DUJS Fall 2003

GEODESIC
DOMES
A Fuller Understanding of Applied Geometry
LAURA GARZON ʻ02
he geodesic dome originates from one man’s effort to
model universal patterns of force. Working in the mid20th 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, threedimensional 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
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images courtesy of the Buckminster Fuller Institue
T
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.
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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).
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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.
a
b
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 17mdiameter “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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images courtesy of the Buckminster Fuller Institue
Figure 5: Dome at Expo ʻ67 in Montreal (left), Ford Rotunda Dome (right)
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
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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
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Bukminster Fullerʼs Carbondale Home
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 straightsided 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
images courtesy of the Buckminster Fuller Institue
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
DARTMOUTH UNDERGRADUATE JOURNAL OF SCIENCE
theoretical models of force, in the eyes of flies, or, indeed,
in constructed space enclosures.
-----. “New directions: The Geodesic Dome and Standard of
Living Package.” Perspecta 1 (1952): 29-37.
BIBLIOGRAPHY
Indlu Geodesic Research Project with R. Buckminster Fuller.
Durban, South Africa: School of Architecture, University of
Natal, 1958.
Baldwin, J. BuckyWorks: Buckminster Fuller’s Ideas for Today.
New York: Wiley, 1996.
Kenner, Hugh. Geodesic Math and How to Use It. Berkeley:
University of California Press, 1976.
“Domes House LA’s Homeless: Geodesic Principles are
Translated into Flexible, Temporary Housing.” Architecture 83:
(1994): 95.
Knauer, Gene. “The Return of the Geodesic Dome.” The
Futurist: 26 (1992): 29-32.
Edmondson, Amy. A Fuller Explanation: The Synergetic
Geometry of R. Buckminster Fuller. Boston: Birkhäuser, 1987.
Marks, Robert. The Dymaxion World of Buckminster Fuller.
Carbondale: Southern Illinois University Press, 1960.
Fuller, Buckminster. Synergetics: Explorations in the Geometry
of Thinking. New York, MacMillan, 1975.
Pawley, Martin. Buckminster Fuller. New York: Taplinger
Publishing, 1990.
-----. Synergetics 2: Explorations in the Geometry of
Thinking. New York, MacMillan, 1979.
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.html
Sieden, Lloyd Steven. “The Birth of the Geodesic Dome: How Bucky
Did It.” The Futurist 23 (1989): 14-19.
image courtesy of the Buckminster Fuller Institue
Bukminster Fuller
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