Tensegrity, tensional integrity or floating compression is a structural principle based on a system of isolated components under compression inside a network of continuous tension, and arranged in such a way that the compressed members (usually bars or struts) do not touch each other while the

prestressed Prestressed concrete is a form of concrete used in construction. It is substantially "prestressed" ( compressed) during production, in a manner that strengthens it against tensile forces which will exist when in service. Post-tensioned concreted i ...

tensioned members (usually cables or tendons) delineate the system spatially. The term was coined by

Buckminster Fuller Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing ...

in the 1960s as a

portmanteau A portmanteau word, or portmanteau (, ) is a blend of wordsKenneth Snelson.

Concept

Tensegrity structures are based on the combination of a few simple design patterns: * members loaded in either pure compression or pure tension, which means that the structure will only fail if the cables yield or the rods buckle. This enables the material properties and cross-sectional geometry of each member to be optimized to the particular load it carries. * preload or tensional prestress allows cables to always be in tension, to maintain structural integrity. * mechanical stability, which allows the members to remain in tension/compression as stress on the structure increases. The structure also becomes stiffer as cable tension increases. Because of these patterns, no structural member experiences a

bending moment In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending mo ...

and there are no shear stresses within the system. This can produce exceptionally strong and rigid structures for their mass and for the cross section of the components. The loading of at least some tensegrity structures causes an auxetic response and negative Poisson ratio, e.g. the T3-prism and 6-strut tensegrity icosahedron. In16695

A conceptual building block of tensegrity is seen in the 1951

Skylon Skylon may refer to: * Skylon (Festival of Britain), a landmark structure of the 1951 Festival of Britain * Skylon (spacecraft), a proposed orbital spaceplane * Skylon Tower, an observation tower in Niagara Falls, Ontario * ''Skylon'' (album), a ...

. Six cables, three at each end, hold the tower in position. The three cables connected to the bottom "define" its location. The other three cables are simply keeping it vertical. A three-rod tensegrity structure (shown to the right) builds on this simpler structure: the ends of each green rod look like the top and bottom of the Skylon. As long as the angle between any two cables is smaller than 180°, the position of the rod is well defined. While three cables are the minimum required for stability, additional cables can be attached to each node for aesthetic purposes or to build in additional stability. For example, Snelson's Needle Tower uses a repeated pattern built using nodes that are connected to 5 cables each. Eleanor Heartney points out visual transparency as an important aesthetic quality of these structures. Korkmaz ''et al.'' has argued that lightweight tensegrity structures are suitable for

adaptive architecture Within reconfigurable computing, an Adaptive architecture is a system which changes its structure, behaviour or resources according to demand. The adaptation made is usually lways?to non-functional characteristics rather than functional ones. ...

Applications

Tensegrities saw increased application in architecture beginning in the 1960s, when Maciej Gintowt and Maciej Krasiński designed

Spodek Spodek (meaning "saucer" in Polish) is a multipurpose arena complex in Katowice, Poland, opened on 9 May 1971. Aside from the main dome, the complex includes a gym, an ice rink, a hotel and three large car parks. It was the largest indoor venue o ...

arena complex (in

Katowice Katowice ( , , ; szl, Katowicy; german: Kattowitz, yi, קאַטעוויץ, Kattevitz) is the capital city of the Silesian Voivodeship in southern Poland and the central city of the Upper Silesian metropolitan area. It is the 11th most popu ...

Poland Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populou ...

), as one of the first major structures to employ the principle of tensegrity. The roof uses an inclined surface held in check by a system of cables holding up its circumference. Tensegrity principles were also used in

David Geiger David H. Geiger (1935 – October 3, 1989) was an American engineer who invented the Air-supported structure, air-supported fabric roof system that at the time of his death was in use at almost half the domed stadiums in the world. Geiger was born ...

's Seoul

Olympic Gymnastics Arena The Olympic Gymnastics Arena (), also known as the KSPO Dome since 2018, is an indoor arena located within the Olympic Park in Seoul, South Korea. It has a capacity of 15,000 and can be extended upto 20,000. It was constructed between 31 Augu ...

