Sunday 20 May 2012

Tensegrity [i].

(... for he was a tidy pachyderm.)
Up, Down, References, Addenda.

What this is:

Hiroshi Watanabe, Arbolito Park Quito, Ecuador.Thought experiment: Put some toothpicks in a baloon with some white glue, blow it up, shake it around so the toothpicks stick to the inside surface of the baloon, let out the air. What happens?

The 'Love Knot' tensegrity, which takes a pentagon (sometimes thought of as a symbol of humanity) into another pentagon and whose fifteen struts form one continuous circuit ... has fascinated me for many years. Recently I was dreaming of a model formed of curved chrome tubes joined with tiny chrome springs. Maybe I will get around to it ... and maybe not. This post is just to document some of the initial design stages (on the umpteenth reiteration).

[Dedicated to two sisters, Phoebe and Tanaquil (and to their father who had passed before I got a chance to meet him).]

          Triangular Prism with 1 layer
          Modified Triangular Prism (O Minimo)
Capturing the imagination:
          Expanded Octahedron
          Truncated Tetrahedron
Odd-numbered Prisms (with continuous circuits):
          Pentagonal Prism with 2 layers
          Pentagonal Prism with 3 layers (Love Knot)
          Heptagonal Prism with 2 layers (7 sided)
          Hendecagonal Prism with 3 layers (Loonie) (11 sided)

Triangular Prism with 1 layer:

Triangular Prism.Triangular Prism.Triangular Prism.

There is a 'twist angle', a rotational displacement in the two polygons at the ends of the prism - which can be computed - the arithmetic involved is described near the end of 'Geodesic Math and How To Use It'.

Modified Triangular Prism (O Minimo):

O Minimo - looking along the axis.O Minimo view.O Minimo view.O Minimo view.

A triangular prism with an additional strut through the centre - four struts held by nine tendons, or 2¼ tendons supporting each strut. Consider the minimum number of guy-wires necessary to hold up a pole ...

Expanded Octahedron:

Expanded Octahedron facing diamond.Expanded Octahedron triangular face.Expanded Octahedron view.Expanded Octahedron view.

You can also think of it as a 2 layer triangular prism.

There is a transformation that takes this tensegrity into the truncated tetrahedron. I had hoped to do a bit of an animation ... but I don't know how, so words will have to do - there is a description in 'Introduction to Tensegrity'.

The divine proportion turns into 1:2 in the process.

Truncated Tetrahedron:

Triangular Face.Hexagonal Face.Edge.


The only model to survive all of the travels and upheavals is this truncated tetrahedron. It lived on the dashboard of my van for some years, got banged around, hung in there somehow. Pugh calls this configuration the 'zig zag' - got that right.

Truncated Tetrahedron.Truncated Tetrahedron.Truncated Tetrahedron.Truncated Tetrahedron.Truncated Tetrahedron.

Pentagonal Prism with 2 layers:

Pentagonal Prism with 2 Layers.Pentagonal Prism with 2 Layers.Pentagonal Prism with 2 Layers.Pentagonal Prism with 2 Layers.

Pentagonal Prism with 3 layers (Love Knot):

Pentagonal Prism with 3 layers - Love Knot.Pentagonal Prism with 3 layers - Love Knot.Pentagonal Prism with 3 layers - Love Knot.Pentagonal Prism with 3 layers - Love Knot.

The notion took root in the late 70's - from a single comment in 'Introduction to Tensegrity': "The struts form one continuous circuit." It turns out there is a family of such structures. Knowing how to tie a Turk's Head led to experiments with rope as well, though it was difficult to join the end to the beginning neatly.

The atomic number of (good old polymorphic) Phosphorus is 15. The Phosphorus molecule P4 (in the white forms) is a tetrahedron and in red is approximately so. I am working at determining what the standard Phosphorus allotropes look like: α is body centered cubic, and β is triclinic - but of course the Internet is not that useful at this level.

Hanging in the window of an apartment in Ipanema is the beginning, in 2002, of a love knot to be made with (possibly) curved & coloured florescent neon tubes - the ballast concealed in clever hubs - for the centre of the arch at Canecão. Didn't get very far (distracted by oil) ... Nor in 2005 with soda straws - a single elastic band serving as the four tendons of each diamond.

