View source for Tensegrity

{{short description|Structural design made of isolated members held in place by tension}}
{{for|the movement system created by Carlos Castaneda|Tensegrity (Castaneda)}}
{{Use dmy dates|date=June 2018}}
[[File:Tensegrity simple 3.gif|thumb|right|The simplest tensegrity structure (a T3-prism). Each of three compression members (green) is symmetric with the other two, and symmetric from end to end. Each end is connected to three cables (red), which provide tension and precisely define the position of that end in the same way as the three cables in the [[Skylon (Festival of Britain)|Skylon]] define the bottom end of its tapered pillar.]]
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'''Tensegrity''', '''tensional integrity''' or '''floating compression''' is a [[structural]] principle based on a system of isolated components under [[compression (physical)|compression]] inside a network of continuous [[tension (mechanics)|tension]], and arranged in such a way that the compressed members (usually bars or struts) do not touch each other while the [[Prestressed structure|prestressed]] tensioned members (usually cables or tendons) delineate the system spatially.{{sfn|Gómez-Jáuregui|2010|p=19}}

Tensegrity structures are found in both nature and human-made objects: in the human body, the bones are held in compression while the connective tissues are held in tension, and the same principles have been applied to furniture and architectural design and beyond.

The term was coined by [[Buckminster Fuller]] in the 1960s as a [[portmanteau]] of "tensional integrity".<ref>{{Cite journal |last1=Swanson |first1=RL |year=2013 |title=Biotensegrity: a unifying theory of biological architecture with applications to osteopathic practice, education, and research-a review and analysis |journal=The Journal of the American Osteopathic Association |volume=113 |issue=1 |pages=34–52 |doi=10.7556/jaoa.2013.113.1.34 |pmid=23329804 |doi-access=free}}</ref>

== Core concept ==

Tensegrity is characterized by several foundational principles that define its unique properties:

# '''Continuous tension''': Fundamental to tensegrity, the tension elements—typically cables or tendons—form a continuous network that encases the entire structure. This allows for the even distribution of mechanical stresses and maintains the structural form, contributing to the overall stability and flexibility of the system.
# '''Discontinuous compression''': The compression components, such as struts or rods, are distinct in that they do not make direct contact with each other but are instead suspended within the tension network. This eliminates the need for rigid connections, enhancing the structural efficiency and resilience of the system.
# '''Pre-stressed''': A key aspect of tensegrity structures is their pre-stressed state, in which tension elements are tightened during the assembly process. Pre-stressing contributes significantly to the structural stiffness and stability, ensuring that all elements are either in tension or compression at all times.
# '''Self-equilibration''': Tensegrity structures are self-equilibrating and so automatically distribute internal stresses across the structure. This allows them to adapt to varying loads without losing structural integrity.
# '''Minimalism and efficiency''': Tensegrity systems employ a minimalist design philosophy, utilizing the minimum amount of materials to achieve maximum structural strength. 
# '''Scalability and modularity''': The design principles of tensegrity allow for scalability and modular construction. Tensegrity structures to be easily adapted or expanded in size and complexity according to specific requirements.

Because of these patterns, no structural member experiences a [[bending moment]] and there are no [[shear stress]]es within the system. This can produce exceptionally strong and rigid structures for their mass and for the cross section of the components.

These principles collectively enable tensegrity structures to achieve a balance of strength, resilience, and flexibility, making the concept widely applicable across disciplines including architecture, robotics, and biomechanics.

== Early example ==
[[File:In16695.jpg|upright|thumb|left|The [[Skylon (Festival of Britain)|Skylon]] at the [[Festival of Britain]], 1951]]

A conceptual building block of tensegrity is seen in the 1951 [[Skylon (Festival of Britain)|Skylon]]. Six [[wire rope|cable]]s, 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 above in a spinning drawing of a T3-Prism) 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 and for [[Redundancy (engineering)|redundancy]]. For example, [[Kenneth 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.<ref>{{ citation | first = Eleanor | last = Hartley | chapter = Ken Snelson and the Aesthetics of Structure | publisher = Marlborough Gallery | type = exhibition catalogue | title = Kenneth Snelson: Selected Work: 1948–2009 | date = 19 February - 21 March 2009 }}</ref> Korkmaz ''et al.'' has argued that lightweight tensegrity structures are suitable for [[adaptive architecture]].<ref name="korkmaz1">{{harvnb|Korkmaz|Bel Hadj Ali|Smith|2011}}</ref><ref name="korkmaz2">{{harvnb|Korkmaz|Bel Hadj Ali|Smith|2012}}</ref>

{{clear}}

== Applications ==

=== Architecture ===
Tensegrities saw increased application in architecture beginning in the 1960s, when [[Maciej Gintowt]] and [[Maciej Krasiński]] designed [[Spodek]] arena complex (in [[Katowice]], [[Poland]]), 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]]'s Seoul [[Olympic Gymnastics Arena]] (for the [[1988 Summer Olympics]]), and the [[Georgia Dome]] (for the [[1996 Summer Olympics]]). [[Tropicana Field]], home of the Tampa Bay Rays major league baseball team, also has a dome roof supported by a large tensegrity structure.

