Tensegrities are appropriate descriptions for plant life, like the bending of a leaf or a tree moving in the wind. Triangulation provides stability and redundancy. A fully triangulated tensegrity cannot axially rotate. Fuller’s discussion of tensegrity pneumatics and hydraulics. A hydraulic structure is equivalent to a highly prestressed inelastic tensegrity whereas a pneumatic structure has some compliancy available. Joints in vertebrate skeletons require fascial envelopes that act as tensegral structures equivalent to invertebral hydrostatic bodies.
Context: Tom Flemons Archive
Here Tom discusses the limited range of motion present in fully-triangulated tensegrity structures such as those created by Kenneth Snelson. Using Tom’s modular approach to tensegrity design, tensegrity structures with increased range of motion are created using tensegral linkages between highly prestressed tensegrity modules.
Tensegrities are appropriate descriptions for plant life
(Aug 31, 2015) Tensegrities model envelopes not interiors and so are more appropriate descriptions for plant life. Internal hydraulic pressures in a plant create tensegrity like columns, more properly understood as a mast, which help raise it into the air. Articulations in tensegrities are gradual and smooth like the bending of a leaf or a tree moving in the wind. This is akin to the incremental articulations found in a spiral tensegrity mast. A grass stem is rigid yet can move slightly in the wind. Any action which exceeds its design parameters creates a joint which is a destruction of the stem’s integrity. Articulations in animals involve hinges with ranges of motion greater than 90° – tensegrities don’t do this intentionally. But if the resolution of a tensegrity mast is sufficiently high, it can follow acute angles but the modelling is more to do with surfaces and less to do with structural mechanics.
Triangulation provides stability and redundancy
(Feb 21, 2017) In a tensegrity, tensioned triangulation provided the stability and allows for multiple paths of redundancy. Cutting any one tension line will not result in a catastrophic failure. Tensioned triangulation means that prestress is key to creating a relatively rigid structure but it is very difficult to approach the stiffness of a compression based truss. Rigidity depends on the geometry as well. For example a rhombic icosadodecahedral tensegrity is inherently less rigid than a fully triangulated sphere. In terms of tensegrity masts – they tend to be more flexible (like a blade of grass or a tree) because the unit of assembly – the tensegrity prism is not usually completely triangulated. Snelson’s masts for example move in the wind quite organically. His very interesting paper on the connection between 3 dimensional weaving and tensegrities [Tensegrity, Weaving and the Binary World] is quite a good read.
Fully triangulated tensegrity structures
On pages 19-22 of Tensegrity, Weaving and the Binary World Kenneth Snelson describes how he creates a surface triangulation out of struts and cables. Using this method of triangulation, a tensegrity structure that contains S struts has to contain 5S-6 cables and 4S-4 triangles. To derive this, define the symbols S Struts, C cables, V vertices, E Edges, T Triangles. Use the following four equalities.
S+C = E (Every edge is either a strut or a cable)
V = 2S (Every strut has a vertex at both ends)
E = 3T/2 (Every triangle has three edges, and each edge participates in forming two triangles)
V-E+T = 2 (Euler’s Formula for planar triangulations)
Combine these to derive that C=5S-6 and T=4S-4.
A fully triangulated tensegrity cannot axially rotate
(Jan 10, 2017. Tom responds to the question “Do you consider the 3-prism to be fully triangulated?”) Re: your question – I don’t really know how to answer. I think prism tensegrities by their nature are not as stable as spherical ones but clearly they can be built as a mast that emulates a grass stalk or tree which has some range of motion. The top and bottom of a 3 prism are of course triangulated but there is something different about the sides. The tension members that make up the sides form rhombuses but if you include the strut they are connected to there is a triangulation. This allows them some axial rotation. However it is possible to add additional tension members connecting the top and bottom of adjacent struts in prism tensegrities and then they are fully triangulated in the sense that they cannot axially rotate.
