Geodesic dome

Geodesic dome

Overview
A geodesic dome is a spherical or partial-spherical shell structure or lattice shell
Thin-shell structure
Thin-shell structures are light weight constructions using shell elements. These elements are typically curved and are assembled to large structures...

based on a network of great circle
Great circle
A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

s (geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s) on the surface of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

. The geodesics intersect to form triangular elements that have local triangular rigidity and also distribute the stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

across the structure. When completed to form a complete sphere, it is a geodesic sphere. A dome is enclosed, unlike open geodesic structures such as playground climbers.

Typically a geodesic dome design begins with an icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

inscribed in a hypothetical sphere, tiling each triangular face with smaller triangles, then projecting the vertices of each tile to the sphere.
Discussion

Recent Discussions
Encyclopedia
A geodesic dome is a spherical or partial-spherical shell structure or lattice shell
Thin-shell structure
Thin-shell structures are light weight constructions using shell elements. These elements are typically curved and are assembled to large structures...

based on a network of great circle
Great circle
A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

s (geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s) on the surface of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

. The geodesics intersect to form triangular elements that have local triangular rigidity and also distribute the stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

across the structure. When completed to form a complete sphere, it is a geodesic sphere. A dome is enclosed, unlike open geodesic structures such as playground climbers.

Typically a geodesic dome design begins with an icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

inscribed in a hypothetical sphere, tiling each triangular face with smaller triangles, then projecting the vertices of each tile to the sphere. The endpoints of the links of the completed sphere are the projected endpoints on the sphere's surface. If this is done exactly, each sub-triangle edge is a slightly different length, requiring links of many sizes. To minimize this, simplifications are made. The result is a compromise of triangles with their vertices lying approximately on the sphere. The edges of the triangles form approximate geodesic paths over the surface of the dome.

Geodesic designs can be used to form any curved, enclosed space. Standard designs tend to be used because unusual configurations may require complex, expensive custom design of each strut, vertex and panel.

History

The first dome that could be called "geodesic" in every respect was designed after World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...

by Walther Bauersfeld
Walther Bauersfeld
Walther Bauersfeld was a German engineer, employed by the Zeiss Corporation, who, on a suggestion by the German astronomer Max Wolf, started work on the first projection planetarium in 1912. This work was stopped by military needs during World War I, but resumed after the war...

, chief engineer of the Carl Zeiss
Carl Zeiss
Carl Zeiss was a German maker of optical instruments commonly known for the company he founded, Carl Zeiss Jena . Zeiss made contributions to lens manufacturing that have aided the modern production of lenses...

optical company, for a planetarium
Planetarium
A planetarium is a theatre built primarily for presenting educational and entertaining shows about astronomy and the night sky, or for training in celestial navigation...

to house his planetarium projector. The dome was patented, constructed by the firm of Dykerhoff and Wydmann on the roof of the Zeiss plant in Jena
Jena
Jena is a university city in central Germany on the river Saale. It has a population of approx. 103,000 and is the second largest city in the federal state of Thuringia, after Erfurt.-History:Jena was first mentioned in an 1182 document...

, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

, and opened to the public in July 1926. Some 20 years later, R. Buckminster Fuller
Buckminster Fuller
Richard Buckminster “Bucky” Fuller was an American systems theorist, author, designer, inventor, futurist and second president of Mensa International, the high IQ society....

named the dome "geodesic" from field experiments with artist Kenneth Snelson
Kenneth Snelson
Kenneth Snelson is a contemporary sculptor and photographer. His sculptural works are composed of flexible and rigid components arranged according to the idea of 'tensegrity', although Snelson does not use the term....

at Black Mountain College
Black Mountain College
Black Mountain College, a school founded in 1933 in Black Mountain, North Carolina, was a new kind of college in the United States in which the study of art was seen to be central to a liberal arts education, and in which John Dewey's principles of education played a major role...

in 1948 and 1949. Snelson and Fuller worked developing what they termed "tensegrity
Tensegrity
Tensegrity, tensional integrity or floating compression, is a structural principle based on the use of isolated components in compression inside a net of continuous tension, in such a way that the compressed members do not touch each other and the prestressed tensioned members delineate the...