2002 Love Knot.2005 Love Knot.2005 Love Knot.

Strut detail.Strut detail.
No struts to be had in Toronto, so it goes. The carpenters are all too busy except one guy who says, "Sorry, not keen to do this," duh. Oh well. ... The only problem with soda straw models is that the elastics soon turn to dust. Ai ai ai.

Prisms with odd numbers of layers tend to be pretty close to actual prisms (where the end polygons align) - with even numbers of layers they are closer to anti-prisms (where the end polygons are rotated half an edge).

Love Knot.Love Knot.Love Knot.

Love Knot - diamond detail.Love Knot - diamond detail.Love Knot - diamond detail.

Love Knot.Love Knot.Love Knot.

Being odd, the circuit pattern love knot is close to a prism, and joining the ends of the struts makes it just about exactly so. This will simplify the metrics since it will not be difficult to compute coordinates for all the joints, and hence angles, radii &etc. The difficulty will be finding a 3D modeller to hold it - AutoCad I guess. Bleah! Expensive and hard to use ... maybe there is something else out there by now. (?)

When the diamond becomes the chord of a curve ... my son quickly saw that it will be not be the chord of a circle but will be twisted - have to watch for that capability in the modelling software.

A tool that would compute a minimal static system is probably too much to ask. (?)

Back in the 70's we stuck a tensegrity into Stardyne and it just went berserkers - infinite displacements with zero loads and so on :-) The smart guys I worked with in those days have cashed out and vanished - and I am wishing I had done the same.

143 mm,107,81,57 - 1:.75:.57:.42 (Happy Birthday Babe!):
Flattened love knot tensegrity.Flattened love knot tensegrity.Flattened love knot tensegrity.

Heptagonal Prism with 2 layers: (7 sided)

Heptagonal Prism with 2 layers.Heptagonal Prism with 2 layers.Heptagonal Prism with 2 layers.Heptagonal Prism with 2 layers.

If you look closely at that last image you can see squares emerging from the murk.

Hendecagonal Prism with 3 layers (Loonie): (11 sided)

11-Prism with three layers - Loonie.11-Prism with three layers - Loonie.11-Prism with three layers - Loonie.

If it looks somewhat cone shaped it is because I was shrinking one of the 11-gons to see what would happen - more on that later.

One thing you can see in this model is how the ends of the struts come closer together as the frequency increases.


Just after my son was born (and a few weeks after I started architecture school) a mathematician friend of mine told me about a figure constructed of three intersecting rectangles. If the sides of the rectangles were in the divine proportion the corners would form a regular icosahedron. So I made one with paper and sticky tape, connected the corners with a needle and thread - sure enough, there was the icosahedron. I hung it up high in the corner of the room.

In the other corner of the room was the bed. I used to lie there sometimes and doze with my new son on my chest, pretending to match our heartbeats as he fell asleep; and one day I thought I saw another figure there - one with the long edges of the rectangles as sticks - and when I made that one (with chop sticks and string and clove hitches) I found that six of the strings seemed to be unnecesssary. (?) So I replaced the strings with rubber bands, removed the redundant ones, and ... Presto Whiffo! There was the Expanded Octahedron. Eureka!

When I ran to show my prof the next day, he gave a glance, said, "Yup, the six-strut tensegrity," and carried on past me down the stairs. ... Someone had been there first. ... Euclid maybe - not Kenneth Snelson though some pawits think so (and definitely not our Bucky).

 :-)And that was about as close as I ever got to an original idea (and a good time to abandon jealousy as well, envy, property - intellectual and otherwise ... other things).

Some thirty years later, as we sat in a sidewalk barzinho a little south of the Tropic of Capricorn one evening (that same son of mine and I), we decided to solve the pesky climate change crisis once and for all.

How to get maximum bang for a policy buck?

Juliana Paes for Antartica (by Ambev).After many Antartica's, first prize went to establishing no-car zones a dozen or so miles in diameter around all substantial city centres. Success would depend upon citizen satisfaction & uptake - so: secure parking at the perimeter, effective public transit within the zone, and set the planners to re-allocate the streetspace.

