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.

Even more chestahedron

The last post about the chestahedron called "Mottled Heart" went a little bit all over the place, as I wrote it in multiple stages before the piece went to its final destination. So let's rewind and start at the beginning.

The artist Frank Chester set out on a mission to find a geometric structure with 7 equally sized faces. After many explorations he discovered the chestahedron, an object with 7 faces (four equilateral triangles, three kites) and 7 vertices. The structure does not qualify as Platonic Solid, as it has two different edge lengths and two types of faces.

As the structure bases on a tetrahedron folded open, it elegantly relates to all Platonic Solids, as well to a sphere surrounding it. According to Chester, the structure represents the geometry of our heart, please check out his talks for a more in depths explanation for this. When I followed a presentation about the genesis of this shape, my mind got blown several times, inspiring to seek some hands-on experiences with it.

In my first experiments I got the length for the top three struts wrong with only slightly satisfying results. Luckily, I found out the proper numbers, so that the latests builds give me better ideas about the qualities of this unique structure.

My 'standard' way of building tensegrities follows this simple algorithm:
1) All edges of the wireframe model become struts.
2) Each strut gets a string roughly 10% longer than the strut length.
3) The string network reflects a truncated version of the base geometry, eg the strings of my 6 strut "tetrahedron" create a truncated tetrahedron.
4) The number of struts converging in a corner determines the slicing, three edges create a triangle, four edges create a square, etc
5) Building of the tensegrity starts with a 'corner', eg connecting three struts with the strings shaping a triangle to begin building tetrahedron, cube or dodecahedron.
6) Each string connects to two more stick ends.
7) Repeat building 'corners' at the second string attachment position and continue until structure completed.

This simplified version works out fine for all Platonic Solids, it seems to fail for complex intersecting geometries like star tetrahedron. It worked well for the chestahedron, although, if you're really pedantic, the strings represent of truncated chestahedron. While geometrically interested people can perceive and identify the Platonic Solids in its representation as truncated tensegrity, the names of these geometric shapes evades a majority of people.

Our consciousness seems to resonate with geometry. The symmetry of it appeals to our perception of beauty, and it doesn't really matter whether we can put a name to a structure we experience. Architecture and engineering rely traditionally on squares, we have on overabundance of distorted cubes arounds us.

Mobile architecture utilises triangles much more, and geodesic domes offer a nice relieve of the geometrical desert which most urban landscapes offer. The chestahedron hides the numbers 1 to 7 in an elegant and surprising way. 1 object created from 2 base structures, a 4 sided tetrahedron, and 3 slices of a 5 pointed pentagram shows 7 corners and 7 faces. 6 edges shape a perfect hexagram through the centre of a sphere surrounding the chestahedron.

I played around a little bit with less symmetrical structures, but the majority of objects I build and sold showed multiple symmetries. I build some bases for spheres, there's often no clear up and down in my objects. The chestahedron breaks this mould - it commands like an obelisk to be put on its base. It invites to have something suspended from the apex.

The effect of a counterweight can be compared to someone pushing the object to the ground. As long as the counterweight doesn't move, which will happen. Without anchoring I could easily topple the structure over by moving the pendulum much out of centre, yet there was quite a lot of range of movement in a stable state possible.

With only about 80 cm height, "Mottled Heart" stands in a relatively sheltered space, surrounded by a planter box and equally high plants. 3 plastic tubes, fitting snugly over the bamboo sticks, anchor it about 10cm into the ground. Most of the time I saw it moving. I wonder how weathering will effect the stretch in the material, I anticipate a vast visual improvement. As I recycled the struts from a first experiment to paint on bamboo, the paint will wash and weather off. The strings will bleach off, the spot will get more and more sun exposure the closer summer gets.

I know how to improve the immediate visual appeal of the materials involved. While I was busking, I experimented a lot with colour, just a learn more about the fierce Australian sun than I wanted to. If something looks good outdoors over time, it works with nature and not against it. Oiling plant surfaces can provide interesting graceful ageing of material.

Instead of being the trickster, stunning by the immediate shineyness of their illusion, I let Mother nature do her part of trickery. If the "Mottled Heart" still beats a year from now, it will look quite different. Until then, I can enjoy seeing the calming movement reminding me of eternal change.

Chestahedron

I came across a very interesting geometric shape, an object with 7 openings (faces). It is composed out of 4 equilateral triangles and 3 kite-shaped openings. The kite is composed by cutting a similar sized in half and arranging the parts along their longest sides.

Mathematically speaking, it would be classified as diminished trapezoid, or as a heptahedron. You can find 7 a lot of times: Number of openings, number of crossings (vertexes), it's entire surface area is 7 times that of an equilateral triangle, there are 3 crossings with 4 trajectories, and 4 crossings with 3 trajectories.