," an engineering principle of continuous tension and discontinuous compression that allowed domes to deploy a lightweight lattice of interlocking icosahedrons that could be skinned with a protective cover. Although Fuller was not the original inventor, he developed the intrinsic mathematics of the dome, thereby allowing popularization of the idea — for which he received U.S. patent 2,682,235 https://www.google.com/search?q=2682235&tbm=pts 29 June, 1954.

The geodesic dome appealed to Fuller because it was extremely strong for its weight, its "omnitriangulated" surface provided an inherently stable structure, and because a sphere encloses the greatest volume for the least surface area.

However, from a practical perspective, geodesic constructions have some disadvantages. They have a very large number of edges in comparison with more conventional structures which have just a few large flat surfaces. Each of the edges must be prevented from leaking, which can be quite challenging for a geodesic structure. Also, spaces enclosed within curved boundaries tend to be less usable than spaces enclosed within flat boundaries. (Since it would be impractical to produce sofas with every possible curved shape, they are normally constructed along straight lines, and so leave wasted space when placed in a curved space.)

The dome was successfully adopted for specialized uses, such as the 21 Distant Early Warning Line
Distant Early Warning Line
The Distant Early Warning Line, also known as the DEW Line or Early Warning Line, was a system of radar stations in the far northern Arctic region of Canada, with additional stations along the North Coast and Aleutian Islands of Alaska, in addition to the Faroe Islands, Greenland, and Iceland...

domes built in Canada in 1956, the 1958 Union Tank Car Company
Union Tank Car Company
Union Tank Car Company or UTLX is a railway equipment leasing company headquartered in metro Chicago, Illinois. As the name says, they specialise in tank cars, and covered hopper cars.Founded in 1866 by J. J...

dome near Baton Rouge, Louisiana
Baton Rouge, Louisiana
Baton Rouge is the capital of the U.S. state of Louisiana. It is located in East Baton Rouge Parish and is the second-largest city in the state.Baton Rouge is a major industrial, petrochemical, medical, and research center of the American South...

designed by Thomas C. Howard of Synergetics, Inc. and specialty buildings like the Kaiser Aluminum
Henry J. Kaiser
Henry John Kaiser was an American industrialist who became known as the father of modern American shipbuilding. He established the Kaiser Shipyard which built Liberty ships during World War II, after which he formed Kaiser Aluminum and Kaiser Steel. Kaiser organized Kaiser Permanente health care...

domes (constructed in numerous locations across the US, e.g., Virginia Beach, VA
Virginia Beach, Virginia
Virginia Beach is an independent city located in the Hampton Roads metropolitan area of Virginia, on the Atlantic Ocean at the mouth of the Chesapeake Bay...

), auditoriums, weather observatories, and storage facilities. The dome was soon breaking records for covered surface, enclosed volume, and construction speed.

Leveraging the geodesic dome's stability, the US Air Force experimented with helicopter
Helicopter
A helicopter is a type of rotorcraft in which lift and thrust are supplied by one or more engine-driven rotors. This allows the helicopter to take off and land vertically, to hover, and to fly forwards, backwards, and laterally...

-deliverable units.

The dome was introduced to a wider audience as a pavilion for the 1964 World's Fair
World's Fair
World's fair, World fair, Universal Exposition, and World Expo are various large public exhibitions held in different parts of the world. The first Expo was held in The Crystal Palace in Hyde Park, London, United Kingdom, in 1851, under the title "Great Exhibition of the Works of Industry of All...

in New York City
New York City
New York is the most populous city in the United States and the center of the New York Metropolitan Area, one of the most populous metropolitan areas in the world. New York exerts a significant impact upon global commerce, finance, media, art, fashion, research, technology, education, and...

designed by Thomas C. Howard of Synergetics, Inc. This dome is now used as an aviary
Aviary
An aviary is a large enclosure for confining birds. Unlike cages, aviaries allow birds a larger living space where they can fly; hence, aviaries are also sometimes known as flight cages...

by the Queens Zoo
Queens Zoo
The Queens Zoo is a zoo located in the New York City borough of Queens, located in Flushing Meadows – Corona Park. The zoo is part of an integrated system of four zoos and one aquarium managed by the Wildlife Conservation Society in partnership with the New York City Department of Parks and...

in Flushing Meadows Corona Park after it was redesigned by TC Howard of Synergetics, Inc.

