# Tensegrity

The simplest tensegrity structure (a 3-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 compression and precisely define the position of that end in the same way as the three cables in the Skylon define the bottom end of its tapered pillar.
 Stereo image
Left frame
Animation A similar structure but with four compression members.

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 (usually bars or struts) do not touch each other and the prestressed tensioned members (usually cables or tendons) delineate the system spatially.[1]

The term was coined by Buckminster Fuller in the 1960s as a portmanteau of "tensional integrity".[2] The other denomination of tensegrity, floating compression, was used mainly by Kenneth Snelson.

## Concept

Tensegrity structures are based on the combination of a few simple design patterns:

• loading members only in pure compression or pure tension, meaning the structure will only fail if the cables yield or the rods buckle
• preload or tensional prestress, which allows cables to be rigid in tension
• mechanical stability, which allows the members to remain in tension/compression as stress on the structure increases

Because of these patterns, no structural member experiences a bending moment. This can produce exceptionally rigid structures for their mass and for the cross section of the components.

The Skylon at the Festival of Britain, 1951

A conceptual building block of tensegrity is seen in the 1951 Skylon. 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 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.

Variations such as Needle Tower involve more than three cables meeting at the end of a rod, but these can be thought of as three cables defining the position of that rod end with the additional cables simply attached to that well-defined point in space.

Eleanor Heartney points out visual transparency as an important aesthetic quality of these structures.[3] Korkmaz et al.[4][5] put forward that the concept of tensegrity is suitable for adaptive architecture thanks to lightweight characteristics.

## Applications

Largest Tensegrity bridge in the world Kurilpa Bridge- Brisbane

On 4 October 2009, the 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 such structure.

A 12m high tensegrity structure exhibit at the Science City, Kolkata.

The idea was adopted into architecture in the 1960s when Maciej Gintowt and Maciej Krasiński, architects of Spodek, a venue in Katowice, Poland, designed it 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.

In the 1980s David Geiger designed Seoul Olympic Gymnastics Arena for the 1988 Summer Olympics. The Georgia Dome, which was used for the 1996 Summer Olympics, was a large tensegrity structure of similar design to the aforementioned Gymnastics Hall. Tropicana Field, home of the Tampa Bay Rays major league baseball team, has a dome roof supported by a large tensegrity structure.

Shorter columns or struts in compression are stronger than longer ones. This in turn led some, namely Fuller, to make claims that tensegrity structures could be scaled up to cover whole cities.

## Biology

Biotensegrity, a term coined by Dr. Stephen Levin, is the application of tensegrity principles to biologic structures.[6] Biological structures such as muscles, bones, fascia, ligaments and tendons, or rigid and elastic cell membranes, are made strong by the unison of tensioned and compressed parts. The muscular-skeletal system is a synergy of muscle and bone. The muscles and connective tissues provide continuous pull[7] and the bones present the discontinuous compression.

A theory of tensegrity in molecular biology to explain cellular structure has been developed by Harvard physician and scientist Donald E. Ingber.[8] For instance, the expressed shapes of cells, whether it be their reactions to applied pressure, interactions with substrates, etc., all can be mathematically modeled if a tensegrity model is used for the cell's cytoskeleton. Furthermore, the 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,[9] and even organs. This view is supported by how the tension-compression interactions of tensegrity minimize material needed, add structural resiliency, and constitute the most efficient possible use of space. Therefore, natural selection pressures would strongly favor biological systems organized in a tensegrity manner.

As Ingber explains:

The tension-bearing members in these structures — whether Fuller's domes or Snelson's sculptures — 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.[8]

In embryology, Richard Gordon proposed that Embryonic differentiation waves are propagated by an 'organelle of differentiation'[10] where the cytoskeleton is assembled in a bistable tensegrity structure at the apical end of cells called the 'cell state splitter'.[11]

## History

Kenneth Snelson's 1948 X-Module Design as embodied in a two-module column[12]

The origins of tensegrity are controversial.[13] 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 domes. Throughout his career, Fuller had experimented incorporating tensile components in his work, such as in the framing of his dymaxion houses.[14]

Snelson's 1948 innovation spurred Fuller to immediately commission a mast from Snelson. In 1949, Fuller developed an 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. Snelson's 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.[15] 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.[16]

Snelson's best known piece is his 18-meter-high Needle Tower of 1968.