[[File:Brisbane (6868660143).jpg|thumb|left|Largest tensegrity bridge in the world, [[Kurilpa Bridge]] – [[Brisbane]]]]
On 4 October 2009, the [[Kurilpa Bridge, Brisbane|Kurilpa Bridge]] opened across the [[Brisbane River]] in [[Queensland, Australia]]. A multiple-mast, cable-stay structure based on the principles of tensegrity, it is currently the world's largest tensegrity bridge.

=== Robotics ===
[[File:NASA SUPERball Tensegrity Lander Prototype.jpg|thumb|right|[[NASA Ames|NASA]]'s [[NASA Institute for Advanced Concepts#2012 NIAC Project Selections|Super Ball Bot]] is an early prototype to land on another planet without an airbag, and then be mobile to explore. The tensegrity structure provides structural compliance absorbing landing impact forces and motion is applied by changing cable lengths, 2014.]]

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,<ref>{{Cite book |last1=Sabelhaus |first1=Andrew P. |last2=Bruce |first2=Jonathan |last3=Caluwaerts |first3=Ken |last4=Manovi |first4=Pavlo |last5=Firoozi |first5=Roya Fallah |last6=Dobi |first6=Sarah |last7=Agogino |first7=Alice M. |last8=SunSpiral |first8=Vytas |title=2015 IEEE International Conference on Robotics and Automation (ICRA) |chapter=System design and locomotion of SUPERball, an untethered tensegrity robot |date=May 2015 |location=Seattle, WA, USA |publisher=IEEE |pages=2867–2873 |doi=10.1109/ICRA.2015.7139590 |isbn=978-1-4799-6923-4|hdl=2060/20160001750 |s2cid=8548412 |hdl-access=free }}</ref> bio-mimicking robots,<ref>{{Cite book |last1=Lessard |first1=Steven |last2=Castro |first2=Dennis |last3=Asper |first3=William |last4=Chopra |first4=Shaurya Deep |last5=Baltaxe-Admony |first5=Leya Breanna |last6=Teodorescu |first6=Mircea |last7=SunSpiral |first7=Vytas |last8=Agogino |first8=Adrian |title=2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) |chapter=A bio-inspired tensegrity manipulator with multi-DOF, structurally compliant joints |date=October 2016 |chapter-url=http://dx.doi.org/10.1109/iros.2016.7759811 |pages=5515–5520 |publisher=IEEE |doi=10.1109/iros.2016.7759811 |isbn=978-1-5090-3762-9|arxiv=1604.08667 |s2cid=4507700 }}</ref><ref>{{Cite journal |last1=Zappetti |first1=Davide |last2=Arandes |first2=Roc |last3=Ajanic |first3=Enrico |last4=Floreano |first4=Dario |date=2020-06-05 |title=Variable-stiffness tensegrity spine |journal=Smart Materials and Structures |volume=29 |issue=7 |page=075013 |doi=10.1088/1361-665x/ab87e0 |bibcode=2020SMaS...29g5013Z |s2cid=216237847 |issn=0964-1726}}</ref><ref>{{Cite journal |last1=Liu |first1=Yixiang |last2=Dai |first2=Xiaolin |last3=Wang |first3=Zhe |last4=Bi |first4=Qing |last5=Song |first5=Rui |last6=Zhao |first6=Jie |last7=Li |first7=Yibin |date=2022 |title=A Tensegrity-Based Inchworm-Like Robot for Crawling in Pipes With Varying Diameters |journal=IEEE Robotics and Automation Letters |volume=7 |issue=4 |pages=11553–11560 |doi=10.1109/LRA.2022.3203585 |bibcode=2022IRAL....711553L |s2cid=252030788 |issn=2377-3766}}</ref> and modular soft robots.<ref>{{Citation |last1=Zappetti |first1=D. |last2=Mintchev |first2=S. |last3=Shintake |first3=J. |last4=Floreano |first4=D. |title=Bio-inspired Tensegrity Soft Modular Robots |date=2017 |work=Biomimetic and Biohybrid Systems |pages=497–508 |place=Cham |publisher=Springer International Publishing |doi=10.1007/978-3-319-63537-8_42 |arxiv=1703.10139 |isbn=978-3-319-63536-1 |s2cid=822747 }}</ref> The most famous tensegrity robot is the [[NASA Institute for Advanced Concepts#2012 NIAC Project Selections|Super Ball Bot]],<ref>{{Cite web |last1=Hall |first1=Loura |date=2015-04-02 |title=Super Ball Bot |url=http://www.nasa.gov/content/super-ball-bot |access-date=2020-06-18 |website=NASA}}</ref> a rover for space exploration using a [[Jessen's icosahedron|6-bar tensegrity structure]], currently under developments at [[NASA Ames]].