Another dome is from Expo 67
Expo 67
The 1967 International and Universal Exposition or Expo 67, as it was commonly known, was the general exhibition, Category One World's Fair held in Montreal, Quebec, Canada, from April 27 to October 29, 1967. It is considered to be the most successful World's Fair of the 20th century, with the...

at the Montreal
Montreal
Montreal is a city in Canada. It is the largest city in the province of Quebec, the second-largest city in Canada and the seventh largest in North America...

World's Fair
World's Fair
World's fair, World fair, Universal Exposition, and World Expo are various large public exhibitions held in different parts of the world. The first Expo was held in The Crystal Palace in Hyde Park, London, United Kingdom, in 1851, under the title "Great Exhibition of the Works of Industry of All...

, where it was part of the American Pavilion. The structure's covering later burned, but the structure itself still stands and, under the name Biosphère
Montreal Biosphère
The Biosphère is a museum in Montreal dedicated to the environment. It is located at Parc Jean-Drapeau, on Île Sainte-Hélène in the former pavilion of the United States for the 1967 World Fair Expo 67.- Expo 67 :...

,
currently houses an interpretive museum
Museum
A museum is an institution that cares for a collection of artifacts and other objects of scientific, artistic, cultural, or historical importance and makes them available for public viewing through exhibits that may be permanent or temporary. Most large museums are located in major cities...

Saint Lawrence River
The Saint Lawrence is a large river flowing approximately from southwest to northeast in the middle latitudes of North America, connecting the Great Lakes with the Atlantic Ocean. It is the primary drainage conveyor of the Great Lakes Basin...

.

A dome appeared in the 1967 James Bond
James Bond
James Bond, code name 007, is a fictional character created in 1953 by writer Ian Fleming, who featured him in twelve novels and two short story collections. There have been a six other authors who wrote authorised Bond novels or novelizations after Fleming's death in 1964: Kingsley Amis,...

film You Only Live Twice
You Only Live Twice (film)
You Only Live Twice is the fifth spy film in the James Bond series, and the fifth to star Sean Connery as the fictional MI6 agent James Bond. The film's screenplay was written by Roald Dahl, and loosely based on Ian Fleming's 1964 novel of the same name...

, inspiring the production designer of Austin Powers
Austin Powers (film series)
The Austin Powers series is a series of action-comedy films written by and starring Mike Myers as the title character, directed by Jay Roach and distributed by New Line Cinema...

The Spy Who Shagged Me to use a dome for Dr Evil's moon base.

During the 1970s, the Cinesphere
Cinesphere
Cinesphere is the world's first permanent IMAX movie theatre, built in 1971. Its design is of a triodetic-domed structure, akin to a Geodesic Dome, similar to that of the later Spaceship Earth, the icon of Walt Disney World's Epcot theme park...

dome was built at the Ontario Place
Ontario Place
Ontario Place is a multiple use entertainment and seasonal waterfront park attraction located in Toronto, Ontario, and owned by the Crown in Right of Ontario. It is administered as an agency of the Ontario Ministry of Tourism and Culture. Located on the shore of Lake Ontario, just south of...

amusement park in Toronto
Toronto
Toronto is the provincial capital of Ontario and the largest city in Canada. It is located in Southern Ontario on the northwestern shore of Lake Ontario. A relatively modern city, Toronto's history dates back to the late-18th century, when its land was first purchased by the British monarchy from...

Canada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...

. In 1975, a dome was constructed at the South Pole
Amundsen-Scott South Pole Station
The Amundsen–Scott South Pole Station is the American scientific research station on the high plateau of Antarctica. This station is located at the southernmost place on the Earth, the Geographic South Pole, at an elevation of 2,835 meters above sea level.The original Amundsen-Scott Station was...

, where its resistance to snow and wind loads is important.

Methods of construction

Wooden domes have a hole drilled in the width of a strut. A stainless steel band locks the strut's hole to a steel pipe. With this method, the struts may be cut to the exact length needed. Triangles of exterior plywood are then nailed to the struts. The dome is wrapped from the bottom to the top with several stapled layers of tar paper, in order to shed water, and finished with shingles. This type of dome is often called a hub-and-strut dome because of the use of steel hubs to tie the struts together.

Panelized domes are constructed of separately-framed timbers covered in plywood. The three members comprising the triangular frame are often cut at compound angles in order to provide for a flat fitting of the various triangles. Holes are drilled through the members at precise locations and steel bolts then connect the triangles to form the dome. These members are often 2x4's or 2x6's, which allow for more insulation to fit within the triangle. The panelized technique allows the builder to attach the plywood skin to the triangles while safely working on the ground or in a comfortable shop out of the weather. This method does not require expensive steel hubs.

