Polish-American scientist and philosopher Alfred Korzybski remarked that “the map is not the territory“, encapsulating his view that an abstraction derived from something, or a reaction to it, is not the thing itself. Korzybski held that many people do confuse maps with territories, that is, confuse models of reality with reality itself.[wikipedia]
Tensegrities appear to defy gravity but also definition. They are art objects and geometric oddities, but also structures that can model complex systems. They have been employed to describe the hidden forces operating in living structures and also enlisted to describe complex social relations between groups(https://en.wikipedia.org/wiki/Syntegrity).
A model is by definition, not as complex as the thing modeled. A model of a system can have great explanatory power by virtue of it’s ability to abstract out salient points and to imply rather than explicate nuance and detail. An orrery is a model of the solar system showing the relative orbits of the planets and their moons as they perambulate around the sun but it is never confused with solar system itself. It is a mechanical clockwork device that is many orders or magnitude smaller than the solar system- for it to be useful it simplifies and highlights certain features and ignores others. Even it’s clockwork nature (though of course now we have computerized orreries which bear no hint of gears and levers) is to be excused as the means necessary to describe a complex system using simple expedient methods. Dissecting the relationship between a model and the system modelled discloses similitudes and often hides metaphors. We can say the orrery is similar to the solar system in that they both detail the nature of circular orbits of objects around a central attractor . This ignores the details of orbital mechanics which actually describe ellipses not circles and foci rather than central attractors in favour of a general description of a grand ballet of planets in their paths around the sun. The metaphor of a clockwork mechanism which governs this complex dance goes unremarked; clearly no one mistakes the procession of the planets and the stars as an actual clockwork mechanism- except for millenia this was exactly how the universe was viewed, as a series of celestial spheres nested within one another with the earth at its centre. And even today with the Copernican revolution 500 years old the 19th century metaphor of a clockwork universe is still the received wisdom in science. Metaphors have real power in shaping our underlying perspective on how the cosmos is understood. And models can have embedded in them metaphors, which are really assumptions as to how things are. The famous mathematician and physicist Freeman Dyson once commented, that while he had great respect for Stephen Hawking’s work, he felt that he didn’t know the difference between a (theoretical) model and reality. He went on to say that this was a habitual hazard for all theoretical physicists.
Tensegrity formulations are a case in point. Because they are complex systems in themselves, they carry an embedded metaphorical view that gets entangled with the things they are describing. For example living structure may be usefully described as organized fractally in similar ways to cascades of nested tensegrities operating collectively at multiple scales from atoms, molecules, tissues, organs and bodies but even further into groups of individuals or even groups of species operating within complex systems of ecologies. It should be clear but often isn’t that this is a completely different and new metaphorical view of life that is far removed from reductionist biology or clockwork biomechanics. But it is still a metaphorical construction and shouldn’t be confused with the actual structure of the body at least at the level of structural anatomy. There are no struts and cables filling the body and bones cannot be reduced to simple compression members with muscles and ligaments acting as the tension network. A model of the spine using stellated tetrahedrons can look on the surface to be a close approximation of the actual mechanism of the spine but a closer look reveals that there are no exact equivalences to struts and cables. In fact it looks as though the fascia which wraps every structure in the body in multilayered strands could be acting as a compression sleeve helping the spine achieve stability and flexibility. This can be modelled using tensegrity principles (see my paper Bones of Tensegrity) but it is a far cry of struts and cables.
“On Exactitude in Science” is a one-paragraph short story by Jorge Luis Borges, about the map-territory relation, written in the form of a literary forgery. The Borges story, imagines an empire where the science of cartography becomes so exact that only a map on the same scale as the empire itself will suffice. “Succeeding Generations… came to judge a map of such Magnitude cumbersome… In the western Deserts, tattered Fragments of the Map are still to be found, Sheltering an occasional Beast or beggar…” wikipedia
Tensegrities stand apart from other kinds of structures in many ways. They can effectively model the fundamental mathematical and geometrical patterns that seem baked into reality and they seem somehow anchored to the basic nature of all structure at the smallest scale of matter and energy. But unlike Greek Platonic Solids, tensegrities are transparent- both in structure and design. You can literally see through them, and through also to how the forces that hold them together are arranged. Even so, how they manage to hold their form is non-obvious without some study and instruction. They must be understood holistically; they challenge us to think systemically and to see how all the parts of a whole act in concert.
They also 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 tensegritiesand 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?
A tensegrity is literally a map of the forces that goes into making it. All tensegrities have this property of being a diagram of their force vectors as well as being a structure- in this sense tensegrities are self referential. They point to the nature of structure in their structure and display fundamental geometrical properties which underlie the basis of all form. This presents unique problems in categorization… It’s easy to make categorical errors: tensegrities are complex systems that can model even more complex systems but the model isn’t equivalent to the thing it models.