=== Anatomy ===
Biotensegrity, a term coined by Stephen Levin, is an extended theoretical application of tensegrity principles to biological structures.<ref>{{cite book |last=Levin |first=Stephen |chapter=16. Tensegrity, The New Biomechanics |editor-first=Michael |editor-last=Hutson |editor2-first=Adam |editor2-last=Ward |title=Oxford Textbook of Musculoskeletal Medicine |chapter-url=https://books.google.com/books?id=u5G1CgAAQBAJ&pg=PA150 |date=2015 |publisher=Oxford University Press |isbn=978-0-19-967410-7 |pages=155–56, 158–60}}</ref> Biological structures such as [[muscle]]s, [[skeleton|bones]], [[fascia]], [[ligaments]] and [[tendons]], or rigid and elastic [[cell membrane]]s, are made strong by the unison of tensioned and compressed parts. The [[Human musculoskeletal system|musculoskeletal system]] consists of a continuous network of muscles and connective tissues,{{sfn|Souza|Fonseca|Gonçalves|Ocarino|2009}} while the bones provide discontinuous compressive support, whilst the nervous system maintains tension in vivo through electrical stimulus. Levin claims that the [[Vertebral column|human spine]], is also a tensegrity structure.<ref>{{Cite journal |last=Levin |first=Stephen M. |date=2002-09-01 |title=The tensegrity-truss as a model for spine mechanics: biotensegrity |journal=[[Journal of Mechanics in Medicine and Biology]] |volume=02 |issue=3n04 |pages=375–88 |doi=10.1142/S0219519402000472 |issn=0219-5194}}</ref>

=== Biochemistry ===
[[Donald E. Ingber]] has developed a theory of tensegrity to describe numerous phenomena observed in [[molecular biology]].<ref name=Ingber>{{cite journal |last=Ingber |first=Donald E. |journal=Scientific American |date=January 1998 |volume=278 |issue=1 |pages=48–57 |pmid=11536845 |title=The Architecture of Life |url=http://web1.tch.harvard.edu/research/ingber/PDF/1998/SciAmer-Ingber.pdf |archive-url=https://web.archive.org/web/20050515040403/http://web1.tch.harvard.edu/research/ingber/PDF/1998/SciAmer-Ingber.pdf  |archive-date = 2005-05-15 |doi=10.1038/scientificamerican0198-48 |bibcode=1998SciAm.278a..48I }}</ref> For instance, the expressed shapes of cells, whether it be their reactions to applied pressure, interactions with substrates, etc., all can be mathematically modelled by representing the cell's [[cytoskeleton]] as a tensegrity. Furthermore, geometric patterns found throughout nature (the helix of [[DNA]], the geodesic dome of a [[volvox]], [[Buckminsterfullerene]], and more) may also be understood based on applying the principles of tensegrity to the spontaneous self-assembly of compounds, proteins,<ref>{{Cite journal |last1=Edwards |first1=Scott A. |last2=Wagner |first2=Johannes |last3=Gräter |first3=Frauke |date=2012 |title=Dynamic Prestress in a Globular Protein |journal=PLOS Computational Biology |volume=8 |issue=5 |article-number=e1002509 |pmid=22589712 |pmc=3349725 |doi=10.1371/journal.pcbi.1002509
|bibcode=2012PLSCB...8E2509E |doi-access=free }}</ref> 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.<ref>{{cite journal |last1=Skelton |first1=Robert |title=Globally stable minimal mass compressive tensegrity structures |journal=Composite Structures |date=2016 |volume=141 |pages=346–54 |doi=10.1016/j.compstruct.2016.01.105 |url=http://www.sciencedirect.com/science/article/pii/S0263822316300174|url-access=subscription }}</ref> Therefore, [[natural selection]] pressures would likely favor biological systems organized in a tensegrity manner.

As Ingber explains:
{{Blockquote
|text=The tension-bearing members in these structures{{snd}}whether Fuller's domes or Snelson's sculptures{{snd}}map out the shortest paths between adjacent members (and are therefore, by definition, arranged geodesically). Tensional forces naturally transmit themselves over the shortest distance between two points, so the members of a tensegrity structure are precisely positioned to best withstand stress. For this reason, tensegrity structures offer a maximum amount of strength.<ref name=Ingber/>}}

