How to build a tensegrity cube

I started shooting some instruction videos for The Affordable Tensegrity Toolkit, which will be available via Big Cartel as soon as I finished documenting how to build the Platonic Solids with it. There's certainly room for improvement when it comes to the production quality of the videos, but so far I'm quite happy with the result. Contact me via this blog or via lutz (at) smart-at (dot) net for further information.

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.

Unholy grail

Although I'm slowly running out of space, I can't stop myself from building more sculptures. Especially as I started to produce bigger pieces. Tensegrity structures scale in surprising ways - it gets easier to build them when you go bigger.

I haven't seen any other class 1 tensegrity with 48 sticks based on an octahedron, and felt glad about the photos I took from my first attempt for a rebuild. Unlike its little brother with 12 sticks, you can use two colours in a symmetrical fashion throughout the structure. Spherical class 1 tensegrities with more than 20 sticks seem to require a minimal length for the struts to bear their own weight. The first 48 stick octahedron only remained without touching struts when it was hanging, its own weight collapsed the corner square it rested on. Time to get bigger.

Using many sticks for a single structure means preparing lots of identical elements before the build can begin. Colour-coding the different elements made my life easier, and helped me to detect a pattern in the 48 stick octahedron that made building easy. Like in the 12 stick version, four sticks meet in a loop where the original corners were. Additionally, three looped sticks represent the original face of the octahedron. Six corners, made of four sticks, equals 24, eight faces, made of three sticks, equals 24.

 Corner square clockwise

Triangular face counter-clockwise

I build the 30 and 90 stick spheres (based on the icosahedron) stick by stick, from bottom to top in a concentric fashion. The 48 stick octa invited me to a more modular build, the triangles and squares joined but didn't have struts in common.

8 triangulated faces and 6 squared corners 

I coloured one side of the square struts to have the coloured side facing outwards in the triangles, and inside in the squares. The squares turn clockwise, the triangles counterclockwise. The square loop connects the 'short' end of the struts, the triangular loop joins the 'long' ends. I guestimated the points of connection and started off building the 14 modules. Assembling the octa from the modules came easy, I only got lost once but found my mistake fast.

One corner module joined with 4 face modules

The final model surprised and disappointed me. It looked rather cubical than spherical, unlike the smaller first version the corners kept 'open'. Well, to a certain degree. I when I started tuning the model to balance on all corners I went through to various stages of behaviour of the model. The structure would balance on some corners, balance after collapse, not balance with or without collapsing corner.

Model before assembling the final corner module

I decided to wait to transform my disappointment about behaviour and looks until the next day. Then, I explored the random behaviour with more tweaking around. By replacing the triangles with smaller ones I could potentially remedy two things in one step. And it did. The structure now balances on at least three of its corners, and only collapsed under additional load.

* * *

As I replaced 8 triangular modules from the octahedron, I thought initially about building another octa with them. Instead, I started playing with them by joining 6 of them into a flat hexagon. One strut of each triangle went perpendicular to hexagon corner it was joined to, so I bring some tension into the model by looping these six struts together. Voila, an 18 stick class 1 tensegrity made in a jiffy.

The structure reminded me of a fence, or a yurt. I took another triangular module to build a roof and stabilise against warping along the hexagonal baseline. Now I had something tent-like, or a sort of bowl when turned around. The 'roof' triangle collapsed the sticks in it when placed on it, so I played with adjusting the roof triangle size and tension.

Even thin air made my bottomless bowl collapse, so I used another tensul, this time rotating in the opposite direction, to get more stability. The big tensul on top removes the resemblance of the yurt, yet when turned around it all of sudden looks a bit like an ancient cup. The bowl and handle use different colours, nylon cords indicate the 'rim' and connection between handle and bowl.

The structure can even take some light loads, and deforms visibly without losing balance. Another tensegrity I haven't even dreamt of before that opened many cans of worms worth exploring. I'm still not sure whether to call this structure 'Holy grail' or 'Unholy grail'. It opened some new avenues to explore modular building that might even upscale easily.