Temporary greenhouse domes have been constructed by stapling plastic sheeting onto a dome constructed from one-inch square beams. The result is warm, movable by hand in sizes less than 20 feet, and cheap.
It should be staked to the ground to prevent it being moved by wind.

Steel-framework domes can be easily constructed of electrical conduit. One flattens the end of a strut and drills bolt holes at the needed length. A single bolt secures a vertex of struts. The nuts are usually set with removable locking compound, or if the dome is portable, have a castle nut with a cotter pin. This is the standard way to construct domes for jungle-gyms.

Concrete and foam plastic domes generally start with a steel framework dome, wrapped with chicken wire and wire screen for reinforcement. The chicken wire and screen is tied to the framework with wire ties. A coat of material is then sprayed or molded onto the frame. Tests should be performed with small squares to achieve the correct consistency of concrete or plastic. Generally, several coats are necessary on the inside and outside. The last step is to saturate concrete or polyester domes with a thin layer of epoxy compound to shed water.

Some concrete domes have been constructed from prefabricated, prestressed, steel-reinforced concrete panels that can be bolted into place. The bolts are within raised receptacles covered with little concrete caps to shed water. The triangles overlap to shed water. The triangles in this method can be molded in forms patterned in sand with wooden patterns, but the concrete triangles are usually so heavy that they must be placed with a crane. This construction is well-suited to domes because there is no place for water to pool on the concrete and leak through. The metal fasteners, joints and internal steel frames remain dry, preventing frost and corrosion damage. The concrete resists sun and weathering. Some form of internal flashing or caulking must be placed over the joints to prevent drafts. The 1963 Cinerama Dome
Cinerama Dome
Pacific Theatres' Cinerama Dome is a movie theater located at 6360 Sunset Boulevard in Hollywood, California. Designed to present widescreen, 70mm Cinerama films, it opened November 7, 1963. Today it continues as a leading first run theater.- History :...

was built from precast concrete
Precast concrete
By producing precast concrete in a controlled environment , the precast concrete is afforded the opportunity to properly cure and be closely monitored by plant employees. Utilizing a Precast Concrete system offers many potential advantages over site casting of concrete...

hexagons and pentagons.

Dome homes

Fuller hoped that the geodesic dome would help address the postwar housing crisis. This was consistent with his prior hopes for both versions of the Dymaxion House
Dymaxion house
The Dymaxion House was developed by inventor and architect Buckminster Fuller to address several perceived shortcomings with existing homebuilding techniques. Fuller designed several versions of the house at different times, but they were all factory manufactured kits, assembled on site, intended...

.

Residential geodesic domes have been less successful than those used for working and/or entertainment, largely because of their complexity and consequent greater construction costs. Fuller himself lived in a geodesic dome in Carbondale, Illinois
Carbondale, Illinois
Carbondale is a city in Jackson County, in the state of Illinois, within the Southern Illinois region. It is located at the junction of Illinois Route 13 and U.S. Route 51, southeast of St. Louis, Missouri, on the northern edge of the Shawnee National Forest...

, at the corner of Forest and Cherry.
Fuller thought of residential domes as air-deliverable products manufactured by an aerospace-like industry. Fuller's own dome home still exists, the R. Buckminster Fuller and Anne Hewlett Dome Home
R. Buckminster Fuller and Anne Hewlett Dome Home
The R. Buckminster Fuller and Anne Hewlett Dome Home is a Modern Movement building in Carbondale, Illinois that was listed on the National Register of Historic Places in 2006....

, and a group called RBF Dome NFP is attempting to restore the dome and have it registered as a National Historic Landmark.

In 1986 a patent for a dome construction technique involving EPS
Polystyrene
Polystyrene ) also known as Thermocole, abbreviated following ISO Standard PS, is an aromatic polymer made from the monomer styrene, a liquid hydrocarbon that is manufactured from petroleum by the chemical industry...

triangles laminated to reinforced concrete on the outside, and wallboard on the inside was awarded to American Ingenuity of Rockledge Florida. The construction technique allows the domes to be prefabricated in kit form and erected by a homeowner. This method makes the seams into the strongest part of the structure, where the seams and especially the hubs in most wooden-framed domes are the weakest point in the structure. It also has the advantage of being watertight.

