Centripetal forces in tensegrities. Stabilizing an object requires that rotations be cancelled out along four vectors. The tensional matrix of a tensegrity structure acts like a localized gravitational field. Fewer struts makes forces travel faster and makes the tensegrity structure brittle. Tensegrities are dynamically stable partly because chiral rotations absorb energy by acting like springs which oscillate in and out to restore balance. Tensegrities are self-referential, disclosing how all the forces acting on them are arranged. Tension forces in a tensegrity create prestress that is a type of artificial gravity

Context: Tom Flemons Archive

### Centripetal forces in tensegrities

(Oct 7, 2015) I’ve been reading up on centripetal forces and the fact that there is no such thing as centrifugal force. The inertia of objects (mislabeled centrifugal force) are constrained by centripetal forces such as gravity or in the case of tensegrities, tensional nets. Given that centripetal force involves constraining rigid bodies to follow curved paths, one way to look at a group of compression components is to see them as multiple instances of a single object in rotation and each strut then becomes a freeze frame of the rotation path of a hyper object. A tensegrity prism is a frozen three dimensional drawing of a four dimensional rotating object. Each strut and its orientation is an aspect of a more complex object that is moving in space and time. This explains why struts are always in a relationship that is threefold fourfold fivefold etc. and for an object to be stable in space all of the rotations must cancel out. i.e. for every threefold rotation clockwise there has to be a threefold rotation counterclockwise. If you look at the six strut tensegrity there are 8 threefold relationships – half are clockwise and half are counterclockwise, thus the structure is stabilized in three dimensions. Tensegrity is like frozen orbital mechanics then. Fuller intuited this when he talked about tensegrity being the model for how planets orbit the sun and suns orbit the galaxy under the influence of gravity which is the centripetal force. In this definition tension and compression are nowhere found because they aren’t a sufficient description – they don’t imply rotation. I suspect this is important because we have been talking about rotations in the body no? But I don’t think anybody has talked about tensegrities in quite this way yet and I’m not yet sure of the implications.

###### Stabilizing an object requires that rotations be cancelled out along four vectors

(Oct 7, 2015 continued) But it’s more complicated. To stabilize an object (or a tensegrally linked cluster of objects) requires that the rotations be cancelled out along four vectors or axes not three. (I can explain later). There’s an old argument that the geodesic list serve hashed out endlessly a few years ago. How many spokes does it take to stabilize the hub of a bicycle wheel around its axle? It centers on the notion how many tension members does it take to stabilize a point in space (one end of the axle for example). Fuller said it took four tension members that are arrayed from that point, ideally tetrahedrally, to fix that point in space with no range of motion. The guys on the listserv and Fuller then added more stabilizing tension members to control the rotation of the point along all 4 vectors. Clockwise and counterclockwise times four vectors equals another eight tension members for a total of 12. Thus for any point in the body there are at minimum 12 vectors which control its position. They all pull in different directions simultaneously but variably. By changing the variance you change the centripetal force and thus the orbital inertia of rigid bodies which allows for controlled dynamic homeostatic movement. Sound good? *A few days later (Oct 13, 2015) Tom repeated this text in a different context in Modular Tensegrity Design*.

Capturing this in a computer simulation might require some astrophysics modeling software. If I’m right about the relationship to orbital mechanics then a very interesting way to model anatomy would be to construct a multi body simulation that is acting in a gravitational field which in the case of the body is represented by the myofascial domain. Movement in a body would be equivalent to a series of freeze frame solutions to a multi-body problem, which is the realm of orbital mechanics. Kind of puts tensegrity into a different light eh?

### The tensional matrix of a tensegrity structure acts like a localized gravitational field

(Oct 8, 2015) Further explorations of the relationship of tensegrity to orbital mechanics:

Posits –

- that centripetal force and the law of inertia (Newton’s first law – An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force) can be useful in describing tensegrity structures.
- centripetal force is what causes an object to follow a curved path (i.e. is an unbalanced force) such as a ball at the end of a string or a satellite around a planet. The string or the gravity field impose an unbalanced force on an inertial object, and causes it to deflect tangentially to the force. The equations which govern gravitational fields describe centripetal forces.
- 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. This is identical to how a gravity field operates on an object in orbit, thus the tensional matrix of a tensegrity structure can be understood in the same way: as acting like a localized gravitational field.
- 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 unbalanced (centripetal) 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.