* * *

Reading about tensegrity contributed lots to my recent explorations. I had no idea how to deal with or implement joints in a tensegrity structure, or whether this would void somehow the 'true tensegrity paradigm'. According to Fuller, everything is a tensegrity structure, according to Snelson its rather structures where 'islands of compression float in a sea of tension'. Skelton and Oliveira came up with a classification of tensegrity structures that even helps applying this idea to the musculo-skeletal system.

Without compression elements articulating in a common joint, Skelton and Oliveira speak of a Class 1 tensegrity systems. If 2 'sticks' meet, it's a Class 2 tensegrity, and so on. As long as tension is required for the stability of the system, it still deserves consideration as tensegrity.

I had still some struts with eyelids at each end lying around, and two eyelids connected make up a simple joint. While connecting strings to eyelids poses sometimes a challenge, hooking them into a model under tension works smoothly. I wanted to explore suspending small tensegrities from bigger ones, and struts with eyelids nicely suit this purpose.

Viewed from the top, the strings of a tensegrity tetrahedron balanced on a corner look hexagonal. I suspended an octahedron from three sides of a large tetra, the suspended structure floats upright above its support. In this case, the octahedron can be collapsed and reacts sensitively to any movement of the larger support.

'Timeless Hourglass' deploys already suspension, strings leading to the centre of the top triangle. It's large enough to experiment with hanging eyelid sticks from its top triangle. At the moment, a dodecahedron hangs about 80 cm below the top, in the lower half of the hourglass. The connection consists of two 40cm sticks joined with eyelids, acting as joints. Technically, this is now a class 2 tensegrity system.

The joints add an interesting dimension to the behaviour. With lots of movement, the joints act as a dampener, similar to systems for earthquake save sky scrapers. Each moves differently The dodecahedron pendulum retains movement for a long time, though, and slightest currents start moving it.

Suspension brings together the three different joining styles I explored so far: knots held by a hole in a tube, knots held by a groove, strings hooked into an eyelid. It combines as well the different materials: hollow bamboo, short diameter bamboo, oak struts, nylon and elastic cord.

How to build a tensegrity octahedron

Besides icosahedra, octahedra can be build to collapse and bounce back. If you use just elastic cords, the model has high mobility, yet I noticed that the cords slide sometimes to easy around in models with just elastic cord. I use elastics for the six 'corner' loops, and nylon for the remaining twelve tendons. 

The less stretchy nylon helps to maintain the overall shape after a collapse. The different materials indicate their function in a 'solid' octahedron: The edges turned into the nylon cords, the corners folded open into a square loop made of elastic cord. 

The nylon cord has a length of 7cm between the knots (made from 10cm cuts), the elastic cord has a length of 30cm tied to a loop. As I loop the the elastic cord around the grooves to secure their position, the effective length between struts comes down to 6cm. The struts itself measure 16cm, with 15 cm between attachment points.

Components for the model

I aim for precision when I prepare the components for a model, yet symmetrical models can do with a bit of variation in length. With a lot of a variation in tendon or strut length, the model looks less symmetrical, it still works as tensegrity system. 

So it starts from 12 struts (15cm effective compression), 6 loops made from elastic cord, 12 tendons made from nylon. The four 'horizontal' edges will be yellow, the eight vertical ones will be orange. Different colours for different functional elements make it easier to keep organised while the model is still two-dimensional.

Step 1: Attaching the tendons

In the finished model, each strut is connected to two tendons and member of two 'corner loops'. To prevent the tendons from slipping out, they get attached first. Four struts get a 'vertical' and 'horizontal' edge attached, four struts get the remaining four verticals and four remain empty.

Step 2: The bottom corner

The struts with only vertical tendons end up in the bottom of the model. The struts are connected into a loop on the end with the tendon, so that the knots point to the 'inside' of the loop, the longer end goes away from the loop. (This means that if the struts fold over clockwise, the knot will be clockwise towards the strut as well). I start usually with the opposing corners, so that I can always half the tendon when connecting an additional strut. The elastic is looped around the groove once to secure it in place. 