Russian artist Viatcheslav Koleichuk claimed that the idea of tensegrity was invented first by Kārlis Johansons (lv), Soviet avant-garde artist of Latvian descent, who contributed some works to the main exhibition of Russian constructivism in 1921.[17] Koleichuk's claim was backed up by Maria Gough for one of the works at the 1921 constructivist exhibition.[18] Snelson has acknowledged the constructivists as an influence for his work.[19] French engineer David Georges Emmerich has also noted how Kārlis Johansons's work (and industrial design ideas) seemed to foresee tensegrity concepts.[20]

## 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).[21]

The stability (“prestressability”) of several 2-stage tensegrity structures are analyzed by Sultan, et al.[22]

### Tensegrity icosahedra

Mathematical model of the tensegrity icosahedron
Different shapes of tensegrity icosahedra, depending on the ratio between the lengths of the tendons and the struts.

The polyhedron which corresponds directly to the geometry of the tensegrity icosahedron is called the Jessen's icosahedron. Its spherical dynamics were of special interest to Buckminster Fuller[23], who referred to its expansion-contraction transformations around a stable equilibrium as jitterbug motion[24].

The following is a mathematical model for figures related to the tensegrity icosahedron, explaining why it is a stable construction, albeit with infinitesimal mobility.[25]

Consider a cube of side length 2d, centered at the origin. Place a strut of length 2l 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 (0, d, l) and (0, d, –l), those of its parallel strut will be, respectively, (0, –d, –l) and (0, –d, l). The coordinates of the other strut ends (vertices) are obtained by permuting the coordinates, e.g., (0, d, l) → (d, l, 0) → (l, 0, d) (rotational symmetry in the main diagonal of the cube).

The distance s between any two neighboring vertices (0, d, l) and (d, l, 0) is

${\displaystyle s^{2}=(d-l)^{2}+d^{2}+l^{2}=2\left(d-{\frac {1}{2}}\,l\right)^{2}+{\frac {3}{2}}\,l^{2}}$

Imagine this figure built from struts of given length 2l and tendons (connecting neighboring vertices) of given length s, with ${\displaystyle s>{\sqrt {3/2}}\,l}$. The relation tells us there are two possible values for d: one realized by pushing the struts together, the other by pulling them apart. For example, for ${\displaystyle s={\sqrt {2}}\,l}$ the minimal figure (d = 0) is a regular octahedron and the maximal figure (d = l) is a quasiregular cubeoctahedron. In the case ${\displaystyle d={\frac {1}{2}}({\sqrt {5}}-1)l}$ we have s = 2d, so the convex hull of the golden ratio figure (d = ${\displaystyle \phi }$) is a regular icosahedron. Since no article on the kinematics of polytopes would be complete without a Coxeter reference, it is appropriate to note here that by 1940 (prior to Jessen's icosahedron or the discovery of the tensegrity icosahedron) Coxeter had already shown how the twelve vertices of the icosahedron can be obtained by dividing the twelve edges of an octahedron according to the golden ratio, as one of the continuous series of (generally irregular) icosahedra with faces consisting of eight equilateral triangles and twelve isosceles triangles, ranging from cuboctahedron to octahedron (as limit cases), that can be produced by such a process of division.[26]

In the particular case ${\displaystyle s={\sqrt {3/2}}\,l}$ the two extremes coincide, and ${\displaystyle d={\frac {1}{2}}\,l}$, therefore the figure is the stable tensegrity icosahedron.

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 2d of the struts.

## Patents

• U.S. Patent 3,063,521, "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.
• U.S. Patent 3,139,957, "Suspension Building" (also called aspension), 7 July 1964, Buckminster Fuller.
• U.S. Patent 3,169,611, "Continuous Tension, Discontinuous Compression Structure," 16 February 1965, Kenneth Snelson.
• U.S. Patent 3,866,366, "Non-symmetrical Tension-Integrity Structures," 18 February 1975, Buckminster Fuller.

