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.

Modular

The triangle seems most obviously the most stable configuration, and it can be found in a lot of my tensegrity structures. Three trajectories building the eight intersections of a cube would be one example, which transform into eight triangular opening of an octahedron. Two triangles that add up to four triangles in three dimensions of the tetrahedron would be another.

While showing my tensegrity structures in the street I skipped the 30-strut structures for pragmatic reasons. I carry about 13 items around - 3 six strut icosahedra in different sizes, 2 octahedra, 2 tetrahedra, 2 cubes, 2 joined tetrahedra, 2 pentagonal prisms that build some donation vessel.

While it's easy enough to perceive the octahedron as a kind of spherical object, it's much less obvious than a 30 strut icosahedron, or the 24 strut vector equilibrium, or it's dual, the rhombic dodecahedron. Just to keep me busy, I started building the latter two while I'm sitting around, and discovered some interesting quality.

Vector equilibrium

Instead of building strut by strut, the VE and its dual invite themselves to be composed of triangular modules. Actually, the same applies to the cube and octahedron.

Rhombic dodecahedron

The triangular module used to build the above model connects the struts in the middle of the attached string, which works easy enough to build the VE and the dodecahedron. The dodecahedron balances quite easily on its 'square' corners, the VE on its 'rhombic' corners.

Usually, the triangles are rather located on a one third/two third ratio, at least when I aimed for maximum symmetry. While the half configuration works quite satisfying for the larger structures, building cube and octahedron from the same 3-strut modules produces rather skewed results.

Skewed octahedron

The four 'modular' triangles are larger than those that emanate by connection, and instead of having square corners, they are rather rhombic. The same applies to the cube - four intersections have the size of the original modules, the other four end up much smaller.

Skewed cube

The reduced symmetry in the skewed versions still has lots of aesthetic appeal, and invites itself to be build in multicolour. I appreciate more and more the opportunity to experiment with different ratios of tendon lengths that elastic string allows for.

Busking tensegrities

Since I started building tensegrity structures, I got hooked. When I succeeded for the first time, I experienced something I couldn't have imagined before. In a way, I encountered the difference of what is called 'book wisdom' and 'experienced wisdom', and it blew my mind.

Now I'm certain that the concept of tensegrity will be part of my life. It is, scientifically speaking, anyway - from the way the cells in my body maintain their shape, to the interaction of the musculo-skeletal system, it's just a fact of life.

As I enjoy the process of building physical tensegrity structures, in various sizes and shapes, using different materials and strategies, I soon ran out of space to keep or at least collect my own work. For more than two years, I managed to tackle this challenge partly by going to a market and sell some of the structures I build.

I even went to live in the bush for a while with the hope to fill space with bigger things. This only meant limiting the amount of audience - instead of familiarising a larger part of my community with this beautiful way of representing fundamental principles within the universe, I engaged in some sort of (not really satisfying) form of artistic wanking.

Being involved in the Friday Free Shop in Melbourne's City Square gave me a better opportunity to gage the attraction of my work - I donated some of my structures each week I went there, and usually all of them found some new owners. Success! Even without knowing the term tensegrity, about thirty or forty tensegrity toys ended up in unknown curious hands.

Encouraged by the ease to sell my work for free, I decided to make City Square my new market place. As often as the weather permits, I will offer a variety of tensegrity structures there, as well as building new ones while I'm there.

The short time of sunlight restricts the amount of time I can spend there, nevertheless I will do my best to be available for a chat, and a build to customers preferences at City Square in Melbourne. Pick up a sample of my work for free on Fridays, or meet me, weather permitting, trying to get some of your hard-earned money for my honest craft on any other day without rain.

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.

More rip-offs

The success with rebuilding the tetrahedron based on Marcelo Pars' idea motivated me to do more as yet unexplored structures. There's still some challenges I want to take on, watch this space.

Caged rainbow (18 strut tensegrity cube)

The simplest of my new structures just adds 6 struts to a cube with changing chirality along its corners. Similar to a dice, where opposite sides add up to seven, colours oppose each other of each side. Snelson managed to build an x-module in a way that outlined a tetrahedron, I was surprised how well my guestimation for the different tendon lengths worked out.