Step 3: The top corner

The top corner is assembled in the same way - connecting four struts at the end with the vertical (orange) tendons, knots points inside, tendons to the outside. All corner need to fold with the same chirality, ie clockwise or counterclockwise. If you lay the top corner above the bottom corner, the yellow tendons should already point to the strut of the bottom corner it connects to.

Step 4: Connecting top and bottom corners with horizontal tendons

Depending on the length of the struts, and the tightness of fit of the cords and grooves, a lot of slipping and sliding can happen while connecting top and bottom corners. Once we yellow tendons were in place, I took care that the loops didn't get entangled. At this stage, the struts hold each other up, like in the match head trick. 

Step 5: Connecting top and bottom corner with remaining corners

This step unveils the precision of planning and preparing of this process. With a good loop length, the model will start to erect itself, becoming three-dimensional and easing the following steps. The inside-outsideness of the remaining corners is still more of an up-downness when the elastic are connected. The inside of the loops goes along the bottom half of the bottom corner strut, and the top half of the top corner strut. If the struts still fold over un-entangled, the top corner starts pushing up the more corners connect to it. Now it's a good time to check that the free ends of the vertical tendons hang in a decent direction.

 Step 6: Connecting the remaining struts to the tendons

Over and under become more important now, and make life easier at the same time. The remaining struts connect between a bottom and top. Each strut goes under a top corner strut, attaches to the bottom corner tendons on the strut in the same direction as in the top corner. It attaches also the top corner tendon of the top corner strut that runs parallel to the one crossed under. (I wonder if this description makes any sort of sense unless you starting these three-dimensional puzzles yourself).

If all worked fine, the top end of the newly connected strut pull on the top corner and float a bit. 

Step 7: Connecting the top part of the middle corners

A lot of movement will happen now. only eight connections finalise the model. Depending on how much uncontrolled movement happens, the model might fall apart or become lose. Working patiently with a steady hands prevents the frustration that arises from the anticipation of success and the sight of an entangled mess of sticks and strings. Like with the other loop connections, the knot of the tendon need to point in the same direction as the other struts in the loop.

Step 8: Connecting the bottom part of the middle corners

If the cords can slide easily, this step can become tricky. If the tendons and loops are too short, the model might 'fly apart', with too long cord lengths the model will still lay collapsed in a pile. Once the remaining connections are made, I take some time to 'tune' each corner. It's more likely to have some trapezoidal corners instead of nice squared ones. If struts were accidentally 'twisted' during build, they can be turned into the proper direction, so that each corner looks similar (all knots oriented the same way, elastic cords length equally from strut to strut.

Step 9: Test

Now the model can be loaded to collapse. With too long loops, a model might still look like this - well, a bit more chaotic as the elastic string would dangle around somewhere. That happened to me with the very first octahedron I build. I went from corner to corner and looped the elastic several times around each corner, and slowly the structure started 'working'. With sufficient pre-tension the model can be handled and tuned easily, and minor hick-ups during the build process corrected.

Sorry for mixing up the photos - two of the steps show a different octahedron. Anyway, if you happened to notice this error it's about time to start building tensegrities. 

How to build a tensegrity icosahedron

The model I sold most so far is a simple 6-strut icosahedron with elastic string. It can collapse entirely and bounces back fiercely afterwards. The cord is much more durable than rubber bands, my favorite product tester is my neighbour's son, who hasn't manage to destroy his toy for more than half a year now.

The bamboo sticks I used have a diameter of 5mm, with a 1mm wide groove at either end. The elastic cord has a diameter of 0.8mm, the knot prevents the string from sliding out of the groove, especially once the model is fully assembled. The tension level can be increased by looping the cord more than once, but unless you will do with all connections the symmetry can be compromised.