Many of the disadvantages described below do no longer apply due to new materials and modern construction techniques.

As a housing system, enjoying a wave of popularity in the late 1960s and early 1970s, the dome can have numerous disadvantages and problems. Former proponent for, and author about, dome homes Lloyd Kahn is the founder of Shelter Publications. He has collected a lot of the criticisms and listed them on his company's Web site:

The shape of a dome house makes it difficult to conform to code requirements for placement of sewer vents
Plumbing drainage venting
In modern plumbing, a drain-waste-vent is part of a system that removes sewage and greywater from a building and regulates air pressure in the waste-system pipes, facilitating flow. Waste is produced at fixtures such as toilets, sinks and showers, and exits the fixtures through a trap, a dipped...

and chimney
Chimney
A chimney is a structure for venting hot flue gases or smoke from a boiler, stove, furnace or fireplace to the outside atmosphere. Chimneys are typically vertical, or as near as possible to vertical, to ensure that the gases flow smoothly, drawing air into the combustion in what is known as the...

s. Off-the-shelf building materials (e.g., plywood, strand board) normally come in rectangular shapes and so much material may have to be scrapped after cutting rectangles down to triangles, thus increasing the cost of construction. Fire escapes are problematic; codes require them for larger structures, and they are expensive. Windows conforming to code can cost anywhere from 5 to 15 times as much as windows in conventional houses. Professional electrical wiring costs more because of increased labor time. Even owner-wired situations are costly, because more of certain materials are required for dome construction.

Air stratification and moisture distribution within a dome are unusual, and these conditions tend to quickly degrade wooden framing or interior paneling. Privacy is difficult to guarantee because a dome is difficult to partition satisfactorily. Sounds, smells, and even reflected light tend to be conveyed through the entire structure.

As with any curved shape, the dome produces wall areas that can be difficult to use and leaves some peripheral floor area with restricted use due to lack of headroom. Circular plan shapes lack the simple modularity provided by rectangles. Furnishers and fitters usually design with flat surfaces in mind, and so placing a standard sofa (for example) results in a crescent behind the sofa being wasted. This is best overcome by purpose-built fittings, though it adds to cost.

Dome builders using cut-board sheathing materials (as was common in the 1960s and 1970s) find it hard to seal domes against rain, because of their many seams. Also, these seams may be stressed because ordinary solar heat flexes the entire structure each day as the sun moves across the sky.

The most effective waterproofing method with a wooden dome is to shingle
Roof shingle
Roof shingles are a roof covering consisting of individual overlapping elements. These elements are typically flat rectangular shapes laid in rows from the bottom edge of the roof up, with each successive higher row overlapping the joints in the row below...

the dome, but even this can be a problem at the top of the dome, where the slope is less than that required by most roofing materials. (One solution is to add a peaked cap to the top of the dome or to modify the dome shape.) One-piece reinforced concrete
Concrete
Concrete is a composite construction material, composed of cement and other cementitious materials such as fly ash and slag cement, aggregate , water and chemical admixtures.The word concrete comes from the Latin word...

or plastic
Plastic
A plastic material is any of a wide range of synthetic or semi-synthetic organic solids used in the manufacture of industrial products. Plastics are typically polymers of high molecular mass, and may contain other substances to improve performance and/or reduce production costs...

domes are also in use, and some domes have been constructed from plastic or waxed cardboard triangles that are overlapped in such a way as to shed water. Buckminster Fuller's former student J. Baldwin
J. Baldwin
James Tennant Baldwin is an American industrial designer and writer...

insists that there is no reason for a properly designed, well-constructed dome to leak, and that some designs cannot leak (Bucky Works: Buckminster Fuller's Ideas for Today). However, Lloyd Kahn
Lloyd Kahn
Lloyd Kahn is the founding editor-in-chief of Shelter Publications, Inc., and is the former Shelter editor of the Whole Earth Catalog. He is also an author, photographer, and pioneer of the green building and green architecture movements....

, after writing two books on the subject (Domebook 1 and Domebook 2), became disillusioned with domes. He calls domes "smart but not wise", and has collected many of the criticisms given above.

Chord factors

 A geodesic sphere and its dualDual polyhedronIn geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another....

