A library of tensegrity modules provides components for modular tensegrity design. The library should include tensegral weave patterns, tensegrity equivalents for Platonic and Archimedean solids, and tensegrity prisms. Creating an alphabet of tensegrity forms requires consideration of chirality, asymmetry, bilateral symmetry, and translational symmetry. The categorization of types of models is still incomplete. The six-strut tensegrity ball is an expanded octahedron, not an icosahedron.

Context: Tom Flemons Archive and Modular Tensegrity Design

### A library of tensegrity modules

(June 21, 2015) What I do believe is worth doing is to build 3D parts as a catalogue for others to use. This is already happening to some extent (Dawn building Big Puppy in NTRT) and I spend a lot of time in Sketchup trying to render my models for future reference. Maybe an archive needs to be formed – my workspace (in my head, workshop and computer) is quite cluttered and a lot of organizing is required. I have a hard time finding things and it’s just getting worse. I have many large boxes in storage filled with dusty, sagging old prototype tensegrities going back in some cases 30 years. I can’t throw anything away because I might forget how I built it if I ever need to again.

###### Library includes weaves, Platonic and Archimedean solids, prisms

(July 15, 2015 repeated from Simulation Could Help) I envision a software package that has a series of 3D geometrical templates built in which can then be altered to suit the design. For example tensegrities are generated from some basic components and simple constructions – three, four, and five fold weave patterns in 3D are built by knitting together simple geometries. Templates of basic Platonic and Archimedean solids and their tensegrity equivalents could be included in a palette of forms that could be distorted and altered to suit the project. Tensegrity prisms are essential to many of my models and would have to be included in such a palette of shapes.

###### An alphabet of tensegrity forms: chirality, asymmetry, bilateral symmetry, translational symmetry

(Oct 14, 2015) An alphabet of forms… how rigorous should this be? I have scads of models lying around but I don’t necessarily have a completely organized alphabet…

I have some doubts about how easy this is going to be over skype to convey the range of shapes and how they can be linked. In fact it’s hard for me to know where to start.

There is the issue of discrete tensegrities vs linkages of discrete tensegrities into a cluster. How to go about this is not simple. There are face to face bonds, edge to edge bonds and point to point bonds. Each of these is quite a bit different than if we were dealing with solids or wire frame armatures. There are connections where chirality reverses from segment to segment – others where it doesn’t.

Symmetry vs asymmetry and bilateral symmetry (mirror symmetry) vs translational symmetry (whereby a form is approximately symmetrical but is offset by one course of components as in my spiral tensegrity mast – a cw spiral is countered by a ccw spiral but offset – this has a bearing in determining what kind of mast should/could serve as a spinal coupler). Chirality is an important concept which would be part of a discussion of rotational vectors.

What constitutes in this metaphor a letter and what is a word? If a tensegrity model is seen as a word then a structure is a sentence. But maybe the base element is a single compression/tension pair. If there is going to end up being a grammar then choosing the wrong scale might mean that some sentences aren’t able to be spoken or some words can be spelled differently to get different results.

### Two stable states for a six-strut tensegrity with 24 cables

(Oct 24, 2014) The range of how to build stable tensegrity models is vast and I am continually discovering new approaches. So the categorization of types of models is still incomplete. Your student’s [Adrian Muresan’s] discovery of a new stable six strut model is interesting and I built one in the spring based on an image you sent me then.

*The left image shows a 6 strut, 24 cable tensegrity in the usual ball-shaped state. The middle image shows the same structure in a different stable state. The third image shows the physical model Adrian built to verify that this bowl shape is stable. (Note that this structure is different than a 6 prism: this structure has 24 cables and a 6 prism has only 18 cables.) To switch the physical model between the ball shape and the bowl shape, it is necessary to temporarily unhook three cables while rearranging the strut positions. In a computer simulation with collision detection turned off, external force can be applied to pull the structure from one stable state to the other.
*

### The six-strut tensegrity ball is an expanded octahedron, not an icosahedron

Tom explains why he prefers the terminology *expanded octahedron tensegrity.* Following that, we briefly summarize why others prefer *tensegrity icosahedron.*

(March 23, 2016) I think what everyone is referring to as a *tensegrity icosahedron* which is then made the centre piece of the biotensegrity argument is not an icosahedron. I am referring to the classic six strut tensegrity which is in fact based on octahedral geometry. It is technically an *expanded octahedron tensegrity –* the three axes of the octahedron which pass through opposite vertices are doubled and pulled apart by the tension system into an expanded octahedron. The nature of the geometry is such that it never reaches the icosahedral symmetry – the distance between two parallel struts is .5 their length whereas the distance along any one of the tension members is .6124. It’s true that if you pull two parallel struts away from each other and the tensegrity has a bit of elasticity then the proportions of the icosahedron are achieved momentarily but it’s worth noting that the force required to achieve this has to come from the outside of the tensegrity.

(April 28, 2016) This 6 bar tensegrity has been referred to so often as a tensegrity icosahedron that the mistake is embedded in common usage now. In fact it is not an icosahedron and its properties are quite other than icoasahedral. This is not a moot distinction because a lot depends on proper definition and assumptions. There is no force acting from within the structure that can force it to assume an icosahedral geometry.