In embryology, [[Richard Gordon (theoretical biologist)|Richard Gordon]] proposed that [[embryonic differentiation waves]] are propagated by an 'organelle of differentiation'<ref>{{Cite journal|doi = 10.1186/s12976-016-0037-2|title = The organelle of differentiation in embryos: The cell state splitter|year = 2016|last1 = Gordon|first1 = Natalie K.|last2 = Gordon|first2 = Richard|journal = Theoretical Biology and Medical Modelling|volume = 13|article-number = 11|pmid = 26965444|pmc = 4785624 | doi-access=free }}</ref> where the [[cytoskeleton]] is assembled in a bistable tensegrity structure at the apical end of cells called the 'cell state splitter'.<ref name="Hierarchical Genome">{{Cite book | doi=10.1142/2755|title = The Hierarchical Genome and Differentiation Waves| volume=3|series = Series in Mathematical Biology and Medicine|year = 1999|last1 = Gordon|first1 = Richard| isbn=978-981-02-2268-0}}</ref>

== Origins and art history ==

[[Image:Snelson XModule Design 1948.png|upright|thumb|right|Kenneth Snelson's 1948 X-Module Design as embodied in a two-module column{{sfn|Gough|1998|p=109}}]]

The origins of tensegrity are not universally agreed upon.<ref name="origins">{{cite journal |last=Gómez-Jáuregui |first=V. |title=Controversial Origins of Tensegrity |journal= International Association of Spatial Structures IASS Symposium 2009, Valencia | year = 2009 | url =http://www.tensegridad.es/Publications/Controversial_Origins_Of_Tensegrity_by_GOMEZ-JAUREGUI.pdf}}</ref> Many traditional structures, such as [[kayak|skin-on-frame kayaks]] and [[shōji]], use tension and compression elements in a similar fashion.

Russian artist [[Viatcheslav Koleichuk]] claimed that the idea of tensegrity was invented first by [[Kārlis Johansons]] (in Russian as German as Karl Ioganson) ([[:lv:Karlis Johansons|lv]]), a Soviet [[avant-garde]] artist of [[Latvians|Latvian]] descent, who contributed some works to the main exhibition of Russian [[Constructivism (art)|constructivism]] in 1921.<ref name=moscowtimes>{{cite web | url = http://context.themoscowtimes.com/stories/2006/08/18/101.html | title = Building Blocks | access-date = 2011-03-28 | last = Droitcour | first = Brian | date = 18 August 2006 | work = [[The Moscow Times]] | archive-url = https://web.archive.org/web/20081007061240/http://context.themoscowtimes.com/stories/2006/08/18/101.html | archive-date = 7 October 2008 | quote = With an unusual mix of art and science, Vyacheslav Koleichuk resurrected a legendary 1921 exhibition of Constructivist art.}}</ref> Koleichuk's claim was backed up by [[Maria Gough]] for one of the works at the 1921 constructivist exhibition.{{sfn|Gough|1998}} Snelson has acknowledged the constructivists as an influence for his work (query?).<ref>In Snelson's article for Lalvani, 1996, I believe.{{full citation needed|date=March 2021|reason="I believe.", really?}}</ref> French engineer David Georges Emmerich has also noted how Kārlis Johansons's work (and industrial design ideas) seemed to foresee tensegrity concepts.<ref>David Georges Emmerich, ''Structures Tendues et Autotendantes'', Paris: Ecole d'Architecture de Paris la Villette, 1988, pp. 30–31.</ref>

In fact, some scientific paper proves this fact, showing the images of the first Simplex structures (made with 3 bars and 9 tendons) developed by Ioganson.<ref>Gómez-Jáuregui, V. et al. (2023) "[https://www.mdpi.com/2076-3417/13/15/8669 Tensegrity Applications to Architecture, Engineering and Robotics: A Review] {{Webarchive|url=https://web.archive.org/web/202401190841/https://www.mdpi.com/2076-3417/13/15/8669 |date=19 January 2024 }}&rdquo. Appl. Sci. 2023, 13(15), 8669; https://doi.org/10.3390/app13158669</ref>

In 1948, artist [[Kenneth Snelson]] produced his innovative "X-Piece" after artistic explorations at [[Black Mountain College]] (where [[Buckminster Fuller]] was lecturing) and elsewhere. Some years later, the term "tensegrity" was coined by Fuller, who is best known for his [[geodesic dome]]s. Throughout his career, Fuller had experimented with incorporating tensile components in his work, such as in the framing of his [[dymaxion]] houses.{{sfn|Fuller|Marks|1960|loc=Ch. Tensegrity}}

Snelson's 1948 innovation spurred Fuller to immediately commission a mast from Snelson. In 1949, Fuller developed a tensegrity-[[icosahedron]] 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]] took place. At the MOMA exhibition, Fuller had shown the mast and some of his other work.<ref>See photo of Fuller's work at this exhibition in his 1961 article on tensegrity for the ''Portfolio and Art News Annual'' (No. 4).</ref>  At this exhibition, Snelson, after a discussion with Fuller and the exhibition organizers regarding credit for the mast, also displayed some work in a [[display case|vitrine]].{{sfn|Lalvani|1996|p=47}}

Snelson's best-known piece is his {{convert|26.5|m|ft|adj=mid|-high|abbr=off|sp=us}} ''[[Needle Tower]]'' of 1968.<ref>{{Cite web |title=Needle tower |url=https://krollermuller.nl/en/page/3806}}</ref>

== Mathematics of tensegrity ==
The loading of at least some tensegrity structures causes an [[Auxetics|auxetic]] response and negative [[Poisson ratio]], e.g. the T3-prism and 6-strut tensegrity icosahedron.