Runner-up was diverting and separating urban sewage at neighbourhood scale to generate methane for fuel, street lights and the like; and fertilizer and water for urban farming (possibly in those re-planned streets).

Which brings me to an upcoming event here in Toronto - hopefully worth the fifty buck ticket: An Evening with the Pembina Institute: Help “fuel” transportation progress, Wednesday 30 May 2012 from 5:30-8:00 PM - more about the Pembina Institute on their website.

It would be fine to see you there gentle reader. Be well.

Kenneth Snelson, and here.
Geodesic Math and How To Use It, Hugh Kenner, 1976.
Introduction to Tensegrity, Anthony Pugh, 1976.

Building Models:

In the event that you decide to make some of these models, a word of caution: WEAR SAFETY GLASSES ... because, as you assemble models with elastic bands they will inevitably slip from time to time and the struts will suddenly come flying at you, and if you are not wearing protection you may put one of your beautiful eyes out.

Then again, in all the years I have been making these things I only began to wear glasses, nevermind safety lenses, when I could no longer see to do the work without them, so ... fatuous phatic flatulence (aka ignis fatuus) with phallic undertones.

A Note on Tools:
If someone offers you an Olympus Stylus pocket camera - JUST SAY NO! ... On the other hand, there is no way to appreciate these structures through photographs, you have to make and handle them (like it says in the communion hymn, "touch and handle things unseen").

Brass wire is not ideal - it is very difficult to get it tight enough to take out the kinks - but for some reason elastic bands these days deteriorate quite quickly, in this apartment at least. Someone said to me that it is my smoking - that could be - or maybe a slow Freon leak in the fridge.

Do not underestimate simplicity - making the struts for example: putting the finishing nails into the ends requires a punch and a drill-press. Really (I had these ones made for me and it was too complicated to explain this step to the cabinet maker), it is better to put the blanks into a pencil sharpener after the holes are drilled and before the nails are driven ... &etc. (Thanks again Glen.)

Addenda & Ruminations: I ran myself out of struts and now I'm stuck finding someone to make some more, and it's Toronto so ... an hiatus ...for the best maybe. My son came and made some space to hang up the models. Trying to visualize the love knot as an actual knot-knot is difficult, frustrating even, and being able to hang the model just above the screen of this computer makes it easier to look at. These kinds of things used to just leap into my head - now they sort of stumble & crouch :-)

Eschenbach and his Parzifal/Parzival/Percival/Percy is on my mind. I remember the first time I read it and visions came of my young wife's gracious innocence ... a cynic might call it a 'sexual connection' with the text, and there is some truth in that - but not the whole truth.

The point on which the story really hangs, sticks, and persists, is Percy's question, his two last chances at finding the grail: "What ails thee?" And the huge dramatic inertia that surrounds not asking the question, and then maybe not asking it again, and then, finally, asking it - and all the lights come on.

You could easily reformulate it as simply: How are you? Such a ubiquitous phrase and so often repeated that it might (going back to the top of the post) be merely phatic. Surely it must have been about the same in the 13th century? People greeting each other with "Hi. How's it goin'?"   "Good day and how are you?"

So ... how to imagine that the warp-speed cosmic worm hole leading to ultimate spiritual fulfilment is nothing more than one of the commonest of human exchanges? (?)

Going along the Queen Street East sidewalk on a sunny Sunday afternoon in May, crowded with all the good burghers walking their children and dogs, each other, and lots of brown girls with white babies in the most expensive strollers available. What if Percy in his armour and the glistening grail are right here in the middle of it?

There is an implicit irony in the movies and TV programs when they portray religious rituals these days, or there seems to be. Priests and rabbis and the like, repeating prayers; choirs of schoolboys singing well worn hymns, the chestnuts that we all remember, or at least that those of my generation remember: "Bring me my chariot of fire."   "All ye who hear, now to his temple draw near."

What if all the sanctimony and doctrine and holy writ (and the irony as well), however "ful of hy senténce" it may be, is ... fol-de-rol and unnecessary nonsense, and it, the shekinah, is all right here in all of our hands all of the time - as I have quoted from Northrop Frye so often: "In the double vision of a spiritual and a physical world simultaneously present ... our life in the resurrection is already here, and waiting to be recognized." [Slightly changed from the original which is here.]