(for the

1988 Summer Olympics The 1988 Summer Olympics (), officially known as the Games of the XXIV Olympiad () and commonly known as Seoul 1988 ( ko, 서울 1988, Seoul Cheon gubaek palsip-pal), was an international multi-sport event held from 17 September to 2 October ...

), and the

Georgia Dome The Georgia Dome was a domed stadium in the Southeastern United States. Located in Atlanta between downtown to the east and Vine City to the west, it was owned and operated by the State of Georgia as part of the Georgia World Congress Center ...

(for the

1996 Summer Olympics The 1996 Summer Olympics (officially the Games of the XXVI Olympiad, also known as Atlanta 1996 and commonly referred to as the Centennial Olympic Games) were an international multi-sport event held from July 19 to August 4, 1996, in Atlanta, ...

Tropicana Field Tropicana Field (commonly known as the Trop) is a multi-purpose domed stadium located in St. Petersburg, Florida, United States. The stadium has been the home of the Tampa Bay Rays of Major League Baseball (MLB) since the team's inaugural sea ...

, home of the Tampa Bay Rays major league baseball team, also has a dome roof supported by a large tensegrity structure. Brisbane (6868660143)

On 4 October 2009, the

Kurilpa Bridge The Kurilpa Bridge (originally known as the Tank Street Bridge) is a 63 million pedestrian and bicycle bridge over the Brisbane River in Brisbane, Queensland, Australia. The bridge connects Kurilpa Point in South Brisbane to Tank Street in the ...

opened across the

Brisbane River The Brisbane River is the longest river in South East Queensland, Australia, and flows through the city of Brisbane, before emptying into Moreton Bay on the Coral Sea. John Oxley, the first European to explore the river, named it after the ...

Queensland, Australia ) , nickname = Sunshine State , image_map = Queensland in Australia.svg , map_caption = Location of Queensland in Australia , subdivision_type = Country , subdivision_name = Australia , established_title = Before federation , established_ ...

. A multiple-mast, cable-stay structure based on the principles of tensegrity, it is currently the world's largest tensegrity bridge. NASA SUPERball Tensegrity Lander Prototype

NASA SUPERball Tensegrity Lander Prototype

Since the early 2000s, tensegrities have also attracted the interest of roboticists due to their potential to design lightweight and resilient robots. Numerous researches have investigated tensegrity rovers, bio-mimicking robots, and modular soft robots. The most famous tensegrity robot is the Super Ball Bot, a rover for space exploration using a 6-bar tensegrity structure, currently under developments at NASA Ames.

Biology

Biotensegrity, a term coined by Dr. Stephen Levin, is an extended theoretical application of tensegrity principles to biological structures. Biological structures such as

muscle Skeletal muscles (commonly referred to as muscles) are organs of the vertebrate muscular system and typically are attached by tendons to bones of a skeleton. The muscle cells of skeletal muscles are much longer than in the other types of mus ...

s, bones,

fascia A fascia (; plural fasciae or fascias; adjective fascial; from Latin: "band") is a band or sheet of connective tissue, primarily collagen, beneath the skin that attaches to, stabilizes, encloses, and separates muscles and other internal organs ...

ligaments A ligament is the fibrous connective tissue that connects bones to other bones. It is also known as ''articular ligament'', ''articular larua'', ''fibrous ligament'', or ''true ligament''. Other ligaments in the body include the: * Peritoneal ...

and

tendons A tendon or sinew is a tough, high-tensile-strength band of dense fibrous connective tissue that connects muscle to bone. It is able to transmit the mechanical forces of muscle contraction to the skeletal system without sacrificing its abilit ...