Tetra ala Snelson

It might look better when slotted into a base, so that only one strut connects to the ground, similar to Earthlon, which shows the 'ascension skylon' out of Burkhardt's collection of structures, and turns it around.

Earthlon

I drafted Earthlon with elastic strings. but it gained much stability with less elastic tendons. It has only little movement, nevertheless looks quite surprising.

Polarised canary

Polarised canary took quite some time to get together, and probably the tendons holding the upright strut could be a bit shorter. The same concept should work with three struts on either end, and looks a bit like a beam capable of absorbing shock along its length.

Explorations

After building a relatively large number of tried and tested icosahedra and octahedra, which so far exceeded the demand by far, I went back to the fun of exploring other shapes and build methods. Floating Spell is an adapted pentagonal prism, with the top struts connecting across the center.

Floating Spell (20 struts)

Five Elements creates different views from each angle, with two of the twelve pentagonal corners appearing copper from the outside, and black from the inside. Each strut looks basically identical, nevertheless a variety of pattern appear throughout the structure.

Five Elements (30 strut tensegrity icosahedron)

12 Meridians belongs to the recycling projects among my latest explorations. After deploying the centrally joined corner tendons, I rebuild a dodecahedron with black and white struts, which failed to impress me in its first incarnation.

12 meridians (30 strut tensegrity dodecahedron)

Balanced Infinity is a 12 strut cube with centrally joined corner tendon, and elastic string for the tendons along the edges. The model feels very floppy, yet when handled gently balances on each of its eight corners. It can go through quite some interesting before losing balance.

Balanced Infinity (12 strut tensegrity cube)

Star Icosa stems from the ambitious idea to build a 30-strut icosahedron entirely with non-elastic string. So far, my attempts usually lacked the precision in tendon length for satisfying stability in 30 strut models. The star connected corners looks especially interesting under UV light.

Star Icosa (30 strut tensegrity icosahedron)

Redfaced Revisited is another recycling project. The original Redfaced had transparent elastic tendons, It has a cuboctahedral shape (Vector Equilibrium), and handles nicely.

Redfaced Revisited (24 strut cuboctahedron)

Polar Symmetry belongs to the experiments with scaling up. Although the nylon string has only little elasticity, the model can collapse on itself and bounce back.

Polar Symmetry (6 strut tensegrity icosahedron)

The next three objects a variations of the same structure, utilising central corner joints. Green Bridge shows the pure concept: 4 20cm struts rising near vertical, two 30cm struts crossing in the center along a horizontal plane. 

Green bridge

Fiercely occupied uses different colours for the different tendons, and has a Pokemon as inhabitant.

Fiercely occupied

Gargoyled Tetra is also inhabited by a Pokemon, and also outlines a tetrahedron with the four orange tendons in its center. The tetrahedral pull towards the center contributes to the overall stability.

Gargoyled Tetra

After finding some many 'merging' structures in my latest tensegrity experiments, I revisited also the idea of the merkaba. The study has some flaws to it, yet it would like see a much larger build to dismiss this concept.

Merkaba study

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.

Playing around

While I still haven't solved the lack of space, restricting my ambitions to go bigger, I continue to experiment with new ideas. I build Tetroid some time ago, and had it with me at the market quite often, but I wasn't too happy with it overall. The tendon length didn't work out properly, so I decided to connect the three strut in a corner in a star shape instead of a triangular loop.

With more tautness than before, each strut could move laterally a lot more, and the network of tendons now distinctly outlines a tetrahedron. I wonder if an octahedron build like this could still collapse....

Marsupial (Large tetrahedron mounted on 3-strut tensul with small tetrahedron suspended)

The new corner configuration increased the appeal straight away, as next step I mounted the tetra on a tensul, using the 'edges' as mount point. Tapping on the top, the structure bounces and rotates a bit. When done carefully, you can rotate it on the spot. The size invited to suspend something in its middle, a 'traditionally' build tetrahedron. The 'baby' tetra swings in its own frequency when the model gets in motion, like a Joey bopping its head out of its mother's pouch. Well, at least with a lot of imagination.