Back in the day there was a gig at Place des Artes, a Jazz Festival - Murray was doing some kind of radio reportage and I was running him around on my motorcycle - and somehow I found myself 'in the wings' with this remarkable noise.

Many years later when Cynthia the shrink thought she had me pinned I would pull out something like "Love is not love which alters when it alteration finds" and she would mutter in frustration, "Damned transcendence!" Eventually she kicked me out. :-)

There he is, our Pierrot le Fou on the hill in his red-yellow-and-blue 'nitramite' outfit, also muttering ... and then the last whispered (transcendental?) exchange, it might be: "We found it again. What? Eternity. That's the sea moving, with the sunlight."

Another dipthong or two, logorrhœa, prolixity:

1. The mid-year round of UNFCCC gum flapping is on at the Marlim hotel in Bonn. Climate Action Network International puts out a daily newsletter. Reading it carefully is an exercise, almost an archæological dig, into layers upon layers of detritus ... psychopathy. My heart goes out to the editor, Kyle Gracey.

He muses on what 'deep concern' might mean (in a blizzard of non-meaning which he does not, probably cannot, acknowledge): Quantified Emission Limitation and Reduction Objective (QELRO), Adaptation Fund (AF), Board of the Adaptation Fund (AFB), Certified Emission Reductions (CERs), Emissions Trading Scheme (ETS), 'key demanders', 'accreditation process', Nationally Appropriate Mitigation Actions (NAMAs), Measuring Reporting and Verification (MRV), Mechanism for Monitoring and Reporting (MMR) - all buffered with fluffy-puff collectible fast facts.

What a joke. Fuct! But you have to do something, so I read carefully, look up the acronyms, try to understand, comiserate ... He could write about what the AF budget looks like - how many additional bureaucrat maggots are suckling in that sector of the trough.

The best news is that Saudi Arabia is making renewable noises - that could work - setting an excellent example for tar sand billionaires in, say, k-k-Canada. (Eh?)

Russia - Dmitri Medvedev, Japan - Yoshihiko Noda, Canada - Stephen Harper, France - Francois Hollande, U.S. - Barack Obama, Germany - Angela Merkel, Britain - David Cameron, Italy - Mario Monti.2. The global economic foofaraw goes on and on. Individuals suffer greatly but we focus on the aggregate, easier that way. The orthodox liberal views seem to cluster around: "... austerity ... is putting the already-weak recovery in the United States at risk and is fueling instability and extremism in Europe," "a euro-zone growth package," "stimulus to spur growth, employment and development," and "a program to promote growth across recession-racked Europe" (from a NYT Editorial and here). Just how did 'austerity' and 'growth' get into the same category? How did they become something like opposites? And then there is Paul Krugman alternately promoting growth and wringing his Apocalypse Fairly Soon bells like a knee-socks altar boy at mass.

3. Invisible Children send me an email titled 'LRA Top Commander Captured Alive', but it's Caesar Achellam not Joseph Kony. Their transparent self-promoting rhetoric just reminds me of the way they sold out Jason Russell - another facet sparkling on their fundamentalist jewellery. The package with the t-shirt and posters is still unopened on my hall table with the bushels of paper flyers from Ikea & Loblaws & tanning salons and all the rest. Bound for use as absorvente at the bottom of a garbage bag.

Dan Wasserman at Austerity is certainly an engine of extremism. The raw (uncooked that is) fuel of the disenfranchised & disentitled - AND - the innate knowledge of waste, incompetence, squandering, stupidity (the 'official' kind).

Ideas (like the one at the right) work, exercise 'traction', because everyone knows what the trough is - government does need a haircut, a para-dig-em shift, a bowel movement.

5. Almost no one speaks to me so when they do, I listen and consider. Someone says, "The alienation that you feel is directly attributed to your attitude towards your own life," and, "It doesn't turn me off to hear how you have given up." I am deeply moved that someone has spoken to me ... but I am soon wondering who has given up on whom? Knowing (imagining?) that amidst total confusion a mirror reflection, 180° backwards, may look like an explanation, an 'answer'.

We make principles of our incapacities and carry on. (Eh?)


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