, or rigid and elastic

cell membrane The cell membrane (also known as the plasma membrane (PM) or cytoplasmic membrane, and historically referred to as the plasmalemma) is a biological membrane that separates and protects the interior of all cells from the outside environment (t ...

s, are made strong by the unison of tensioned and compressed parts. The

musculoskeletal system The human musculoskeletal system (also known as the human locomotor system, and previously the activity system) is an organ system that gives humans the ability to move using their muscular and skeletal systems. The musculoskeletal system provid ...

consists of a continuous network of muscles and connective tissues, while the bones provide discontinuous compressive support, whilst the nervous system maintains tension in vivo through electrical stimulus. Levin claims that the

human spine The vertebral column, also known as the backbone or spine, is part of the axial skeleton. The vertebral column is the defining characteristic of a vertebrate in which the notochord (a flexible rod of uniform composition) found in all chordates ...

, is also a tensegrity structure although there is no support for this theory from a structural perspective. Donald E. Ingber has developed a theory of tensegrity to describe numerous phenomena observed in

molecular biology Molecular biology is the branch of biology that seeks to understand the molecular basis of biological activity in and between cells, including biomolecular synthesis, modification, mechanisms, and interactions. The study of chemical and phys ...

. For instance, the expressed shapes of cells, whether it be their reactions to applied pressure, interactions with substrates, etc., all can be mathematically modeled by representing the cell's

cytoskeleton The cytoskeleton is a complex, dynamic network of interlinking protein filaments present in the cytoplasm of all cells, including those of bacteria and archaea. In eukaryotes, it extends from the cell nucleus to the cell membrane and is co ...

as a tensegrity. Furthermore, geometric patterns found throughout nature (the helix of DNA, the geodesic dome of a

volvox ''Volvox'' is a polyphyletic genus of chlorophyte green algae in the family Volvocaceae. It forms spherical colonies of up to 50,000 cells. They live in a variety of freshwater habitats, and were first reported by Antonie van Leeuwenhoek in 170 ...

Buckminsterfullerene Buckminsterfullerene is a type of fullerene with the formula C60. It has a cage-like fused-ring structure (truncated icosahedron) made of twenty hexagons and twelve pentagons, and resembles a soccer ball. Each of its 60 carbon atoms is bonded ...

, and more) may also be understood based on applying the principles of tensegrity to the spontaneous self-assembly of compounds, proteins, and even organs. This view is supported by how the tension-compression interactions of tensegrity minimize material needed to maintain stability and achieve structural resiliency, although the comparison with inert materials within a biological framework has no widely accepted premise within physiological science. Therefore,

natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Cha ...

pressures would likely favor biological systems organized in a tensegrity manner. As Ingber explains: In embryology, Richard Gordon proposed that embryonic differentiation waves are propagated by an 'organelle of differentiation' where the

is assembled in a bistable tensegrity structure at the apical end of cells called the 'cell state splitter'.

Origins and art history

The origins of tensegrity are controversial. Many traditional structures, such as skin-on-frame kayaks and

shōji A is a door, window or room divider used in traditional Japanese architecture, consisting of translucent (or transparent) sheets on a lattice frame. Where light transmission is not needed, the similar but opaque ''fusuma'' is used (oshiire/ ...

, use tension and compression elements in a similar fashion. Russian artist

Viatcheslav Koleichuk Viacheslav Koleichuk ( rus, Вячеслав Колейчук; 16 December 1941 – 8 April 2018) was a Russian sound artist, musician, architect and visual artist. Koleichuk mainly made installation art that involves tensegrity. Sometimes these s ...

claimed that the idea of tensegrity was invented first by Kārlis Johansons (in Russian as German as Karl Ioganson) ( lv), a Soviet

avant-garde The avant-garde (; In 'advance guard' or ' vanguard', literally 'fore-guard') is a person or work that is experimental, radical, or unorthodox with respect to art, culture, or society.John Picchione, The New Avant-garde in Italy: Theoretica ...

artist of Latvian descent, who contributed some works to the main exhibition of Russian constructivism in 1921. Koleichuk's claim was backed up by Maria Gough for one of the works at the 1921 constructivist exhibition. Snelson has acknowledged the constructivists as an influence for his work (query?). French engineer David Georges Emmerich has also noted how Kārlis Johansons's work (and industrial design ideas) seemed to foresee tensegrity concepts. In 1948, artist Kenneth Snelson produced his innovative "X-Piece" after artistic explorations at Black Mountain College (where

was lecturing) and elsewhere. Some years later, the term "tensegrity" was coined by Fuller, who is best known for his

geodesic dome A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the structural stress throughout the structure, making geodesic do ...