=== 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).<ref name="RWBurkhardt2008">
{{citation |last= Burkhardt |first=Robert William Jr. |year=2008 |title=A Practical Guide to Tensegrity Design |url=https://www.angelfire.com/ma4/bob_wb/tenseg.pdf |archive-url=https://web.archive.org/web/20041220180510/http://www.angelfire.com/ma4/bob_wb/tenseg.pdf |archive-date=2004-12-20 |url-status=live}}</ref>

The stability ("prestressability") of several 2-stage tensegrity structures are analyzed by Sultan, et al.<ref name="Sultan2001">
{{cite journal |last=Sultan |first=Cornel |author2=Martin Corless |author3=Robert E. Skelton |title=The prestressability problem of tensegrity structures: some analytical solutions |journal=International Journal of Solids and Structures |url=http://www.aoe.vt.edu/people/webpages/csultan/publications-pdfs/journalarticleijss2001.pdf |year=2001 |volume=26 |page=145  |archive-url=https://web.archive.org/web/20151023184609/http://www.aoe.vt.edu/people/webpages/csultan/publications-pdfs/journalarticleijss2001.pdf |archive-date=23 October 2015}}</ref>

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 π/6. The formula for its unique self-stress state is given by,<ref>{{Cite journal |last1=Aloui |first1=Omar |last2=Flores |first2=Jessica |last3=Orden |first3=David |last4=Rhode-Barbarigos |first4=Landolf |date=2019-04-01 |title=Cellular morphogenesis of three-dimensional tensegrity structures |url=https://www.sciencedirect.com/science/article/pii/S0045782518305814 |journal=Computer Methods in Applied Mechanics and Engineering |language=en |volume=346 |pages=85–108 |doi=10.1016/j.cma.2018.10.048 |issn=0045-7825|arxiv=1902.09953 |bibcode=2019CMAME.346...85A |s2cid=67856423 }}</ref><math display="block">\omega = \omega_1 [-\sqrt{3}, -\sqrt{3}, -\sqrt{3}, \sqrt{3}, \sqrt{3}, \sqrt{3}, 1, 1, 1, 1, 1, 1]^T</math>Here, the first three negative values correspond to the inner components in compression, while the rest correspond to the cables in tension.

=== Tensegrity icosahedra ===

[[File:Tensegrity icosahedron.png|thumb|Mathematical model of the tensegrity icosahedron]]
[[File:Tensegrity icosahedron shapes.png|thumb|Different shapes of tensegrity icosahedra, depending on the ratio between the lengths of the tendons and the struts]]
[[File:Icosahedral tensegrity structure.png|thumb|A tensegrity icosahedron made from straws and string]]
The tensegrity [[icosahedron]], first studied by Snelson in 1949,<ref>{{cite thesis
 | last = Cera | first = Angelo Brian Micubo
 | page = 5
 | publisher = University of California, Berkeley
 | type = Ph.D. thesis
 | title = Design, Control, and Motion Planning of Cable-Driven Flexible Tensegrity Robots
 | url = https://escholarship.org/uc/item/2fj2b242
 | year = 2020}}</ref> has struts and tendons along the edges of a polyhedron called [[Jessen's icosahedron]]. It is a stable construction, albeit with infinitesimal mobility.{{Sfn|Kenner|1976|pp=11-19|loc=§2. Spherical tensegrities}}<ref>{{cite web|url=http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/sammlung/ten.htm|title=Tensegrity Figuren|publisher=Universität Regensburg|access-date=2 April 2013|archive-url=https://web.archive.org/web/20130526153151/http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/sammlung/ten.htm|archive-date=26 May 2013}}</ref> To see this, consider a cube of side length {{math|2''d''}}, centered at the origin. Place a strut of length {{math|2''l''}} 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 {{tmath|(0, d, l)}} and {{tmath|(0, d, -l)}}, those of its parallel strut will be, respectively, {{tmath|(0, -d, -l)}} and {{tmath|(0, -d, l)}}. The coordinates of the other strut ends (vertices) are obtained by permuting the coordinates, e.g., {{tmath|(0, d, l) \rightarrow (d, l, 0) \rightarrow (l, 0, d)}} (rotational symmetry in the main diagonal of the cube).

