelters for the homeless, and enterprising builders choose hemispheres for lower energy bills (“Domes” 1994: 95). Radomes still exist in remote stations, having weathered severe climates and the harsh test of time. Nonetheless, domes have not become conspicuously widespread. Mass-produced housing follows standard rectilinear blueprints and custom architecture comes at high costs. Also, domes require tactful truncation to fit flatly on the ground. Their internal layouts must accommodate straight-sided furnishings, and their curved form complicates expansion possibilities. The placement of windows and doors can prove problematic as well. These practical difficulties do not undermine the logic behind the Fuller’s dome, however. Geodesic domes derive from spherical polyhedra that can function as efficiently in

Bukminster Fullerʼs Carbondale Home

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theoretical models of force, in the eyes of flies, or, indeed, in constructed space enclosures.

BIBLIOGRAPHY

Baldwin, J. BuckyWorks: Buckminster Fuller’s Ideas for Today. New York: Wiley, 1996.

“Domes House LA’s Homeless: Geodesic Principles are Translated into Flexible, Temporary Housing.” Architecture 83: (1994): 95.

Edmondson, Amy. A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller. Boston: Birkhäuser, 1987.

Fuller, Buckminster. Synergetics: Explorations in the Geometry of Thinking. New York, MacMillan, 1975.

-----. Synergetics 2: Explorations in the Geometry of Thinking. New York, MacMillan, 1979.

-----. “New directions: The Geodesic Dome and Standard of Living Package.” Perspecta 1 (1952): 29-37.

Indlu Geodesic Research Project with R. Buckminster Fuller. Durban, South Africa: School of Architecture, University of Natal, 1958.

Kenner, Hugh. Geodesic Math and How to Use It. Berkeley: University of California Press, 1976.

Knauer, Gene. “The Return of the Geodesic Dome.” The Futurist: 26 (1992): 29-32.

Marks, Robert. The Dymaxion World of Buckminster Fuller. Carbondale: Southern Illinois University Press, 1960.

Pawley, Martin. Buckminster Fuller. New York: Taplinger Publishing, 1990.

Pugh, Anthony. Polyhedra: A Visual Approach. Berkeley: University of California Press, 1976.

Rader, Michael. “Geodesic Domes: Structures and Homes.” http://www.dnaco.net/~michael /domes/domes.htmlSieden, Lloyd Steven. “The Birth of the Geodesic Dome: How Bucky Did It.” The Futurist 23 (1989): 14-19.

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GEODESIC DOMES A Fuller Understanding of Applied Geometry

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