Seven seven seven

I build already a few versions of (truncated) chestahedra, one of them most likely still on display in Queensland, the rest of them embellishing my home. My "quick and dirty" way of transforming a geometric structure into a tensegrity basically cuts off the corners, with as many struts as the geometry had edges. The network of strings therefor doesn't properly reflect the original geometry, with the exception of the 4-strut tetrahedron.

4 strut tensegrity tetrahedron

6 strut (truncated) tetrahedral tensegrity

Truncation basically produces the "dual" of a Platonic solid. Cutting the corners of a cube creates the octahedron, cutting the corners of an octahedron brings back the cube. The number of faces becomes the number of vertices, while the number of edges remains the same. However, this beautiful relation does only really exist between hexahedron (cube) and octahedron, and between dodecahedron and icosahedron. Applying the same algorithm to non-Platonic solids creates still very interesting transformations.

The chestahedron, which can into human consciousness just very recently, also has a dual, the decatria. I'm surprised that it took me two years from finding out about the chestahedron to learn about its dual, which still is more than a mystery to me. I know it has 13 faces, 19 corners and 30 edges, mostly likely three different kind of faces. I still struggle to understand the 2d images I saw so far, how many different edge length are involved, so I delayed the ambition to "tensegrify" the decatria.

I got inspired, however, to build a chestahedron similar to the 4 strut tetrahedron, using the tension elements to outline the wireframe version of its geometry. I struggled a lot when tried this for the octahedron, failed completely for the cube so far. The six strut icosahedron doesn't need the additional six strings to unveil it's "true" geometry, my ten strut dodecahedron usually ends up slightly imperfect, with most pentagons not being really symmetric. 

10 strut dodecahedron

6 strut icosahedron

I ruminated a lot before getting hands on, using my experiences of building asymmetric structures to have a plan which made sense to me. I got frustrated on earlier attempts to construct things which seemed initially possible, but then turned out quite different. The idea to use seven struts for building a seven-sided object with seven corners kept me going. As the chestahedron has an unfolded tetrahedron at its base, the first tensegrity shape conceived in modern times might provide a great starting point.

This constellation was build in the 1920s before the term "tensegrity" was coined.

My first attempt followed my intuition. I chose three different length for the struts: 30cm for the base, 20cm for the vertical riser, and 15 cm for the middle section. The length for the outer tension network were simple, using the edge length relations Frank Chester published for the chestahedron. 9 strings were knotted to 30 cm, 3 more to 16cm, for a 0.53 ratio between top and base edge. In the truncated version, the top seemed to sink a bit in, distorting the beautiful relation of the solid object.

Healing heart (made of yarrow with suspended copper wire spiral) 

I started off as minimal as possible, connecting the three base struts with a string loop which served to received the three shorter struts for the middle section as well. The central riser was supposed to connect to the outer string network, and three pieces of elastic string connected to the top end of the base struts.

It was relatively straight forward to get everything together. All I needed to do was to connect the bottom of the vertical riser to the top of the base struts, creating an expansion from bottom to top through the inside which should be limited by the tension network on the outside. It got a bit fiddly, all seven sticks come together fairly close in the centre, but there were only two connections to go.... and then everything fell apart in a tangle of sticks and strings.

So decided to use some transparent elastic string to stabilise the base, making a classic nine string, three strut tensul out of it. It still took some dexterity to finish it, yet this the little deviation from making it as minimal as possible provided a satisfying prove of the concept which emerged less than 24 hours in my mind.

Very first seven strut tensegrity chestahedron as prove of concept

Elastic string always allows a bit of leeway, and I used it sometimes to draft models. Some of the four strut tetrahedra combine elastic string in the centre, and non-elastic on the outside. Non-elastic string requires much more precision than elastic, but besides this, I love the "invisibility" aspect of it. Frank Chester mentioned that geometric shapes act as scaffolding to manifest physical objects, so I'm perfectly happy to have some transparent scaffolding still in place.

I probably stopped using non-stretchy string for smaller objects after having some careless punters breaking my sculptures. I think it was Edison who mentioned that "you cannot make things foolproof, because fools are so damn inventive". I liked the idea to show the framework of a chestahedron with the outer tension network of a tensegrity, while hiding the supporting inside tension with transparent string. 

When I measured the draft I made, I noticed some variations of lengths, so I chose some very similarly prepared struts and dedicated some time to prepare my strings with as much precision as possible. The second model looked promising already in its early stages.

Unfolded tetrahedron, four equilateral triangles

All the supporting tension elements are now made with transparent string, symbolising the invisible forces. I still needed two attempts to find a good length for the strings supporting the vertical riser. The final version has a relaxed amount of tension. As it's not really meant to be stressed heavily, I'm quite confident that it will maintain its shape for years to come.

Seven strut chestahedron

Here you go. An object with seven vertices, seven faces made with seven sticks and seven supporting transparent strings. Can it get any better? Most certainly. I used three different length for the struts, introduced new length for the invisible support. The perceived centre moved up, although it still seems to divide the structure with the golden ratio.

Now that I know how to build a version of it, I'm curious how to explore this shape even more. It's close to my heart.... as it is the scaffolding needed to create a heart in first place. Stay tuned.

Inner strength

I got asked to build a tensegrity model that shows excitability on the outside, and has some core adding resistance against heavy loads. That's the inspiration for the Inner Strength models, both of which have already arrived in Queensland, with one unfortunately broken in transport. 

Inner Strength - Purple Heart

Inner Strength Purple Heart served as concept study for this idea. I used an already build up octahedron and experimented with ways to install an icosahedron in its centre. The most difficult part was to figure out the length for the tendons leading to the centre, they needed enough tension to withstand forces. Probably too much, one of the purple struts broke while the model was squashed by something heavy during transport.

Inner Strength Hard Core

I used the experiences from the first build to play with the variables a bit - a heavy, solid core held by low diameter struts for the octahedral shell. The weight of the core adds to overall tension on the outer shell in seemingly stabilising way, with still an amazing amount of varying movement throughout the structure.

Cube in octa

I already discovered that I didn't need struts in the centre, tetrahedron, octahedron and cube can be constructed by tendons joined at the corners. The Cube in octa is suspended from the center of the triangular faces of the octahedron, creating redundant network paths. The nylon tendons highlight that the cube was made without struts, and embed a familiar shape in a less conventional surrounding.

Copped Hypercube

Copped Hypercube was my first attempt to suspend the frame of a Platonic Solid in a small scale model. The struts of this model were recycled from a models that lost most of its tension, after being very floppy to begin with. The current emanation is extremely springy, and it seems like the cube adds resistance against total collapse, while some its edges slack off.

It's fun to build a framework entirely from string, so that it entirely depends on tension to hold its shape. I will have to upscale to find out more about the change in behaviour depending on the relation of inner and outer structure. The large scale model on display at the moment has a tiny tetrahedron in relation to its out shell, which still seems to contribute to its overall stability and movement.

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.