(May 19, 2016) Take for example a six strut expanded octahedron. It is commonly and incorrectly referred to as a tensegrity icosahedron because if you add 6 additional tension members joining adjacent parallel struts, a twenty faceted figure appears which at first glance seems to be similar to an icosahedron which has 20 equilateral triangular facets. But a quick check of the lengths disclose that the distances between the nodes vary. Given a strut length of 1.0 the coloured tension members will necessarily have a length of 0.6124 between nodes (and to answer you question 4 decimals is sufficient for most purposes) but the distance between any two parallel struts is 0.5. This is necessarily true of any expanded octahedron because the geometry requires it. If you add those 6 tension members onto the structure they are going to be only 0.5 length and thus the structure is not composed entirely of equilateral triangles and thus is not an icosahedron.

###### Fuller’s jitterbug: tensegrities do not change phase in this manner

(March 23, 2016) We can talk about the oscillatory nature of tensegrities and throw in mention of the jitterbug effect whereby 12 vertices link 8 solid triangles which can oscillate and rotate either in to the octahedron form or rotate out past the icosahedron to the cubo-octahedron or vector eqilibrium form, but this is not what a tensegrity does. There is no way that the six strut tensegrity can assume icosahedral symmetry without a force from the outside. No combination of shortening or lengthening struts or cables makes this possible.

(April 28, 2016) I think it is irrelevant to invoke the jitterbug oscillation that transforms a wireframe octahedron through a phase change to an icosahedron and beyond to a cubo-octahedron because tensegrities do not phase change in this manner. They reach a homeostasis which is endogenous to the geometry that underlies them and resist deformation. They don’t move from the octahedron through the icosahedron to the cubo-octahedron by rotating around their vertices because the tension net restrains all rotations. It is possible of course to start with a 3 strut octahedron tensegrity and double each strut which nets the expanded octahedron tensegrity. But the next doubling of struts results in a tensegrity cube. There is no icosahedron in the picture…

###### Importance of proper terminology

(March 23, 2016) This is more than a moot point. I have had this argument with Steve [Levin] for about 17 years now. It was the first thing I brought up when we first talked on the phone and it never got resolved. I assumed I was missing something and just went along with his nomenclature. But you can’t found a theory of biology on a misnamed geometrical structure. A proper icosahedral tensegrity requires 30 struts and depending on how the vertices are arranged it either looks more like an icosahedron, or a dodecahedron or if the struts are spaced equally along the tension member an icosadodecahedron tensegrity. Technically they are all the same from a tensegrity point of view as there are no absolute vertices.

(March 23, 2016) Incidentally if you slide the struts along the tension members of an expanded octahedron tensegrity you can derive a tensegrity tetrahedron. Also if you then double the number of struts from six to twelve you get a tensegrity cube. There are essentially two families of geometrical symmetries – the octahedral/hexahedral (cube) family with 3,4 fold symmetries and the icosahedral, dodecahedral with 3,5 fold symmetries. The common structure that fits into both families is the tetrahedron – 3 fold symmetry but they don’t mix except in more complex compound forms. So it is incorrect and misleading to refer to the six strut tensegrity as icosahedral.

###### Terminology should reflect the bilateral symmetry of the tensegrity ball; symmetry is important for biological modeling

(April 28, 2016) Further, the expanded octahedron belongs to the family of geometric objects which have bilateral symmetry. This includes the cube, the octahedron and the cubo-octahedron. There are no other families of geometric objects which can be cleaved bilaterally (i.e. along edges). Icosahedral geometry is not bilaterally symmetrical – it’s symmetry is five fold and three fold not four fold and three fold as in the cubo-octahedron. Similarly the tetrahedron and dodecahedron do not possess mirror symmetry. Given that embryology seems dependent on mitosis occurring bilaterally it is hard to see how invoking icosahedral geometry has any descriptive value here.

###### Reasons for using the terminology t*ensegrity icosahedron*

*In October 2020, Steve Levin wrote: My argument with Tom was always whether it was an expanded octahedron or a collapsing icosahedron. Because the icosahedron is the polyhedron with the largest volume for surface area, it more closely matched the bubble in the foam, so seems best-suited for a “cell”. The “tensegrity icosahedron” is the vector diagram of the forces when an icosahedron is loaded and represents the least energy state between the cuboctahedron and the octahedron [illustrated by the jitterbug]. A cuboctahedron needs external energy to expand to the tensegrity state, I think, (I am just speculating on this), that an icosahedron uses internal energy to collapse to a tensegrity icosahedral state. Fuller saw the tensegrity structure as never existing in nature because it was the middle of an ongoing oscillating event.
*

Borge Jessen’s orthogonal icosahedron The shape described by Borge Jessen in 1967 closely matches that of the tensegrity icosahedron. He named it the *orthogonal icosahedron* because all the faces meet at 90 degrees.

A paper model will compress down to the octahedron but distorts the faces slightly, whereas the thin card model will expand and contract only within a certain range – like the tensegrity icosahedron – but then becomes too stiff when trying to push further. Jessen’s icosahedron is illustrated on youtube and described in Wikipedia and Wolfram MathWorld. A correction is reported here, with detailed discussion here.

Scientific investigation of these topics is continuing.