The distance ''s'' between any two neighboring vertices {{math|(0, ''d'', ''l'')}} and {{math|(''d'', ''l'', 0)}} is
:<math display="block">s^2 = (d - l)^2 + d^2 + l^2 = 2\left(d - \frac{1}{2} \,l\right)^2 + \frac{3}{2} \,l^2</math>

Imagine this figure built from struts of given length {{math|2''l''}} and tendons (connecting neighboring vertices) of given length ''s'', with <math>s > \sqrt\frac{3}{2}\,l</math>. 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 <math>s = \sqrt\frac{3}{2}\,l</math> the two extremes coincide, and <math>d = \frac{1}{2}\,l</math>, 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]], for which the ratio of <math>d</math> and <math>l</math> would be the [[golden ratio]], rather than 2. However both sets of coordinates lie along a continuous family of positions ranging from the [[cuboctahedron]] to the [[octahedron]] (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<ref>{{Cite book|title=Regular Polytopes|title-link=Regular Polytopes (book)|last=Coxeter|first=H.S.M.|publisher=Dover|year=1973|orig-date=1948|edition=3rd|location=New York|pages=51–52|chapter=3.7 Coordinates for the vertices of the regular and quasi-regular solids|author-link=Harold Scott MacDonald Coxeter}}</ref> and later called the "jitterbug transformation" by Buckminster Fuller.<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211211/9sM44p385Ws Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20141022184744/http://www.youtube.com/watch?v=9sM44p385Ws Wayback Machine]{{cbignore}}: {{Citation|last=Fuller|first=R. Buckminster|title=Vector Equilibrium|date=2010-10-22|url=https://www.youtube.com/watch?v=9sM44p385Ws|access-date=2019-02-22}}{{cbignore}}</ref><ref>{{Cite journal|last=Verheyen|first=H.F.|date=1989|title=The complete set of Jitterbug transformers and the analysis of their motion|journal=Computers & Mathematics with Applications|volume=17, 1-3|issue=1–3|pages=203–250|doi=10.1016/0898-1221(89)90160-0}}</ref>

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.{{Sfn|Kenner|1976|pp=16-19|loc=Elasticity multiplication}}

== Patents ==

* {{US patent|3063521}}, "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.
* {{US patent|3139957}}, "Suspension Building" (also called aspension), 7 July 1964, Buckminster Fuller.
* {{US patent|3169611}}, "Continuous Tension, Discontinuous Compression Structure," 16 February 1965, Kenneth Snelson.
* {{US patent|3866366}}, "Non-symmetrical Tension-Integrity Structures," 18 February 1975, Buckminster Fuller.

== Basic tensegrity structures ==

<gallery>
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 [[Karlis Johansons|Karl Ioganson]], 1921<ref group="gallery">{{harvnb|Gómez-Jáuregui|2010|p=28. Fig. 2.1}}</ref>
File:Tensegrity Icosahedron.png|Tensegrity Icosahedron, [[Buckminster Fuller]], 1949<ref group="gallery">{{harvnb|Fuller|Marks|1960|loc=Fig. 270}}</ref>
File:Tensegrity Tetrahedron.png|Tensegrity Tetrahedron, Francesco della Salla, 1952<ref group="gallery">{{harvnb|Fuller|Marks|1960|loc=Fig. 268}}.</ref>
File:Tensegrity X-Module Tetrahedron.png|Tensegrity X-Module Tetrahedron, [[Kenneth Snelson]], 1959<ref group="gallery">{{harvnb|Lalvani|1996|p=47}}</ref>
</gallery>

== Tensegrity structures ==

<gallery>
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|Science City]], [[Kolkata]].
File:AfrikaBurn 2015 Dissipate.JPG|''Dissipate'', an hourglass tower art sculpture including tensegrity structure, constructed at [[AfrikaBurn]], 2015, a [[Burning Man Festival|Burning Man]] regional event
</gallery>

== In chemistry ==

Sometimes highly strained molecules can be surprisingly stable due to lack of easily accessible decomposition pathways, like [[cubane]] or [[1.1.1-propellane]].

== See also ==

* {{annotated link|Cloud Nine (tensegrity sphere)|Cloud Nine}}, giant sky-floating tensegrity spheres named by Buckminster Fuller
* {{annotated link|Hyperboloid structure}}
* {{annotated link|Interactions of actors theory}}
* {{annotated link|Saddle roof}}
* {{annotated link|Space frame}}
* {{annotated link|Synergetics (Fuller)|Synergetics}}
* {{annotated link|Tensairity}}
* {{annotated link|Tensile structure}}
* {{annotated link|Thin-shell structure}}
* {{annotated link|Kinematics of the cuboctahedron}}, the geometry of the motion of the tensegrity icosahedron