I build it easily as tensegrity structure, with my simplest construction method. 3 of the 12 edges are shorter (with a factor of sqr(3)/2 ), which I guesstimated for the first build. The model tends towards a circular shape, the elegant elongation of Frank Chester's models gets a bit lost. I experiment with using different spins of the 4- and the 3-trajectory crossings, yet set on its triangular base, it tends to 'go bubbly'.

A larger model, with a better approximation of the strut length comes closer to the desired appearance of an elongated object when suspended from the top corner. Maybe there's a simple way of keeping it 'slim' by ways of an internally suspended structure.

 Chester demonstrates in his presentation how his chestahedron relates to 4 of the 5 Platonic Solids, embeds the Golden Ratio and how it fits into the Flower of Life.

PS: I found a document having the angles and strut length relationships. The shorter struts have a 0.53 factor in relation to the base length. The latest models use a 0.5 factor, which increased their optical appeal and structural stability. The slimness I missed once I found in the proper proportions.

I recycled 75cm bamboo struts for the largest version so far. Standing on its triangular base, the structure resembles an obelisk. A teardrop shaped former bed post top is suspended from the top three struts. At the moment, it's suspended using the same type of string used overall. I will replace it with fishing line, and adjust the length so that the centre of the object indicates the centre of the hidden hexagram.

PPS: While the teardrop/bell shaped centre piece isn't probably in the centre of hidden hexagram, it attached it already in a 1:1.61 relationship (height from ground:length to the top). As the object has three points of contact with about 30 degree from vertical I plan to use some hollow plastic tubes as support anchors for them in the ground. The relatively high mount point of the bell will topple the object if it is too far from the centre.

It's fun to play a bit with this piece - the pendulum creates interesting patterns of movement, even in a still state of the pendulum. The visual effect of white paint peeling off, combined with pink string, appears very harsh. In outdoor conditions, the original bamboo will reappear, the strings will bleach. It will grow over as well - the patch I want use is fertilised with three mouse corpses, mulch and saw dust, with heaps of mustard seeds.

The 4-strut tetrahedron in my front yard turned invisible. A ranking plant took it over, and attacked the two brugmansias next to it. I expected this plant to die back in winter, but I noticed only the comfrey and chamomile to die back. I want to prune the rosemary next to the patch where the 'Mottled Heart' will live.

Most of my outdoor creations didn't survive more than some months. The first 'garden model' still lives, more than I want to. Mold has taken hold of the repurposed broomsticks, so I need consider treatment for materials meant to sustain outdoor conditions. The fierce sun bleaches lots of colour, which is why I'm curious curious about the change in colour especially with the pink string.

The dodecahedron above the office block still twirls around on a string. The prevailing wind often nails it to the eastern wall, but a change in wind direction brings it back to a floaty space. The nylon string I use mostly stretches a little bit over time, yet it still looks sufficiently tense. I use the same string for 'Mottled Heart', which might need readjustment over time. My estimations for string length meant it's not too easy to take a cm out of the overall length.

I already fell in love with the interactivity of this object. Anchors will hopefully provide a minimalist way of preventing being blown away by the wind, or toppled over by over ambitious experimentalists. The wind can mainly attack from one side, so the movement shouldn't get out of control. With spring on the door step, plants will use the support to grow intro different spaces.

The last garden sculpture was destroyed during a party. The project gained some useful insights, yet even without the destructive effort it wasn't meant to last. Replacing the stinky compost place with a beating heart appeals to me.

Best thing ever

Using phone and computer to document my latest piece has been frustrating, to say the least. Working on 'Diamond DNA' got me exhilarated, and the results exceeded my expectations after the idea came to me.

Base joined tetrahedral tensigrity

The diamond structure, two tetrehedra joined at the base, shows an interesting balance along its central edges. I presented the smaller versions upright, along the axis of compressability, suspending two thirds of the material with only three points of contact to the ground.

It's the Illuminati tensegrity - 2 triangular and 3 square corners, the geometry strengthening diamond. Build properly, it will balance on the square corners around its girth, albeit very delicately. Suspended, it should show horizontal stability between the triangular corners.

I experimented with connecting the former corners (crossings) centrally instead of outlining triangle, squares and pentagons. I'm not too sure about the classification of tendons connecting in a hub without compression elements, but it adds functional and aesthetic qualities.

The central junction of three corner tendons can act as a mount point for anything suspended in the centre. I had a spiral made of six shorter sticks which needs a pull from either ends to maintain a 3d shape. I looped string around the centre of the outer sticks and attached the elastic string to the junction of the two triangular corners in the structure.