## References

1. ^ Gómez-Jáuregui, V (2010). Tensegrity Structures and their Application to Architecture. Servicio de Publicaciones Universidad de Cantabria. p. 19. ISBN 978-8481025750.
2. ^ Swanson, RL (2013). "Biotensegrity: a unifying theory of biological architecture with applications to osteopathic practice, education, and research-a review and analysis". The Journal of the American Osteopathic Association. 113 (1): 34–52. doi:10.7556/jaoa.2013.113.1.34. PMID 23329804.
3. ^ Eleanor Hartley, "Ken Snelson and the Aesthetics of Structure," in the Marlborough Gallery catalogue for Kenneth Snelson: Selected Work: 1948–2009, exhibited 19 February through 21 March 2009.
4. ^ Korkmaz, et al. (June 2011)
5. ^ Korkmaz, Bel Hadj Ali & Smith 2011
6. ^ Levin, Stephen (2015). "16. Tensegrity, The New Biomechanics". In Hutson, Michael; Ward, Adam (eds.). Oxford Textbook of Musculoskeletal Medicine. Oxford University Press. pp. 155–6, 158–160. ISBN 978-0-19-967410-7.
7. ^ Souza et al. 2009
8. ^ a b Ingber, Donald E. (January 1998). "The Architecture of Life" (PDF). Scientific American. 278 (1): 48–57. doi:10.1038/scientificamerican0198-48. PMID 11536845. Archived from the original (PDF) on 15 May 2005.
9. ^ Edwards, Scott A.; Wagner, Johannes; Gräter, Frauke (2012). "Dynamic Prestress in a Globular Protein". PLoS Computational Biology. 8 (5): e1002509. Bibcode:2012PLSCB...8E2509E. doi:10.1371/journal.pcbi.1002509. PMC 3349725. PMID 22589712.
10. ^ Gordon, N.K. and Gordon, R. The organelle of differentiation in embryos: the cell state splitter [invited review.] Theor. Biol. Med. Model. 13(Special issue: Biophysical Models of Cell Behavior, Guest Editor: Jack A. Tuszynski), #11. 2016
11. ^ Gordon, Richard (1999). The Hierarchical Genome and Differentiation Waves. Series in Mathematical Biology and Medicine. 3. doi:10.1142/2755. ISBN 978-981-02-2268-0.
12. ^ Gough, Maria (Spring 1998). "In the Laboratory of Constructivism: Karl Ioganson's Cold Structures". October. 84: 90–117 See p. 109. doi:10.2307/779210. JSTOR 779210.
13. ^ Gómez-Jáuregui, V. (2009). "Controversial Origins of Tensegrity" (PDF). International Association of Spatial Structures IASS Symposium 2009, Valencia.
14. ^ Fuller & Marks 1960, Ch. Tensegrity
15. ^ See photo of Fuller's work at this exhibition in his 1961 article on tensegrity for the Portfolio and Art News Annual (No.4).
16. ^ {harvnb|Lalvani|1996|p=47}}
17. ^ Droitcour, Brian (18 August 2006). "Building Blocks". The Moscow Times. Archived from the original on 7 October 2008. Retrieved 28 March 2011. With an unusual mix of art and science, Vyacheslav Koleichuk resurrected a legendary 1921 exhibition of Constructivist art.
18. ^ Gough 1998, pp. 90–117
19. ^ In Snelson's article for Lalvani, 1996, I believe.
20. ^ David Georges Emmerich, Structures Tendues et Autotendantes, Paris: Ecole d'Architecture de Paris la Villette, 1988, pp. 30–31.
21. ^ Burkhardt, Robert William, Jr. (2008), A Practical Guide to Tensegrity Design (PDF)
22. ^ Sultan, Cornel; Martin Corless; Robert E. Skelton (2001). "The prestressability problem of tensegrity structures: some analytical solutions" (PDF). International Journal of Solids and Structures. 26: 145. Archived from the original (PDF) on 23 October 2015.
23. ^ Fuller, R. Buckminster (22 October 2010), Vector Equilibrium, retrieved 22 February 2019
24. ^ Verheyen, H.F. (1989). "The complete set of Jitterbug transformers and the analysis of their motion". Computers & Mathematics with Applications. 17, 1-3 (1–3): 203–250. doi:10.1016/0898-1221(89)90160-0.
25. ^ "Tensegrity Figuren". Universität Regensburg. Archived from the original on 26 May 2013. Retrieved 2 April 2013.
26. ^ Coxeter, H.S.M. (1973) [1948]. "3.7 Coordinates for the vertices of the regular and quasi-regular solids". Regular Polytopes (3rd ed.). New York: Dover. pp. 51–52.

### Gallery

1. ^ Gómez-Jáuregui 2010, p. 28. Fig. 2.1
2. ^ Fuller & Marks 1960, Fig. 270.
3. ^ Fuller & Marks 1960, Fig. 268.
4. ^ Lalvani 1996, p. 47