Kenneth Snelson, the sculptor who designed and built some of the first tensegrity structures felt that his discovery had no useful purpose beyond an artistic rendering of matter. But there are hidden features to tensegrities that seem relevant to many disciplines so a closer look at what can be done with them is perhaps relevant.
For example, when studying math it’s often important to not only get to a right answer but to show how you got there. How something gets solved is as important as the solution itself. In the ‘real world’ a similar metric can apply.
Consider building a table as a mathematical problem: any ‘solution’ requires a structure that has a solid planar horizontal surface parallel to and some distance above the base that can support a book and a cup of coffee without them sliding off. If it’s too flimsy or incorrectly supported, these conditions won’t be met and down goes the coffee.There are of course many ways to build a functional table, but the triangulation necessary to stabilize any three-dimensional structure is usually hidden in the material and not visible. A table could have four solid legs that are fixed in place by some form of joinery or angle brackets, but these techniques do not explicate how the forces such as shear or torque are behaving inside the table. Or a table could be built from sheets of plywood that are glued or screwed together and the hidden triangulation necessary to stabilize it, is built into the material itself. But showing your work in this case means demonstrating in the materials and construction how the forces are handled so that the table does the work it’s designed to.
A tensegrity table is different; it does the work required but embeds within, it’s own raison d’être. For example, it can be built minimally from four isolated floating legs stabilized only by means of tensional cables in a configuration known as a tensegrity prism. (fig. 2)
Tensegrity prisms do not look like solid prisms that are derived from2D extrusions of simple shapes. The legs do not stand vertically, rather they tilt at an angle such that the base of the table is rotated 45° from the top. The tension lines attached to the ends of the struts, constrain this rotation and prevent them from falling outward. As the entire structure is prestressed, it is rigid and can easily support a table top. All of the forces that go into stabilizing it are clearly on display, and easily visible to the observer. The legs are in pure axial compression even though they are rotated at an angle, and the triangulation is readily apparent in the tension network that links the legs. There are no redundant parts, every strut and cable is needed to stabilize it; thus it is a minimal description solution to the problem of a table: you can not build a simpler stable structure.
Fuller claimed that, “all structures properly understood from the solar system to the atom are tensegrity structures.” (https://fullerfuture.files.wordpress.com/2013/01/buckminsterfuller-synergetics.pdf)
And so tensegrities can diagram macro scale models of molecules and atoms – modeling the strong and weak nuclear forces that bind all matter and energy into coherence.In this, tensegrities are not so much things as physical descriptions of instantiated events in ongoing mass/energy transactions. So much for metaphors…
To illustrate this claim we can apply a tensegrity description to Newton’s laws of motion. It will become apparent that the descriptor and the described at some point are interchangeable though. Newtons first law of motion-also known as the law of inertia, states that an object in motion will continue on a path or vector, unless it is acted upon by an outside force. Gravity is such a force, causing objects to drift together through mutual attraction. As objects close on one other, conservation of angular momentum compels them to describe elliptical paths whereby they circle each other according to the laws of orbital mechanics. Fuller felt, that such events could be understood tensegrally including the motions of the planets and stars as they make their grand orbits around their centers of masses. This is a different kind of description of tensegrity that doesn’t directly involve tension and compression elements. Rather, gravity and tensegrity relationships are described by a single centripetal force. Centrifugal force is characterized as the inertia of objects not bounded by a contracting force.
The tensional envelope that describes the boundary of a tensegrity system could be defined as a centripetal force that constrain compression elements to behave very much like objects caught in a gravity field. In a tensegrity structure the tensional net which pulls the entire structure taut is doing so by trying to make it smaller. The tensional net is the centripetal force which causes the compression elements to deflect tangentially from the center of the force. Thus, tension forces in a tensegrity create a level of prestress on the system that is a type of artificial gravity. Compression members are bound by this field- if you disconnect one end of a compression member from the tensional net, it springs outward, behaving very much like an object escaping from orbit. Struts do not connect to each other directly, rather they create chiral groups that behave like orbital objects conserving angular momentum rotating past each other. This is why tensegrity structures do not look like other geometrical solids where edges meet at definite points and surfaces are formed by coplanar edges. As it turns out Platonic solids are imaginary forms that do not exist outside of our imaginations and machinations. Plato’s idealized forms turns out to be tensegrities which are real (enough) descriptions of how structure and thus space time is organized.
Living structure is organized organically and can be described partially (or wholly) as tensegrity systems scaling from atoms to visible form. This applies to all living structure both plants and animals: all are tensegral, bound centripetally by an artificial gravity field (equivalent) caught inside the larger gravity field of a planet. This is why in weightless conditions, a body remains integral to it’s own essential force field, doesn’t collapse and can operate outside a gravity field.