== Notes ==

{{reflist|group=gallery}}

== References ==

{{Reflist}}

== Bibliography ==

{{More footnotes needed|date=March 2009}}
* {{cite journal |first=R. Buckminster|last=Fuller|year=1961 |title=Tensegrity |journal=Portfolio and Art News Annual |issue=4 |pages=112–127, 144, 148 |url=http://www.rwgrayprojects.com/rbfnotes/fpapers/tensegrity/tenseg01.html}}
* {{cite book |first=R. Buckminster|last=Fuller|author-mask=1 |title=Synergetics: Explorations in the Geometry of Thinking |url=https://books.google.com/books?id=AKDgDQAAQBAJ |year=1982 |publisher=Macmillan |isbn=978-0-02-065320-2 |orig-date=1975 |volume=I}}
* {{cite book |first=R. Buckminster |last=Fuller |author-mask=1 |title=Synergetics 2: Further Explorations in the Geometry of Thinking |url=https://books.google.com/books?id=Op4qDwAAQBAJ |year=1983 |publisher=Macmillan |volume=2 |isbn=978-0-02-092640-5 |orig-date=1979}} [http://www.rwgrayprojects.com/synergetics/s07/toc07.html Online]
* {{cite book |first1=R. Buckminster |last1=Fuller |author-mask=1 |last2=Marks |first2=Robert W. |title=The Dymaxion World of Buckminster Fuller |publisher=Anchor Books |year=1973 |isbn=978-0-385-01804-3 |at=Figs. 261–280 |url=https://books.google.com/books?id=X3pRAAAAMAAJ |orig-date=1960 |ref={{harvid|Fuller|Marks|1960}}}} 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.
* {{cite book |first=Hugh |last=Kenner |title=Geodesic Math and How to Use It |publisher=University of California Press |year=1976 |isbn=978-0-520-02924-8 |url=https://archive.org/details/geodesicmathhowt0000kenn |url-access=registration }} 2003 reprint {{ISBN|0520239318}}. This is a good starting place for learning about the mathematics of tensegrity and building models.
* {{cite book |last=Gómez-Jáuregui |first=Valentin |year=2007 |title=Tensegridad. Estructuras Tensegríticas en Ciencia y Arte |location=Santander |publisher=Universidad de Cantabria |isbn=978-84-8102-437-1|language=es}}
* {{cite book |last=Gómez-Jáuregui |first=Valentín |author-mask=1 |year=2010 |title=Tensegrity Structures and their Application to Architecture |location=Santander |publisher=Servicio de Publicaciones de la Universidad de Cantabria |isbn=978-84-8102-575-0 }}
* {{cite journal |last=Gough |first=Maria |title=In the Laboratory of Constructivism: Karl Ioganson's Cold Structures |journal=October |volume=84 |pages=90–117 |date=Spring 1998 |jstor=779210 |doi=10.2307/779210 }}
* {{cite journal | last1 = Juan | first1 = S. J. | last2 =Tur | first2 = J M | title = Tensegrity frameworks: Static analysis review | journal = Mechanism and Machine Theory | volume = 43 | issue = 7 |date=July 2008 | pages =859–81 | doi=10.1016/j.mechmachtheory.2007.06.010| citeseerx = 10.1.1.574.7510 }}
* {{cite journal |last1=Korkmaz |first1=Sinan |last2=Bel Hadj Ali |first2=Nizar |last3=Smith |first3=Ian F.C. |title=Determining Control Strategies for Damage Tolerance of an Active Tensegrity Structure |journal=Engineering Structures |doi=10.1016/j.engstruct.2011.02.031 |volume=33 |issue=6 |pages=1930–1939 |date=June 2011 |bibcode=2011EngSt..33.1930K |citeseerx=10.1.1.370.6243 |url=http://infoscience.epfl.ch/record/164609/files/Korkmaz%20et%20al,%20Determining%20Control%20Strategies%20for%20Damage%20Tolerance%20of%20an%20Active%20Tensegrity%20Structure,%20Engineering%20Structures%20(2011)_2.pdf |archive-url=https://web.archive.org/web/20110929164401/http://infoscience.epfl.ch/record/164609/files/Korkmaz%20et%20al,%20Determining%20Control%20Strategies%20for%20Damage%20Tolerance%20of%20an%20Active%20Tensegrity%20Structure,%20Engineering%20Structures%20(2011)_2.pdf |archive-date=29 September 2011 }}
* {{cite journal |last1=Korkmaz |first1=Sinan |last2=Bel Hadj Ali |first2=Nizar |last3=Smith |first3=Ian F.C. |author-mask1=1 |author2-mask=1 |author3-mask=1 |title=Configuration of Control System for Damage Tolerance of a Tensegrity Bridge |journal=[[Advanced Engineering Informatics]] |doi=10.1016/j.aei.2011.10.002 |date=January 2012 |volume=26 |issue=1 |pages=145–155 |url=http://infoscience.epfl.ch/record/175523}}
* {{cite journal |editor-first=Haresh |editor-last=Lalvani |title=Origins of Tensegrity: Views of Emmerich, Fuller and Snelson |journal=International Journal of Space Structures |volume=11 |issue=1–2 |pages=27–55 |year=1996 |doi=10.1177/026635119601-204|s2cid=114004009|url=http://journals.sagepub.com/toc/spsa/11/1-2|url-access=subscription }}
* {{cite journal |first1=Thales R. |last1=Souza |first2=Sérgio T. |last2=Fonseca |first3=Gabriela G. |last3=Gonçalves |first4=Juliana M. |last4=Ocarino |first5=Marisa C. |last5=Mancini |title=Prestress revealed by passive co-tension at the ankle joint |journal=Journal of Biomechanics |volume=42 |issue=14  |pages=2374–80 |date=October 2009 |doi=10.1016/j.jbiomech.2009.06.033 |pmid=19647832 |doi-access=free }}