The mathematical object "chord" of the "geodesic sphere" corresponds to the structural "strut" of the physical "geodesic dome". A chord
Chord (geometry)
A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...

is a (straight) line segment joining two points on a curve. For simple geodesic domes, curves follow the surface of a sphere circumscribing a regular polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

with triangular faces, (tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

, icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

, or octahedron
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

). The desired frequency of the subsequent geodesic sphere or dome is the number of parts or segments into which a side (edge) of the underlying polyhedral triangle is subdivided. The frequency has historically been denoted by the Greek letter "" (nu
Nu (letter)
Nu , is the 13th letter of the Greek alphabet. In the system of Greek numerals it has a value of 50...

). By connecting like points along the subdivided sides, a natural triangular grid is formed on each face of the polyhedron. Each segment of the grid is then projected as a "chord" onto the surface of the circumscribing sphere. The technical definition of a chord factor is the ratio of chord length to the radius of the circumscribing sphere. It is therefore convenient to think of the circumscribing sphere as scaled to radius = 1 in which "chord factors" are the same as "chord lengths", (fractional values less than one).

For geodesic spheres, a well-known formula for calculating any "chord factor" is:

where "" is the corresponding angle of arc for the given chord, that is, the "central angle" spanned by the chord with respect to the center of the circumscribing sphere. Determining the central angle usually requires some non-trivial spherical geometry
Spherical geometry
Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....

.

In Geodesic Math and How to Use It Hugh Kenner
Hugh Kenner
William Hugh Kenner , was a Canadian literary scholar, critic and professor.Kenner was born in Peterborough, Ontario on January 7, 1923; his father taught classics...

writes, "Tables of chord factors, containing as they do the essential design information for spherical systems, were for many years guarded like military secrets. As late as 1966, some 3ν icosa figures from Popular Science Monthly were all anyone outside the circle of Fuller licensees had to go on." (page 57, 1976 edition). Other tables became available with publication of Lloyd Kahn's Domebook 1 (1970) and Domebook 2 (1971). With advent of personal computers, the mathematics became more solvable. Rick Bono's Dome software outputs a script that can be used with the POV-ray
POV-Ray
The Persistence of Vision Raytracer, or POV-Ray, is a ray tracing program available for a variety of computer platforms. It was originally based on DKBTrace, written by David Kirk Buck and Aaron A. Collins. There are also influences from the earlier Polyray raytracer contributed by its author...

raytrace to produce 3D pictures of domes. Domes based on the frameworks of different underlying polyhedra along with various methods for subdividing them will produce quite different results. Mathematical formulas developed by Peter W. Messer for calculating chord factors and dihedral angles for the general geodesic sphere appear in the Appendix of the 1999 Dover edition of Spherical Models by Magnus J. Wenninger.

Related patterns

Similar geodesic structures may be based upon the pattern of edges and vertices of certain platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

s, or upon various expansions of these called Johnson solid
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...

s. Such structures may be composed of struts of uniform length while having faces other than triangles such as pentagons or squares, or these faces may be subdivided by struts of other than the basic length. Plans and licenses for such structures derived from licenses of the Fuller patents were produced during the 1970s by Zomeworks (now a manufacturer of solar tracker
Solar tracker
A solar tracker is a generic term used to describe devices that orient various payloads toward the sun. Payloads can be photovoltaic panels, reflectors, lenses or other optical devices....

s). Both geodesic and non-geodesic structures can be derived similarly from the archimedean solid
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

s and catalan solid
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....

s.

The building of strong stable structures out of patterns of reinforcing triangles is most commonly seen in tent
Tent
A tent is a shelter consisting of sheets of fabric or other material draped over or attached to a frame of poles or attached to a supporting rope. While smaller tents may be free-standing or attached to the ground, large tents are usually anchored using guy ropes tied to stakes or tent pegs...

design. It has been applied in the abstract in other industrial design
Industrial design
Industrial design is the use of a combination of applied art and applied science to improve the aesthetics, ergonomics, and usability of a product, but it may also be used to improve the product's marketability and production...

, but even in management science and deliberative structure
Structure
Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...

s as a conceptual metaphor
Conceptual metaphor
In cognitive linguistics, conceptual metaphor, or cognitive metaphor, refers to the understanding of one idea, or conceptual domain, in terms of another, for example, understanding quantity in terms of directionality . A conceptual domain can be any coherent organization of human experience...