###### Fewer struts makes forces travel faster and makes the tensegrity structure brittle

(Oct 8, 2015, continued) The smaller a tensegrity structure is, i.e. the fewer the number of struts it is made from, the faster forces travel through it and the more brittle it is. For example, in a three strut tensegrity prism, there is no redundancy built in and a failure of one component is catastrophic. Its range of possible tensions is limited, meaning small changes in tension results in large changes in rigidity. The three struts rotate and pass close to the centre of structure. 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. A six strut figure is in a ‘slightly higher orbit’ but each strut still bisects 25% inward from it’s approximate circumference. A twelve strut cubo-octahedron or even better, a thirty strut icosa-dodecahedron begin to approximate high orbit tensegrities – the struts each cut a shallow chord through the interior space and are closer to the tension envelope. A higher orbit tensegrity means greater potential energy, which translate to greater resilience, greater redundancy and a slower moment of reaction to a change of force. The range of movement is more elastic and less brittle.

###### Tensegrities are dynamically stable partly because chiral rotations absorb energy by acting like springs which oscillate in and out to restore balance

(Nov 15, 2015) A second insight was to realize that describing tensegrities in terms of tension and compression elements is limiting and not a complete description. For one thing it does not account for the chiral rotations built into their fundamental structure. Except for tensegrity prisms which are in a sense monads because their handedness is absolute, most tensegrities contain within them clockwise and counterclockwise rotations which additively cancel each other out. In a six strut expanded octahedron tensegrity there are eight triangular facets formed by the ends of any three struts spiralling around each other. Four have a clockwise rotation, four rotate counterclockwise. Tensegrities are dynamically stable partly because these rotations absorb energy by acting like springs which oscillate in and out to restore balance. This led me to investigating the nature of centripetal forces whereby a continuous force acting upon an object compels it to alter its motion and changes its acceleration. Gravity is a centripetal force that describes the relationship between discrete bits of matter and causes them to fall into elliptical orbits around each other. Newton’s three laws of motion describe this very well. It occurred to me that the tensional envelope that bounds a tensegrity acts like a centripetal force field exactly like gravity. Objects in orbit around larger objects tend to find stable configurations which can last for a very long time presuming other forces like friction are absent or minimal. I wondered if tensegrities could be looked at (at least metaphorically) through the lens of orbital mechanics.

The inertia of objects (mislabeled centrifugal force) is constrained by centripetal forces such as gravity or in the case of tensegrities, tensional nets. Given that centripetal force involves constraining rigid bodies to follow curved paths, one way to look at a group of compression components is to see them as multiple instances of a single object in rotation and each strut then becomes a freeze frame of the rotation path of a hyper object. A tensegrity prism is a frozen three dimensional drawing of a four dimensional rotating object (see images above). Each strut and its orientation is an aspect of a more complex object that is moving in space and time. A lot of implications fall out of this assumption and I’m still working them out.

**Tensegrities are self-referential, disclosing how all the forces acting on them are arranged**

(Nov 15, 2015, continued) Tensegrities are the only structures I’m aware of that disclose how all the forces acting on them are arranged at any given time. In that sense they are meta objects – self referential. In orbital mechanics a higher orbit takes more energy to achieve, it is more sedate and has more stability. This might be equivalent to a tensegrity sphere with a large number of struts. There is lots of redundancy built in. In contrast a low orbit is faster and takes less energy to achieve. There is less redundancy which is less room for error. A 3 strut tensegrity prism will collapse if any one member is removed, not so a more complex one.

### Tension forces in a tensegrity create prestress that is a type of artificial gravity

(Feb 19, 2016) I think I had a real insight when I invoked orbital mechanics and centripetal force to describe how we as living beings contain our own artificial gravity field. … While I like praise as much as the next human, I’m really more interested in having a discussion with like minded folks – riffing on ideas and seeing where it leads. It’s like fishing – I feel like I keep throwing out leads and getting few bites back.

(How Tensegrity Models Reality, April 2018) 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. Newton’s 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 constrains 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 its own essential force field, doesn’t collapse and can operate outside a gravity field.