Fractal and Membrane Structures

The fractal dimension of tensegrities needs further study. A fractal tensegrity strut can lengthen, shorten or bend, as illustrated by a helical tensegrity mast with three fractal levels.  Membrane tensegrities are a better match for modeling muscles with multiple insertion angles and locations, because membranes model tensional forces more comprehensively than single lines. A compression rod can be replaced by a planar tensegrity membrane bounded by 3, 4 or 5 edges. Tensegrity sheets can be rolled into cylinders to create torsos and appendages. Membranes can be actuated across their surface.

Context: Tom Flemons Archive

Fractal dimension of tensegrities

How Tensegrity Models Reality (2018) Tensegrities float in a kind of fractal dimension – they are not solids, there are no surfaces, edges are composed of tensional members, and nodes where tension meets compression clearly show how forces are mitigated to maintain structural integrity. Any of the components, either struts or cables, can be further understood to be made from arrays of smaller tensegrities and so on down, in smaller and smaller scales like Mandelbrot sets. It’s not turtles but tensegrities all the way down… But what fractal dimension are they? Do they operate in a boundary region somewhere between two and three dimensions? Or is it more useful to consider them as four or even five dimensional objects?

Fractal struts and masts

A fractal tensegrity strut can lengthen, shorten or bend; this could be used to actuate a larger model

(March 9, 2016) Attached are some new Sketchup jpegs [the two images below] of a fractal tensegrity strut. I’ve also attached the sketchup file if you have sketchup in your computer it can be fun to dive in deep… I anticipate that there may be some way to effect changes to a fractal model at the gross scale by tweaking the elements at the fractal levels – more on that in another post – still working out the implications…


(April 4, 2016) I’ve uploaded some sketchup fractal files to the drive. I wonder if it is possible to build in a fractal component to the NTRT software. If a fractal tensegrity strut could be actuated it would allow for the possibility of changing the shape of the strut. Because the strut is now a tensegrity it can lengthen, shorten or even bend – which would be another unique way to actuate a larger model. In the examples [the two images above], I’ve taken the 6 bar X-octa and face bonded a number of them to form a tensegrity strut. I then made a larger X-octa tensegrity using six tensegrity struts [the two images below]. If the lines were given a compliant (passive elasticity) quality then actuating the interior strut lines would change the strut dimensions and thus the larger tensegrity. Just another way of thinking about how to actuate a tensegrity linkage. There’s an even more complex fractal I’ll upload soon – it’s really taxing my computer to build it as I’m up to 3 fractal levels…!


A helical  tensegrity mast with three fractal levels


Membranes model tensional forces more comprehensively than single lines

(Dec 4, 2015 repeated from Tensegral Linkages) Membranes – a potential network of prestressed lines can be replaced in all or part by a membrane which attaches to all of the nodal ends of all of the struts. Membranes disperse forces through many pathways not available to a line based tensegrity. Generally for ease of construction and clarity membranes are left out of models but they should not be discarded as in some cases it may be important that the tension network is expressed by membranes and not the cables.

Membrane tensegrities are a better match for muscles with multiple insertion angles and locations

(June 6, 2014 repeated from Tensegrity Weave Encasing a Joint) As for how you wire all of this together… As Vytas has pointed out any muscle attaches in multiple locations along a bone via the periosteum. Thus I would think you can make multiple attachment points along a bone similar to how membranes can model tensional forces more comprehensively than single lines.

(March 9, 2015) The best way to imagine the state changes from one tensegrity configuration to another as the body moves from one position to another is to model tensegrity tension systems as membranes and not individual lines. In a tensegrity that employs membranes forces are dispersed throughout the membrane radially. In an expanded octahedron tensegrity (6 struts) the six dihedral membranes diffuse the forces in a much more complex way than lines do. I hypothesize that membrane tensegrities are a better match for muscles with multiple insertion angles and locations.

Fractal membrane tensegrity structures


Membrane tensegrities that adjust to wind

(Sept 22, 2016) I have some models of membrane tensegrities which adjust to some extent when exposed to wind. Attached are some images of various ‘tensegrity turbines’ spinning away on my porch. The windmill was designed with a particular type of sailing vessel in mind – a trimaran powered by a windmill that ‘sails’ directly into the wind.  The problem was to design a windmill that could be feathered in a strong wind to depower the blades. I proposed a tensegrity type windmill that could be calibrated to spill wind proportionate to the power. Never got past a working prototype which has spun on my deck for the past 6 years. Also I am not sure what he is referring to re: SPOF achilles heel of tensegrities. They in fact do not collapse with the failure of a single component. I could envision a scenario where if the prestress was high enough and an impact was large enough to overwhelm the system the failure could be catastrophic enough to behave like an explosion which could damage the entire structure but generally tensegrities don’t behave this way. The more complex the structure, the more redundancy is built in – the more pathways to distribute stress and avoid failure…

A compression rod can be replaced by a planar tensegrity membrane bounded by 3, 4 or 5 edges

(Oct 24, 2014) I notice that you mention membrane forms of Tensegrities which excites me as that is one of the areas I am investigating at the moment. Attached are some images [the six images below] that I am working out. and this is similar to Graham’s work on the skull. Instead of working with pure linear compression rods I am investigating what happens when the compression member is actually a surface linked to other surfaces tensegrally. By surface I mean a planar membrane figure bounded by 3,4,5 etc. edges.

Surfaces that are fixed, solid sheets when edge bonded form e.g. Platonic solids which are thus not considered to be tensegrities. A set of 8 triangular surfaces when attached spherically by their vertices forms the well known vector equilibrium or jitterbug cubo-octahedron.

Tensegrity sheets can be rolled into cylinders to create torsos and appendages

(Oct 24, 2014 continued) But membrane surfaces can be built as complexes of 3,4,5 etc. strut tensegrity modules which can be linked to form tensegrity arrays or tensegrity sheets which then have depth as well. These sheets can in turn be rolled up into cylinders (second image above) which I expect can be useful to generate elements such as torsos, appendages etc. useful for their multilayered stability and many ways to link up with other components.

Membranes can be actuated across their surface

(Oct 24, 2014) In terms of investigating the properties of pure tensegrities I think it is important to make sure the subcomponents are tensegrities as well. For example my vertebral tetrahedral tensegrity mast uses tetrahedral components that are themselves not tensegrities. This limits the kinds of actuating forces that are available to use as a fixed tetrahedron (or octahedron) structure cannot be articulated itself. Instead, I envision these membranes can be actuated across their surface such that the overall shape or thickness of the larger structure is affected. This in turn may create useful means to extrude or extend segments as appendages to create locomotion.

There is some fascinating research being done, which you probably are aware of using monofilament line coiled tightly to create heat activated artificial muscles 100X stronger than human ones. I’ve built some of them in my shop and hope to employ them in some of my future models. Imagine a fabric composed of such coiled ‘muscles’ stretched across the base of a tensegrity module to form a heat actuatable membrane! They make mention of incorporating electrical resistance filaments into the fabric which generated the requisite heat when a current is applied, to cause the muscles to contract or expand. The possibilities are intriguing…