Wednesday, 20 April 2011

And that's TATT!

Different times have different toys, different ways to explore constructive creativity. Mechano or Lego come to mind to name some of those amazing influences on the development of creative minds around the globe.

Lego provide the 'atomic' toolkit, solid pieces of matter, stacked together. The Affordable Tensegrity Toolkit brings us closer to the unpredictable nature of quantum mechanics. The regular structure come with a twist, or a wobble, a bit of surprise based on very simple rules of construction. You build atoms, which basically consist of plenty of empty space. The empty space and the lightness of 'solid parts of matter' become apparent in a finished tensegrity structure.

Two tetrahedra with opposing chirality
With six struts, you can build the first platonic solid, the tetrahedron.

Trigonal dypyramid 
With nine struts, a structure with surprising properties emerges. Three tensegrity prisms (or tensuls) stacked on each other form the trigonal dypyramid.

Flattened dypiramid
This structure flattens under pressure and bounces back happily.

Trigonal Prism

The trigonal prism offers less excitement, due to lack of symmetry across its corners. It squeezes down, but doesn't bounce back too spectacular.

The cube can be build in two ways. Above you see a cube where all the corners rotate in the same direction. The faces appear square, yet while lying on a face the vertical struts 'lean' to the side. With mixed chirality, the struts cross each other orthogonally, yet each face looks rectangular.

Two stacked orthogonal cubes
Of course, it takes 24 elements to stack to cubes together.

Octahedron (view onto triangular face)
Yet it just take 12 struts for the third platonic solid, the octahedron. Its symmetry in combination with elastic tendons provides lots of bounce.

Octahedron (corner view)
Pushing the corner towards each other flattens the model.

Cuboctahedron (view on triangular face)
With 24 struts you can build the cuboctahedron, or Vector Equilibrium. This structure shows the transition from cube to octahedron, and has thus six square and eight trigonal faces.

Cuboctahedron (view on square face)
All of that (and things I haven't thought of) can be constructed with a maximum of 24 toolkit elements. The remaining two platonic solids, icosahedron and dodecahedron, require 30 struts. The simple joining methods allows it to put smaller models together (like the two stacked cubes) to create larger ones.

How to build an octahedron with the Affordable Tensegrity toolkit

You can build an octahedron (eight triangular faces, twelve edges and six corners joining four edges each) out of twelve identical elements. The cord gives the strut an orientation, a back and front. The knotted end points in clockwise direction when viewed from the front.

Single toolkit element
This orientation determines the twist of the tension element, and helps following simple rules during the build phase. Here goes the first one:
Tendons go along the front of the strut, which means the 'outside' of the finished structure.

The first connection
First, an element connects to the tendon of another. The distance is about one third of the total tendon, for simplicity I call this the 'short end'. Both struts lie on their 'back', the knot in the connection points towards the short end, the knot in the short end points clockwise.

Continuing the pattern
The third strut repeats the same idea: The knot points towards the short end, the short end of the newly connected strut rests upon the strut it threaded in. The next rule becomes apparent:
The knots point towards the end of the strut, not the center.

Four struts form one 'corner'
With the fourth strut one corner of the octahedron is finished. All knots should now point towards the short end of the next strut in the square, and the struts intersect in clockwise direction like in the photo.

Two corners
Preparing top and bottom corner makes the final assembly easier. Simply connect four more struts exactly the way you did before. Put it aside until later

The next vital connection
From now on, things get more three-dimensional and require a bit confidence that everything holds together. Each of the tendons connects to two other struts in the final structure (hence short and long end). The next four struts connect to the long ends, and introduce the second rule of building tensegrities with TATT:
When viewed from the front, the two struts connecting to the tendon of a toolkit element, arrive from opposing sides.

The pattern for the second stage
The long end of the new connection connects to the nearest strut of the corner.

Two struts of the second level
The next strut follows the same idea: Connecting to the open long end of a corner strut, having the long end connected to the next corner strut.

Three struts of the second level
Connect the third strut, remember that tendons go outside, the knots point to the short end, struts connecting the same tendon come from opposite directions.

Eight struts of the octahedron connected
After connecting the second level of the octahedron, the structure slightly bends itself into shape. To get the final shape, more bending needs to be done for the final shape. Eight unconnected ends and eight spaces in tendons remain for the last few connections to be made.

Corner with second level turned around next to top corner 
After turning the eight stick module around, the top corner finishes the build.

First connection of the top corner
The first connection comes easy.

Second connection of the top corner
The second connection bends the model into a skewed shape, it follows the same pattern as before.

Three connection of the top corner
The tension increases while the connections aren't balanced, yet the model get more bounciness and stability during the final steps.

Model with four missing connections
The open ends of the top corner now connect to the rest of the structure, with four attachment spots remaining. The four remaining open ends (from the eight struts of the first build phase) connect into this open spots.

