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.

Thingamabobs

I took up residency with my Magical Thingamabobs nearly a year ago, and many things happened in the open air shop in Melbourne's prestigious CBD since then. I encountered all four seasons now, and could observe the change in angle of the sun towards our planet.

Some people claim that my art work isn't 'emotional', which might hold true to a certain degree. Nevertheless, I experienced the gamut of fear, anger, grief, love, sadness and worry while patiently and persistently doing my work.

The 6 strut icosahedron and the 12 strut octahedron remain my most popular objects, but there's some shapes I discontinued, and a new set of objects to reassemble each time around. As my camera equipment doesn't work too well, I probably missed the opportunity to document some objects I haven't build since then.

Close encounter - 6 strut tetrahedron

I noticed that reassembling some of the objects creates some wear, and a loss of tension. The model above has reworked to have single-string connections to each corner. The 20cm Pars inspired tetrahedron has been painted a little bit to look even more distinct.

Japanese Butterfly - 12 strut tetrahedron

I love the shape of the 24 strut cuboctahedron, and build some colour variation with it. While four colours bring out the four intersection double triangles in the structure, using black as only colour works aesthetically well.

Checkerboard - 24 strut cuboctahedron

An icosahedron with suspended wireframe tetrahedron act as the current 'eye-catcher' for the front row. The transport made the doubled strings a bit weaker, the suspended centrepiece brings it back into good shape. 

Captured tetra - 6 strut icosahedron with suspended tetrahedron

The dodecahedron gets really wobbly as 30 strut structure. I found two variations for it, though. The first one consists of an 6 strut icosahedron, joined like the one above. By connecting the ends of the parallel sticks the strings outline 12 pentagonal faces. The other variation starts as 5 strut prism, with five more sticks outside the girth of it. 

10 strut dodecahedron

Painting the end of the sticks black adds another dimension to the structure, and works well for many basic shapes. The model below has already found a new home, but as I enjoy playing with this one most, it will have a rebirth soon.

12 strut octahedron with elastic and non-elastic string

The amount of 'un-playable' objects increases. The latest one of those offers plenty of potential for all sort of variation. The outer tendons shape a tetrahedron, the sticks are attached to its corner and miss each other in the central hub made of four tendons.

Black flag post 4 strut x-module based tetrahedron

I have two of 4 strut pyramids on display - the 15 cm version above, and a 30 cm one. And some more at home, one version with aluminium tube as struts. I want to explore whether I can use these tetrahedral modules to build a star tetrahedron, that might be a nice project for a rainy day.

I took some of the photos here from the blog Hello Mrs. Morris, who wrote a nice article about me and my work.

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.

Is this street art?

You can touch this has been rebranded for the street market. As I still want to popularise the concept of tensegrity, I started busking my work in a public place, at City Square in Melbourne's CBD. 

Besides bringing about 10 build up structures along, I have string, scissors, ruler and sticks to build more. It's still only few sculptures per day that change hands for money, yet so far I haven't had any no-sales day.

Working in public is fun, and gains quite some attraction. I had plenty of inspiring conversations, and witnessed mass outbreaks of awe when school classes passed by. From of those, I heard the most interesting question so far: Is this street art?

My answer was simple: "If you think it's art, and it's definitely on the street, so yeah, you can call it street art." It has a bit the character of an exhibition as well, a temporary of course, with new objects every day.

I made friends with a chalk artist who is quire regularly nearby, and enjoy that he shares his experience on the street. It must be about the worst time to start this experiment - short daylight hours and not really pleasant weather. Especially the wind keeps me busy, but with a limited amount of models on display I still managed to chase up anything blown away.

The 6-strut icosahedron, plain or in 3 colours, and the octahedron go best, but I sold also the 12 strut tetrahedron, and some 9 strut joined tetras.

Besides some chalking, my setup is very basic. I still would like better display options. I could probably chalk some info, but a good looking solution that portable and easy to transport could improve my sales.

Sunny days bring out interesting patterns casted by the structures, and generally bring more smiles and interactions. I'm still hoping for the rain to cease, I build a tetrahedron as donation hat, and crave to try it out.

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.

Filling the void

I started experimenting with combining tensegrity structures from the very start of my explorations. Most of the results used models of similar dimension and base geometry and some sort of quick and dirty approach. It took me a while, and some larger sculptures to deviate from this idea, and the initial set of combined structures had a clear defined orientation in space.

The idea of core and shell allowed me to create more complex structures which still can be placed in a variety of orientations. It also taught me experimentally that Buckminster Fuller's idea about the tetrahedron as smallest 'building block' of the universe can be demonstrated with tensegrity models.

Although I haven't attempted yet to exactly model the volumetric relations between the Platonic Solids as described in Synergetics, the 'compatibility' of all these highly symmetric structure becomes very apparent. Tetrahedra fits easily into 6-strut icosahedra, 12 strut cubes and octahedra, and 30 strut dodecahedra. Most likely also in 30 strut icosahedra (the most common tensegrity sphere), I just haven't bothered yet to build one.

Cubic merkaba

Octa Octa

Icosa Icosa

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.

Octoids

I haven't been that busy during a week with building tensegrities ever, and I enjoyed it thoroughly. I needed models I could transport without a car. It will be lots of fun to find out how practical my idea of transport will be tomorrow, unicycling with a backpack from which twenty or so tensegrity structures dangle....

Besides a set of octoids, octahedra in which colours enhance aspects of their structure, I build a number of colourful icosahedra. I hope I can source more of the material I used for the latest icosa builds, it saved me lots of work. Doing the computer related things, like preparing and uploading the slideshow turned out much less pleasurable. Nevertheless, all is prepared for an interesting market day tomorrow, including the update of the market datea for the first three month of 2012.

The next season

Many things happened since I last had a stall at the Rose Street Market, most unfortunately, my car got totalled so I now face the challenge to transport enough models either on PT or on a unicycle.

As most of my toy octahedra were gone, I prepared the components for another batch of octoids, spending lots of time on sawing, cutting and a bit of spray painting. Luckily, I didn't forget how to build tensegrities, determined to produce enough portable material for next sunday's market I got into a bit of a rush - six octahedra, one icosahedron and one cube provide the first yield of two days work.

By chance I came up with a new colour combination for square struts, which works amazingly well. I might need to prepare another batch of struts to use up all the coloured ones I have now, and getting the photo and documenting job done.

I nearly forgot the joy of bringing tensegrities to life, especially those with unique looks.

Revisited

Red Star (Intertwined tensegrity tetrahedra)

I revisited my attempts to combine tetrahedra to build a tensegrity representation of the merkaba. Red Star comes so far closest to joining two tetrahedra in this fashion, yet I had to take two different sizes. The tendons of the red tetra are suspended from the struts of the larger one, it can vibrate while the outer structure is extremely solid.

Red Star

The tendons of the larger tetra connect to each other, instead of directly from strut to strut. I haven't tried out this way of connecting a corner with elastic string, with nylon it works really well, and is optically very appealing.

Merkaba (Tensegrity octahedron with 8 tensuls attached to form two tetrahedra)

With 36 sticks the sculpture Merkaba offered initially hardly any depth to it, as all struts had the same colour. I rebuild the structure using a green octahedron, and using bicolour struts for one of the surrounding tetrahedra. The model balances on each of its eight tetrahedral corners, and folds down along the axis connecting opposite corners of the octahedron.

Merkaba

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.