United Duals (Octahedron with a cube intersecting the edges)

I still want to build a tensegrity merkaba, and discover how slight variation produce very amazing outcomes. I started with an octahedron, and added 3-strut moduls to the edges of each triangular face. I moved the strut close together, so that the 24 struts surrounding the octahedron appear like 12 struts in a cube. The struts of the octahedron are a bit less twice the length of the cube struts. The corners of the cube are too small to provide balance for the whole structure, but the model can be 'suspended' from each octahedral corner, which stands slightly out from the cubic faces.

Merkaba (stellated octahedron or octangula)

Having cube and octahedron united was nice, but unexpected. I went back to my small merkaba model and noticed that I had join the tensuls to the corner, and not the edges of an octahedron. It still folds along opposing corners of the octahedron, but the two intersecting tetrahedra remain hidden in the chaos of 36 struts. Having the octahedron in a different colour could bring out more interesting pattern, it's fun to play with, yet a bit visually overwhelming.

Hyper Tetra

Hyper Tetra has a green tetrahedron at its core, surrounded by four tensuls connecting to the edges of it. I made the corner triangles quite large to allow balance on each corner. Now I realise that this model comes closest to the idea of the merkaba: two intersecting tetrahedra. Of course, the 'outer' tetrahedron is roughly twice the size, same sized tetrahedra intersect along their edges. This idea invites to a bigger rebuild, using a 6 strut outer tetrahedron with center holes.

Bell Tower

I played around with models of orthogonal cubes, trying to stack them together. I'd love to build a mega-cube with 20 cubes connecting a larger one, yet I still doubt whether I could easily balance and tune such a large structure.

I would need three columns of tree cubes stacked on top of each other, so I experimented to find out how to stabilise three cubes. Four struts of each cube connect top and bottom vertically, and come very close with simple stacking. I decided to have four central struts, and complete them in the pattern of three stacked cube models.

Bell Tower early draft

The build turned out quite awkward, with some engineering challenges on the path. It took me some hours just to tune the model into a more symmetric shape, and achieved hardly any decent balance. Cubes without diagonal support are inherently instable, so I decided to add cross-bracing tendons to the sides of the structure.

Finally, I was getting more stability and balance, and I began marvelling how to finalise the now sturdy base in a satisfying way. I could easily stack my two cube model over the base, and have an octahedron to top it off.

Bell Tower last draft

Although this experiment created the first 6-level tower I ever made (if you count the base of three levels), balancing was rather unpredictable, and probably hardly stable over time. I had enough leaning towers, and an octahedron on top simply jumped the shark. Finishing the cube-based structure with a 'twisted' cube combined with half an octahedron rather mimics traditional building methods, no need for a wobbly extension on top.

I achieved the final bit of stabilisation by suspending a bell from the four central struts, bringing the centre of gravity down, and dampening lateral movements. I had all aspects together I wanted for that sculpture: a lean base with four central struts, a pointy roof, and a suspended weight. The devil hid in the detail. A cube combined with half of an octahedron can be build with 12 struts, using four joined corners, instead of simply wedging an 8 strut half octahedron on top of a 12 strut cube. Also, the bell had the right kind of weight, but not the looks I wanted.

Bell Tower nearly finished

I spend the first day on building the base structure, the second day with finding a top and suspension, as well as cleaning up the model and fine-tuning. On the third day, it was time to bring everything together. 

Once the bell was in place, the model stood easily balanced, and I installed the last struts without bothering about the wobbly base to work with. The top can now be bend, and the weight of the bell prevents the model from falling over on the rebound. Once in motion, top, base and bell swing in different, connected rhythms, with pressure from the top it rather bends away instead of collapsing.

Bell Tower

Even while blogging, I couldn't stop tinkering. The finishing touches included a cross-bracing of the central struts, similar to the sides, and a tip made with red struts, rotating opposite to the smaller half octahedron underneath. I like the distinct shapes emerging from the red struts from a distance, as well as the sturdiness with all sides cross-braced. 

I used four oak struts (60cm) and 44 bamboo struts with three different length (24 @ 20cm, 16 @ 16cm, 4 @ 28 cm), three types of elastic cord totalling about 12 metres in length, and about 6 metres of nylon cord. Bell Tower measures 90 cm, with a base of 20 x 20 cm, encapsulating a volume of about 35 litres with an estimated weight of about 300 grams. There's roughly at 2:3 ratio in the added length of struts and tendons.

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.