Three missing connections 
The same rules as before apply. The knots point towards the short ends, struts connect from opposing directions to a shared tendon.
Two missing connections
Due to the increased tension, unsecured connection might easily slip during this phase.

One step away from finishing
The increased tension makes building a bit trickier. At the same time, the tension guides you towards making the 'right connections'.

Tensegrity octahedron balancing on a corner
Once we last connection is made, you can test the symmetry by balancing the model on each of its six corners
Flattened model
The elasticity of the cords allows the model to squeeze down, the size of the squared loops in the corners determines to flat the model can get. You can adjust the model by reattaching one stick at a time in a more symmetrical way.

Don't take the rules for the build as eternal truth, for other models other rules (although similar) apply. There's more than one way of building any tensegrity structure, only experimentation can improve any construction method.

The Affordable Tensegrity Toolkit just has hatched and needs now good documentation. Please contact me via this blog if you're interested in more details, or have specific requests or comments.

Saturday, 16 April 2011

Toolkit epiphany

Tomorrow, I'll be on the market again. As I didn't have a sale last time, I had no real need to produce more. However, I prepared masses of bamboo struts, mainly thought for octahedra. I used one tendon per strut successfully for spherical models (24 or more struts), yet used a variety of tendon strategies in smaller models.

In a Class 1 tensegrity, the tendons connect to a continuous network. Ideally, a structure deploys (at least) three separate tendons from each 'knot' to a compression member. A six-strut icosahedron would need 24 tendons, yet a continuos tendon can shape one as well (but that's another story...).

A six-strut tetrahedron still has 18 tendons, which means a 3:1 ratio of tendons to strut (at least three tendons, 18 must be the smaller number of connections between 12 points). I struggled a lot building those, using 4 triangular loops and 6 tendons. Maximising tendons proved well for large structures with few compression elements, but smaller models can be build more economical.

Tensegrities derived from regular geometrical shapes 'cut' the corners off, with as many struts joining as the number of edges joining in a corner. Three edges join the corner of a tetrahedron, hence the corner is opened into a triangle. Having the four corners equally sized balances the structure well, and, to simplify matters more, only six tendons remain for final tuning of the model.

Using 10 instead of 18 hypothetical tendons make life already easier, but somehow I insisted to complicate my tensegrity build experiments more than necessary so far. Besides the x-module, the tensul and the six-strut icosahedron, all symmetrical shapes I encountered only need one tendon per strut, weaving another to either end to complete three tendons per joint.

Toolkit variation with 6, 9 and 12 struts

After revisiting the java app at xozzox I wanted to test the versatility of a single elastic tendon per strut approach by building a 'trigonal dipyramid'. Basically, it's a 3-stage tensul tower, yet with fewer connections than I build in larger scale so far. It has 5 corners, 2 triangular and 3 squared, and squishes down nicely when pushed along the triangular corners.

I went to build the trigonal prism using similar components, and finished much faster than the still quite laborious first go. Can I really build all platonic solids (tetrahedron, cube, octahedron, icosahedron and dodecahedron) with the simple one tendon per strut strategy?

One approach to convert a platonic solid into a tensegrity structure transforms each edge into a compression element, and the corners into tension loops. Following this definition, I already made an icosahedron and a dodecahedron, using one tendon for each of the thirty struts required. (The six strut icosahedron connects two of its twelve corners, using a 'shortcut' through the centre of the structure)

After studying some of my tetrahedral models I concluded that it can be done, and again got stunned by the simplicity of the process. Once I figured out the 'weaving' pattern, I just needed some trust in the stability of these structures, and everything came together easily.

Building a cube and an octahedron as final proof of concept happened after some side explorations. I used physical models as template, and re-used the components of prior experiments.

Toolkit top view
Above you see 48 identical struts, connected with 48 tendons, comprising five different structures with a maximum of twelve and a minimum of six compression elements. You need 24 struts for a vector equilibrium, 30 for icosahedron and dodecahedron, 90 for a Fullerene.

When I did the last bit of prototyping, I used the same measures as for structures with more compression elements. I doubt it would hold up spheres with more than 48 struts, yet it works great for the platonic solids, combinations thereof, and seemingly many other composite structures. Assembling any of the above models (even with some additional stability measures and tuning) took much less than 30 minutes.

Using single tendons can confuse easily, but following simple rules and pattern make it easy to build all platonic solids with just 30 identical elements. The elastic cord offers enough tension for quite a high number of elements (towers still tend to be floppy), and hooking/unhooking the tendons took little effort. The models end up in a size desktop suitable, I have to see how the market reacts tomorrow.

