Editing Tensegrity (section)

== 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}}