(Jan 10, 2017) It’s clear that tensegrities are more difficult to analyze than traditional space frames. The only completely triangulated 3D space frame truss is the octet truss which consists of face bonded octahedrons interspersed with tetrahedrons. It is triangulated along 6 axes in 3 dimensions. Many trusses are only fully triangulated in 2 dimensions (think of roof trusses). So again tensegrities fall into some fractal space between 2 and 3 dimensions. The alternative [bowl-shaped] orientation of the 6 strut tensegrity is not fully triangulated because the hexagonal bottom is free to distort. If additional tension members were added to form a star of David then you could say it is more fully triangulated but there is still the problem of the rhombic sides…
Triangulating in tension versus triangulating in tension and compression
(Feb 1, 2017) I have been giving a lot of thought to the matter of triangulation and have some new thoughts to share. Mostly they concern the difference between triangulating in tension versus triangulating in tension and compression. A triangle composed of three struts fixed at the corners for example can be put either in tension or compression where generally a tensegrity triangle is either pure tension or a hybrid of tension and compression. The expanded octahedron for example has two types of triangulation – 8 pure tension triangles bounding the circumference and 12 hybrid diamond triangles (both sides of six diamond rhombuses) which have two sides of tension and one of compression. I think that this complicates the analysis of tensegrity structures when it comes to stability because tension triangulation still allows for a range of motion absent from rigid structures.
Snelson illustrates these two types of triangles on page 19 of Tensegrity, Weaving and the Binary World. The red type 1 tension/compression triangles are formed by one strut and two tension lines. The green type 2 pure-tension triangles are formed by three tension lines.
Modelling hydraulic and pneumatic forms
(Sept 1, 2015) Regarding the difference between modelling pneumatic and hydraulic forms: liquids are not compressible, gases are – therefore the difference would be in the level of prestress and elasticity. A hydraulic structure is equivalent to a highly prestressed inelastic tensegrity whereas a pneumatic structure has some compliancy available. A soccer ball is stiff depending on how much air is pumped in but it can still deform slightly whereas a ball completely filled with water has nowhere to go and thus can’t change its shape.
Fuller: Tensegrities are equivalent to pneumatic or hydraulic structures
(July 26, 2015) As Fuller says all structure properly understood is tensegral. This means that there are material properties that allow compressive forces whether skeletal (hard tissue) cartilaginous (semi hard) or hydrostatic to operate on them and materials which accept tension forces. At some level these two and only these two forces are required. In engineered structures the tensegrity is at the atomic/molecular levels. In soft matter (life) it happens at all levels.
In answer to your question about tensegrity pneumatics and hydraulics see Fuller in Synergetics (700.01 to 703.16) where he talks directly about tensegrities being equivalent to pneumatic or hydraulic structures.
Selected text from Fuller Synergetics:
700.011 The word tensegrity is an invention: it is a contraction of tensional integrity. Tensegrity describes a structural-relationship principle in which structural shape is guaranteed by the finitely closed, comprehensively continuous, tensional behaviors of the system and not by the discontinuous and exclusively local compressional member behaviors. Tensegrity provides the ability to yield increasingly without ultimately breaking or coming asunder.
701.00 Pneumatic Structures 701.01 Tensegrity structures are pure pneumatic structures and can accomplish visibly differentiated tension-compression interfunctioning in the same manner that it is accomplished by pneumatic structures, at the subvisible level of energy events. 701.02 When we use the six-strut tetrahedron tensegrity with tensegrity octahedra in triple bond, we get an omnidirectional symmetry tensegrity that is as symmetrically compressible, expandable, and local-load-distributing as are gas-filled auto tires.
Joints in vertebrate skeletons require fascial envelopes that act as tensegral structures equivalent to invertebral hydrostatic bodies
(Aug 8, 2015) In the case of a hydrostatic or pneumatic structure the pressure inside the membrane provides the compressive element. This can be represented by a great number of tangential compression chords that travel close to the outer skin and thus leave the majority of the interior space empty and not part of the structure. So in most tensegrities there is no possibility of a large range of motion or many degrees of freedom because the envelope is continuous. The only instance of an articulation would be in the failure of a part of the envelope allowing a partial collapse of the structure. This reinforces the notion that from a tensegrity point of view a joint is really a failure of integrity – a disjoint or deflation of the form. Spiral tensegrity masts may be special cases where their length to breadth ratio allows for incremental changes in positioning components resulting in gradual distortions in form, like snakes, cat’s tails, elephant’s trunks etc. These changes are allowed only where there is incomplete triangulation as found in the rhombic patterns created in such spiral masts. Adding a diagonal component freezes any movement at this point.
Given this argument I would suggest that lever mechanics are clearly involved whenever there are long stiffened segments (long bones) that have ends which allow articulations to occur (condyles and eminences, convexities and concavities in the geometry of the bone ends). Vertebrate skeletons allow for large land animals to resist the pull of gravity and reduce interior pressures which means less heat production. But fascial envelopes which act as tensegral structures are necessary to assist and augment joint complexes and allows the skeletal framework to articulate effectively. The equation I am making is that fascial complexes are equivalent to invertebral hydrostatic bodies which can be analyzed as tensegrity structures because tensegrities describe envelopes not interiors.