Thursday, 14 April 2011

Frog in the well

Frog in the well
I went to the local hardware store to find a small table saw. It could make preparing wooden struts much faster and even more precise than the Dremel, but I'll have to wait to find out. Instead, I brought a 3m aluminium tube home and got busy.

The first hole I drilled was too wide to hold a knotted nylon string, yet I decided to worry about this fact later and prepared three alu struts for a prism, each with three equidistant holes about 2cm from the end. Doubling the tendon should provide additional strength, and a knot of two strings of nylon held safely in the hole.

I used the measures taking from a bamboo strut structure with nearly 1m struts. I adjusted the final tension by pulling the vertical tendons overly tight, and then knotting off between 2 and 5 cm of it. I ended up with the box shape I wanted, with decent tautness. I did a stress test by putting a zabuton on top, and it took the load nicely.

I pushed the end of the strings back, and riveted the end of the tubes. The construction details hidden, not perfectly but nicely balanced, evenly spaced holes made it a satisfying piece by itself. Yet it invited me to do more, to use it as base for a more complex tensegrity.

After installing two bamboo tensuls into the top triangle, and playing around with spheres on top of them, I build a 30-strut icosahedron with nylon strings. I chose a strut length I didn't use before, and estimated the string length required, with nasty consequences. The overall tension split one after another strut, as could drops from just one meter height.

The drop resistance might not have improved, yet in its current installation the structure is unlikely to be exposed to excess stresses. The sphere now rests on a single smaller tensul, with a 3:1 ratio of strut length from basis to middle part, and a 3:1 ratio of strut length and upper triangle of the support for the sphere.

Instead of a tension triangle, the base of the support prism get its tension from three connected nylon string loops, connected to the lower end of the struts. A small Chinese bell with a frog sitting on it is suspended from the center.

Now all I need are three columns to elevate the model some more. The aluminium struts were easier to work with than I thought, and invite bigger structures.

Sunday, 10 April 2011

Class 2 tetrahedron

Tensegrity structures still wait for more popularity. While my fellow market traders got used to them, and spend some time playing with them, it's still rather one in hundred passers-by that identifies my work as tensegrity.

I shied away from investigating tensegrity theory since I started building sculptures, but I renewed my research lately with some surprising findings. I realised that I reinvented the wheel - the Unholy Grail uses the structure of Bob Burkhard's Wheel 2. When I browsed Bob's great site again I stumbled upon a Class 2 tensegrity tetrahedron.

I still have some models using eyebolts available for recycling, so I considered rebuilding a tensegrity structure with joints. I threaded the eyebolts of two struts together, which creates a multi-directional joint. A class 1 tetrahedral tensegrity needs 6 struts, and a bit of imagination to detect the tetrahedral shape. The hinged class 2 tetrahedron only requires 4 struts, and seven tendons.

I had no idea about the precise tendon length, nor how the joints would affect the build process. I started with elastic tendons for the edges, and a fixed tendon between the joints. The nightmare began. I hoped that the elastic cord would allow me to 'stretch' the model into a stable position, but the jointed struts kept turning and unhinging some outer tendons. The mobility of my improvised joints backfired, and after some variations of central tendon length, outer tendon length and order of attaching outer tendons I gave up.

Unlike many nicely rendered tensegrity structures one can find on the web, Bob Burkhard showed two photos of actual models showing this class 2 tensegrity. Knowing for sure that this idea can be build, using quite familiar connection types, I reflected on my difficulties during the failed attempts and devised a new strategy.

I used nylon cords with little bowline knots at either end - this should limit slippage of the outer tendons, and give equal length. Even with fewer components than most class 1 models, this build remained challenging. At first, I used a metal hook as connection between joints, with little luck. Then I limited the mobility of the joints by tying elastic cord around it several times, replacing the hook as central tendon.

After several attempts all outer tendons got connected, and shaped a tetrahedron. I didn't let go of the struts, the model didn't feel self-sustaining yet. The elastic cord made it easy to shorten the central tendon, and gave the model stability. It still collapsed, and sometimes outer tendons became loose when I played with it. It still takes me patience to pop it back into a stable 3d state after a collapse. I closed the eyebolts, so that the tendons stay in place.

The final result stunned me. The outer tendons clearly delineate a tetrahedron, and two pairs of joined struts, held together by a short central tendon, connect the corners to its central area. The joined struts give the model optically more substance, and the behaviour provided pure fun. The model balances on a triangular face, so one strut always points up. If you push back this strut, the tendon connecting it to its joint member slacks off. Once you release it, it springs forward, easily with enough momentum to tilt the model over.

If I use joints again, I make sure I use a hinge joint for the tetrahedron. Two joined eyebolts offer too much freedom of movement, which might contribute to a collapse as well. The structure feels different from most class 1 tensegrities I build, and show a surprising dynamic movement under little external stress.