s. Throughout his career, Fuller had experimented with incorporating tensile components in his work, such as in the framing of his

dymaxion Dymaxion is a term coined by architect and inventor Buckminster Fuller and associated with much of his work—prominently his Dymaxion house and Dymaxion car. Dymaxion, a portmanteau of the words ''dynamic'', ''maximum'', and ''tension''; ...

houses. Snelson's 1948 innovation spurred Fuller to immediately commission a mast from Snelson. In 1949, Fuller developed a tensegrity-

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

based on the technology, and he and his students quickly developed further structures and applied the technology to building domes. After a hiatus, Snelson also went on to produce a plethora of sculptures based on tensegrity concepts. His main body of work began in 1959 when a pivotal exhibition at the

Museum of Modern Art The Museum of Modern Art (MoMA) is an art museum located in Midtown Manhattan, New York City, on 53rd Street between Fifth and Sixth Avenues. It plays a major role in developing and collecting modern art, and is often identified as one of t ...

took place. At the MOMA exhibition, Fuller had shown the mast and some of his other work. At this exhibition, Snelson, after a discussion with Fuller and the exhibition organizers regarding credit for the mast, also displayed some work in a vitrine. Snelson's best known piece is his 18-meter-high '' Needle Tower'' of 1968.

Stability

Tensegrity prisms

The three-rod tensegrity structure (3-way prism) has the property that, for a given (common) length of compression member “rod” (there are three total) and a given (common) length of tension cable “tendon” (six total) connecting the rod ends together, there is a particular value for the (common) length of the tendon connecting the rod tops with the neighboring rod bottoms that causes the structure to hold a stable shape. For such a structure, it is straightforward to prove that the triangle formed by the rod tops and that formed by the rod bottoms are rotated with respect to each other by an angle of 5π/6 (radians). The stability (“prestressability”) of several 2-stage tensegrity structures are analyzed by Sultan, et al. The T3-prism (also known as Triplex) can be obtained through form finding of a straight triangular prism. Its self-equilibrium state is given when the base triangles are in parallel planes separated by an angle of twist of pi/6. The formula for its unique self-stress state is given by,

\omega = \omega_1 \sqrt(3), -\sqrt(3), -\sqrt(3), \sqrt(3), \sqrt(3), \sqrt(3), 1, 1, 1, 1, 1, 1 T

Here, the first three negative values correspond to the inner components in compression, while the rest correspond to the cables in tension.

Tensegrity icosahedra

The tensegrity

, first studied by Snelson in 1949, has struts and tendons along the edges of a polyhedron called

Jessen's icosahedron Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same numbers of vertices, edges, and faces as the regular icosahedron. It is named for Børge Jessen, who studied it in 1967. In 1971, ...

. It is a stable construction, albeit with infinitesimal mobility. To see this, consider a cube of side length , centered at the origin. Place a strut of length in the plane of each cube face, such that each strut is parallel to one edge of the face and is centered on the face. Moreover, each strut should be parallel to the strut on the opposite face of the cube, but orthogonal to all other struts. If the Cartesian coordinates of one strut are and , those of its parallel strut will be, respectively, and . The coordinates of the other strut ends (vertices) are obtained by permuting the coordinates, e.g., (rotational symmetry in the main diagonal of the cube). The distance ''s'' between any two neighboring vertices and is :

s^2 = (d - l)^2 + d^2 + l^2 = 2\left(d - \frac \,l\right)^2 + \frac \,l^2

Imagine this figure built from struts of given length and tendons (connecting neighboring vertices) of given length ''s'', with

s > \sqrt\frac\,l

. The relation tells us there are two possible values for ''d'': one realized by pushing the struts together, the other by pulling them apart. In the particular case

s = \sqrt\frac\,l

the two extremes coincide, and

d = \frac\,l

, therefore the figure is the stable tensegrity icosahedron. This choice of parameters gives the vertices the positions of Jessen's icosahedron; they are different from the

regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It ...

