Exploring other things...

I guess I'll need some more practise in producing videos, there's still some unclear instructions and a couple of hang-ups in it. I still hope for decent light conditions in my improvised video studio to shoot the how-tos for icosahedron and dodecahedron, with the potential to redo the intro part as well. However, this video gives you an impression how fast models can be build: Only seven minutes show what's happening between the first connection and last connections being made, without time lapse.

Getting wild

I received the elastic cord I wanted to use for The Affordable Tensegrity Toolkit, and prepared the first 30 stick prototype with it. The diameter of the cord fits nicely to the groove width, it wedges in and form a stable connection (within limits).

I build first a 30-strut icosa with it, and was amazed about the bounce the final structure had. Instead of using a structure as template of the build, I had a generic weaving pattern in mind, following two simple rules. Once finished, I played with the icosa like a football, producing some domino effects with other structures.

30 strut icosahedron

The next test consisted of timing the transformation from icosahedron into dodecahedron. That meant disassembling the icosa completely, and reusing the components in a different pattern. Again, I navigated through the build by its pattern, creating triangular corners around pentagonal faces. The structure warped itself in shape already while completing the third of twelve pentagons, and after eight minutes the transformation was complete.

I threw the dodecahedron quite lot around, which opened sometimes a corner. Playing it hard goes the limits of the attachment technique. This time I decided to time the disassembly by itself, less than two minutes to undo the sixty connections.

30 strut dodecahedron

As expected, building the 6 strut tetrahedron proved most difficult, but cube and octahedron provided a fast, straight forward build. In a room without other sculptures, I started throwing the cube and octahedron quite hard against the wall. At some point, a tendon in the octahedron snapped, though I wasn't sure whether it was the impact or the way I held it before throwing.

After I replaced the tendon, I continued to bounce the models madly from wall to wall. This time I took care of holding the model mainly at the struts. I guess I limited the vigour I used for my experiments, although I used enough force to hear the tendons swishing during flight. Anyway, no more breakage occurred. The octahedron can safely be used for throwing games and bounced off walls. With all the fun I had finding out the stability limits by relatively brutal force, I look forward to more swishing, clicking and hitting sounds while doing some stress testing for the tensegrity toolkit.

30 struts in three different models

You can reconfigure the model easily. Each single cord gets used as three tendons, two for the corner and one for the connection between corners. While building a model, aiming for similar length makes building easiest. Of course, as there are no markers each connection has to be guesstimated. When I played with different configuration of cube and octahedron, I noticed the dual quality. As two struts connect to each cord, you can place them very close together. The model can't collapse any more, yet seems more robust when thrown around.

Effectively, the total number of tendons reduces from 36 to 24. I'm not certain whether the proximity of the struts converts the 'missing' tendon into a kind of joint, however, by ignoring this tendon the remaining 24 tendons outline a cuboctahedron, the intersection between cube and octahedron. Both physical models look and behave similar in this configuration. By moving the struts together, they shaped four entwined triangles, like faces of a tetrahedron twisted inside and around. Reminds me of the jitterbug transformation, so I don't think I discovered something 'new', just new for me.

Four intertwined triangles in a 12strut pseudo cuboctahedron

Intermezzo

I think the tetrahedron represent the number 2, the basic duality in universe. It contains as well the number 3. I see more three-ness in the 6 faces of a cube and the 6 vertices of an octahedron, the 2by2-ness appears in 4 edges constituting a face (cube) or converging into an edge (octahedron).  Somehow, five-ness appears in the shapes observable. From a specific perspective, pentagonal outlines appear, all the while of hexagram and pentagram can be inscribed to some struts. Is there already the five-ness of the icosahedron in cube and octahedron?

In the 'orthogonal' cube, eliminating the 'middle' tendon doesn't create entwined triangles, yet brings two struts together along their length. The closer I moved the parallel struts together, the more familiar the structure appeared: it's a kind of 12-strut icosahedron.

Orthogonal cube morphed into 12-strut icosahedron

The new cord material requires a bit more work to prepare the toolkit elements, but so far looks extremely promising to combine easy build methods with lasting tendons.

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.

Cube

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.

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.