, especially in the work of Stafford Beer, whose transmigration method is based so specifically on dome design that only fixed numbers of people can take part in the process at each deliberation
Deliberation
Deliberation is a process of thoughtfully weighing options, usually prior to voting. In legal settings a jury famously uses deliberation because it is given specific options, like guilty or not guilty, along with information and arguments to evaluate. Deliberation emphasizes the use of logic and...

stages.

Largest geodesic dome structures

Many geodesic domes built are still in use. According to the Buckminster Fuller Institute, the world's ten largest domes are:
• Fukuoka Dome
Fukuoka Dome
The is a baseball field, located in Chūō-ku, Fukuoka, Japan. Built in 1993, the stadium can accommodate 35,695 spectators and was Japan's first stadium with a retractable roof....

(福岡ドーム): Fukuoka
Fukuoka
Fukuoka most often refers to the capital city of Fukuoka Prefecture.It can also refer to:-Locations:* Fukuoka, Gifu, a town in Gifu Prefecture, Japan* Fukuoka, Toyama, a town in Toyama Prefecture, Japan...

, Japan
Japan
Japan is an island nation in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...

, 710 ft (216 m)
• Nagoya Dome
Nagoya Dome
Nagoya Dome , constructed in 1997, is a baseball field, located in the city of Nagoya, Japan. The dome has the capacity to seat up to 38,414 people . It is an example of a geodesic dome....

(ナゴヤドーム): Nagoya, Japan, 614 ft (187 m)
• Tacoma Dome
Tacoma Dome
The Tacoma Dome is an indoor arena located in Tacoma, Washington, USA, approximately 30 miles south of Seattle.-History:...

: Tacoma, Washington
Tacoma, Washington
Tacoma is a mid-sized urban port city and the county seat of Pierce County, Washington, United States. The city is on Washington's Puget Sound, southwest of Seattle, northeast of the state capital, Olympia, and northwest of Mount Rainier National Park. The population was 198,397, according to...

, USA, 530 ft (161.5 m)
• Superior Dome
Superior Dome
The Superior Dome, which opened as the "world’s largest wooden dome" on September 14, 1991, is a domed stadium on the campus of Northern Michigan University in Marquette, Michigan. It is home to the Northern Michigan Wildcats football team, as well as a variety of campus and community events...

: Northern Michigan University
Northern Michigan University
Northern Michigan University is a four-year college public university established in 1899 located in Marquette, in Michigan's Upper Peninsula. With a population of nearly 10,000 undergraduate and graduate students, Northern Michigan University is the Upper Peninsula's largest...

. Marquette, Michigan
Marquette, Michigan
Marquette is a city in the U.S. state of Michigan and the county seat of Marquette County. The population was 21,355 at the 2010 census, making it the most populated city of the Upper Peninsula. Marquette is a major port on Lake Superior, primarily for shipping iron ore and is the home of Northern...

, USA, 525 ft (160 m)
• MSC Dome: Bolivia, 140m diameter, by Geometrica, Inc.
• Ruwais dome Abu Dhabi, 133m diameter, by Geometrica, Inc.
• Walkup Skydome
Walkup Skydome
The J. Lawrence Walkup Skydome is an indoor multi-purpose stadium in Flagstaff, Arizona, on the campus of Northern Arizona University. Opened in September 1977, it is the home of the NAU Lumberjacks football and basketball teams of the Big Sky Conference. The seating capacity is 16,230, with...

: Northern Arizona University
Northern Arizona University
Northern Arizona University is a public university located in Flagstaff, Arizona, United States. It is accredited by the North Central Association of Colleges and Schools, and has 39 satellite campuses in the state of Arizona. The university offers both undergraduate and graduate degrees.As of...

. Flagstaff, Arizona
Flagstaff, Arizona
Flagstaff is a city located in northern Arizona, in the southwestern United States. In 2010, the city's population was 65,870. The population of the Metropolitan Statistical Area was at 134,421 in 2010. It is the county seat of Coconino County...

, USA, 502 ft (153 m)
• Round Valley High School Stadium: Springerville-Eagar, AZ, USA, 440 ft (134 m)
• Former Spruce Goose Hangar: Long Beach, California
Long Beach, California
Long Beach is a city situated in Los Angeles County in Southern California, on the Pacific coast of the United States. The city is the 36th-largest city in the nation and the seventh-largest in California. As of 2010, its population was 462,257...

, USA, 415 ft (126 m)
• Formosa Plastics Storage Facility: Mai Liao, Taiwan, 402 ft (122 m)