, for which the ratio of

d

and

l

would be the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

, rather than 2. However both sets of coordinates lie along a continuous family of positions ranging from the

cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...

to the

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

(as limit cases), which are linked by a helical contractive/expansive transformation. This kinematics of the cuboctahedron is the ''geometry of motion'' of the tensegrity icosahedron. It was first described by H. S. M. Coxeter and later called the "jitterbug transformation" by Buckminster Fuller. Since the tensegrity icosahedron represents an extremal point of the above relation, it has infinitesimal mobility: a small change in the length ''s'' of the tendon (e.g. by stretching the tendons) results in a much larger change of the distance 2''d'' of the struts.

Patents

* , "Tensile-Integrity Structures," 13 November 1962, Buckminster Fuller. * French Patent No. 1,377,290, "Construction de Reseaux Autotendants", 28 September 1964, David Georges Emmerich. * French Patent No. 1,377,291, "Structures Linéaires Autotendants", 28 September 1964, David Georges Emmerich. * , "Suspension Building" (also called aspension), 7 July 1964, Buckminster Fuller. * , "Continuous Tension, Discontinuous Compression Structure," 16 February 1965, Kenneth Snelson. * , "Non-symmetrical Tension-Integrity Structures," 18 February 1975, Buckminster Fuller.

Basic tensegrity structures

File:3-tensegrity.svg, The simplest tensegrity structure, a 3-prism File:Tensegrity 3-Prism.png, Another 3-prism File:4-tensegrity.svg, A similar structure but with four compression members File:Proto-Tensegrity by Ioganson.jpg, Proto-Tensegrity Prism by Karl Ioganson, 1921 File:Tensegrity Icosahedron.png, Tensegrity Icosahedron,

, 1949 File:Tensegrity Tetrahedron.png, Tensegrity Tetrahedron, Francesco della Salla, 1952. File:Tensegrity X-Module Tetrahedron.png, Tensegrity X-Module Tetrahedron, Kenneth Snelson, 1959

Tensegrity structures

File:Kenneth Snelson Needle Tower.JPG, Kenneth Snelson's Needle Tower art sculpture. File:Tensegrity Dome.jpg, A tensegrity dome made of garden stakes and nylon twine built in the yard of a house, 2009 File:Tensegrity Structure - Science Park - Science City - Kolkata 2010-02-18 4567.JPG, A 12m high tensegrity structure exhibit at the Science City,

Kolkata Kolkata (, or , ; also known as Calcutta , the official name until 2001) is the capital of the Indian state of West Bengal, on the eastern bank of the Hooghly River west of the border with Bangladesh. It is the primary business, comme ...

. File:AfrikaBurn 2015 Dissipate.JPG, ''Dissipate'', an hourglass tower art sculpture including tensegrity structure, constructed at

AfrikaBurn AfrikaBurn is an official Burning Man regional event, held at Quaggafontein ( "fountain of Quagga") in the Tankwa Karoo, in the Western Cape Province of South Africa. It is centred on the building of a temporary creative community in a semi-desert ...

, 2015, a

Burning Man Burning Man is an event focused on community, art, self-expression, and self-reliance held annually in the western United States. The name of the event comes from its culminating ceremony: the symbolic burning of a large wooden effigy, referred ...

regional event

Notes

References

Bibliography

* * *
Online
* A good overview on the scope of tensegrity from Fuller's point of view, and an interesting overview of early structures with careful attributions most of the time. * 2003 reprint . This is a good starting place for learning about the mathematics of tensegrity and building models. * * * * * * * *

External links

Scientific Publications in the Field of Tensegrity
by Swiss Federal Institute of Technology (EPFL), Applied Computing and Mechanics Laboratory (IMAC)
Stephen Levin's Biotensegrity site
Several papers on the tensegrity mechanics of biologic structures from viruses to vertebrates by an Orthopedic Surgeon. {{Buckminster Fuller Buckminster Fuller Tensile architecture