We can expand on this metaphor to see where it takes us. The closer a satellite orbits a planet, the greater the gravitational pull and the faster the satellite must move tangentially to keep from falling out of orbit. The further out an object is from a gravitational field the less gravity acts upon it (falling off inversely as the square of the distance) and the slower it needs to move to remain in a stable orbit. Putting aside issues such as atmospheric friction for low elevation orbits, an object placed in orbit will stay in that orbit unless acted upon by an outside force. It takes more energy to put an object in a higher orbit- thus the higher the orbit, the more potential energy it has stored.
How does this relate to tensegrity structures?
Simple tensegrity prisms composed of three or four struts are more vulnerable to catastrophic failure if any one element fails. The fewer the number of struts, the faster forces travel and the more brittle it is. In a simple tensegrity prism there is little redundancy built in and a failure of one component is catastrophic. The range of possible states is limited, meaning small changes in tension results in large changes in rigidity. Three or four struts rotate and pass close to the centre of structure. Perhaps this corresponds to an object in low orbit- close to the centre of gravity – it’s moving faster and small forces have big, quick consequences. Falling out of orbit means in this case a collapse of the structure.
A more complex tensegrity, having more components might equates then to an object in a higher orbit. Six or twelve strut tensegrities are much less likely to catastrophically fail if one component breaks- their overall resilience is higher and like an object in higher orbit more likely to realign into a new elliptical path than catastrophically fail and ‘fall out of orbit’.
Objects in geosynchronous orbit (whereby a satellite achieves a balance between escaping from the bounds of earths gravity field and a lower orbit that slowly degrades over time) cause objects to appear to hang motionless in the sky which is the basis for the satellite communications system that globally links all of us. A tensegrity displays this same balance of forces. A 30 strut spherical tensegrity, has six pentagonal great circle rings of struts interwoven and intersecting in a regular geometric pattern described as an icosa-dodecahedron. Unlike an Archimedean solid of the same name, there are no fixed points, no planar surfaces and no solid interior. Given that centripetal force involves constraining rigid bodies to follow curved paths, the five struts composing a great circle path can be seen as multiple instances of a single object in rotation: a pentagonal ring of struts then becomes a freeze frame example of the rotation path of a particular orbit. A physical tensegrity model is then, a frozen three dimensional instance of a four dimensional rotating object. Or a four dimension space/time example of a fifth dimensional object. Combining all possible great circle paths describes a hyper object possessing super position-thus a tensegrity can even model quantum events as a single instance of all possible solutions of an equation- the act of observation instantiates a multidimensional object as a condensed slice of space/time.
This expanded view of tensegrities allows the possibility of a prominent role in current scientific efforts to describe reality as quantum events. We make models of what we observe to better understand them. Tensegrities are useful abstract models because they are complex and versatile enough to describe reality independent of scale. They are fractal systems that illustrate both atomic and cosmic scale events as well as homeostatic systems such as life.
It is proving to be a versatile metaphor- many forms of clinical practice today refer to tensegrity or biotensegrity approaches to treatment. But as I have argued here and in my paper ‘The Bones of Tensegrity’, it’s easy to confuse metaphor and reality. When that happens, facts are bent to fit the theory instead of the reverse. A great strength of our species is the ability to cognate from patterns we pick out in the environment around us. We can build generalized principles from these patterns and that allow us to make predictions and anticipate future outcomes. Educated guesses are corrected by testing new data against our theories.
However we frequently become invested in our descriptions and fail to incorporate or even see contradictory facts. Cognitive dissonance describes the schizoid process where we refuse to put one and one together and decline to make a new story that fits all the facts. Richard Feynman, the famous physicist, in one of his lectures admonished his students that no matter how beautiful a hypothesis is, if it doesn’t fit the facts, it’s wrong.
This holds true especially in new fields of endeavor where all the facts aren’t yet known and the field itself has a certain fuzzy logic to it. Things are still uncertain and a lot of attention is paid to definitions and the language used to describe phenomena. The territory now claimed by tensegrity is vast by virtue of it’s metaphorical power and most likely things will only get more complex and perhaps more obscure.
This short paper attempts to illustrate the reflexive nature of tensegrities and give a new example of their metaphorical power. It is worth the candle to forge into this particular darkness, because the rewards of tensegrity modeling can be great and because that is what we as a species do-modeling what we do not yet understand to explore what we do not yet know.
As Gregory Bateson said, paraphrasing Robert Burns, ‘A man’s reach should exceed his grasp, else what’s a metaphor?’
copyright T. Flemons 2016