== Further reading ==

* Edmondson, Amy (2007). [https://web.archive.org/web/20031002084349/http://www.angelfire.com/mt/marksomers/40.html "A Fuller Explanation"], Emergent World LLC
* {{cite book |first=Peter |last=Forbes |chapter=9. The Push and Pull Building System |title=The Gecko's Foot: How Scientists are Taking a Leaf from Nature's Book |date=2010 |publisher=Harper Collins |isbn=978-0-00-740547-3 |pages=197–230 |orig-date=2006 |chapter-url=https://books.google.com/books?id=Vm7MK6Zp6S0C}}
* {{cite book |last=Hanaor |first=Ariel |chapter=13. Tensegrity: Theory and Application |editor-first=J. François |editor-last=Gabriel |title=Beyond the Cube: The Architecture of Space Frames and Polyhedra |chapter-url=https://books.google.com/books?id=FkM0945nFV8C&pg=PA385 |date=1997 |publisher=Wiley |isbn=978-0-471-12261-6 |pages=385–408}}
* {{cite journal |last1=Masic |first1=Milenko |first2=Robert E. |last2=Skelton |first3=Philip E. |last3=Gill |title=Algebraic tensegrity form-finding |journal=International Journal of Solids and Structures |volume=42 |issue=16–17 |pages=4833–4858 |date=August 2005 |doi=10.1016/j.ijsolstr.2005.01.014}} They present the remarkable result that any [[linear transformation]] of a tensegrity is also a tensegrity.
* {{Cite journal|author=Morgan, G.J. |year=2003 |title= Historical Review: Viruses, Crystals and Geodesic Domes| journal=[[Trends in Biochemical Sciences (journal)|Trends in Biochemical Sciences]] |volume=28 |issue=2 |pages=86–90 |doi=10.1016/S0968-0004(02)00007-5 |pmid=12575996 |author-link=Gregory J Morgan|doi-access=free }}
* {{cite journal |first=R. |last=Motro |title=Tensegrity Systems: The State of the Art |journal=International Journal of Space Structures |volume=7|issue=2|pages=75–84 |year=1992 |doi=10.1177/026635119200700201 |s2cid=107820090}}
* {{cite book |first=Anthony |last=Pugh |title=An Introduction to Tensegrity |publisher=University of California Press |year=1976 |isbn=978-0-520-03055-8 |url=http://www.antiqbook.com/boox/vel/29501.shtml |access-date=9 May 2008 |archive-url=https://web.archive.org/web/20080504081841/http://www.antiqbook.com/boox/vel/29501.shtml |archive-date=4 May 2008 }}
* {{cite journal |first=Kenneth |last=Snelson |title=Letter to R. Motro |journal=International Journal of Space Structures |date=November 1990 |url=http://www.grunch.net/snelson/rmoto.html}}
* Vilnay, Oren (1990). ''Cable Nets and Tensegric Shells: Analysis and Design Applications'', New York: Ellis Horwood Ltd. {{ISBN?}}
* {{cite journal |first=Bin-Bing |last=Wang |title=Cable-strut systems: Part I – Tensegrity |journal=Journal of Constructional Steel Research |volume=45 |issue=3 |pages=281–89 |year=1998 |doi=10.1016/S0143-974X(97)00075-8 }}
* Wilken, Timothy (2001). ''Seeking the Gift Tensegrity'', TrustMark {{ISBN?}}

== External links ==

{{Commons category}}

* [http://imac.epfl.ch/ Scientific Publications in the Field of Tensegrity] by Swiss Federal Institute of Technology (EPFL), Applied Computing and Mechanics Laboratory (IMAC)
* [http://www.biotensegrity.com/ Stephen Levin's Biotensegrity site] Several papers on the tensegrity mechanics of biologic structures from viruses to vertebrates by an Orthopedic Surgeon.

{{Buckminster Fuller}}

[[Category:Buckminster Fuller]]
[[Category:Tensile architecture]]