I did my best with a random spray job for the outer struts, balanced on one triangular corner it measures more than 2 metres. I have no idea about its durability - it uses elastic strings around its girth (which I might replace), and elastic to suspend the central spiral.

While I liked to call my sculptures 'kinetic', this one fits the description. The slightest amount of wind gets movement into it, either by rotating the centre or the entire structure. It's like a mothership of randomness, another turn will happen, unpredictably. Different perspectives provide different colour aspects, just like the rotation. Any capture seems unique, like an elephant felt up by the blind.

So I indulged in making a clip, which is still rendering in the background while I type. No idea whether I'll get in copyright trouble when uploading it, I haven't even bothered watching the final result besides bits of preview. If you're in a hurry, don't bother watching. It'll be about 15 minutes without anything spectacular happening,

The raw footage still captured lots of detail I was interested in than the GoPro footage I took before. I love the balance the single point of suspension provides, as well as the independent movement of the spiral in the centre, even though the twist pulls the entire structure together.



PS: This piece brought me most excitements from any of my creations. I had some pieces hanging, and enjoyed the movement. Many delicately balanced pieces were blown over, exploring the rotary power will bring another dimension into my work.

And on and on...

It's now nine month since I last busked out my tensegrities. While I accidentally sold a few pieces this year, I'd love the business to pick up again. Still, I took part of Polymer Dreams Lab installation in Coburg, and build a bit bigger.

I still haven't organised better camera gear, the mobile phone camera has many limitations, as well as my ageing IT gear. This made me a bit lazy in maintaining this site, as it has become a bit tedious to add decent photos in an acceptable time frame.

Goodbrew Vector Equilibrium

One of the first bigger pieces is based on the 24 strut cuboctahedron aka vector equilibrium. The struts are brush painted, an area where I can still improve my skills. The structure has survived some drops of a few metres and still hangs out in a large warehouse.

Garden decoration

I came across some nice looking broom handles, with a hole on one side, which I could quickly convert into a 4-strut tetra. I used the pre-existing holes for a continuos inner tendon, and guestimated the length for the outer tendons on the first shot. Unlike larger tetras with slightly flexible struts, the result behaves very sturdy and without the tendency to collapse through its centre. It's now an eye catcher in our front yard, surrounded with two brugmansias.

Eclectic tetra

I rediscovered some dowels I prepared for my very first tensegrity builds, when I still deployed hooks to secure the strings. The dowels probably had the larger diameter I used for small-size models, while the additional 12 sticks through the centre have about the smallest diameter I used. Unless I get another dremel, I can't prepare similar small diameter struts anymore, but combination of materials with different girths in larger scale are well worth contemplating.

Another lucky find were nicely carved bamboo chopsticks. Finding the right length for this one took much longer than building the larger version garden decoration, I nevertheless like the small version a lot.

I sourced some old trampoline springs, and a friend cut some old pipes to lengths for me. I totally underestimated the strengths of the springs, and had to settle down with a slightly asymmetric version of an icosahedron shape. I haven't dared to try to collapse the structure, as I realised how easily working with heavy struts and high tension can lead to serious injury. The springs and the weight allow any impulse to reverberate through the structure for a long time, hitting struts and springs produces quite some interesting sounds. 

Steampunk tensegrity

As the metal started rusting slightly, I decided to give the dull grey with bits of rust a colourful makeover. Different perspectives show now different colours dominating, yet I'm not too sure yet how long it'll survive outdoors.

The great outdoors

With the formats and materials I used so far, my sculptures suit indoors much better than outdoors. Yet bamboo and nylon can withstand outdoor conditions, it just needs some more considerations.

I was a bit surprised when I brought a larger structure to the market on a rainy day. It lost considerable amount of tension, and also its delicate balance. Some similar happened when I spray painted a larger structure and left it outdoors for drying, on the next day it had gone into a dangerously floppy state.

My last experiment involved a 4-strut tensegrity, which basically could be balanced upside-down, with one strut fixed into the ground, and three struts floating in tension. The model reaches about 160 cm up, which gives me a bit of leeway for the tension. The first build felt okay, could be handled without disintegrating and lots of movement throughout.

I installed just before a rain storm broke out, with some significant winds. The next day, I found it still in place, yet the top three struts had folded down. The rain must have allowed the strings to stretch more than a healthy amount, although no connection become undone, the overall tension didn't suffice anymore.

With the strings still wet, I simply tuned the model by looping the strings in their grooves, taking care that the overall symmetry wasn't gone. Only 12 strings are needed to keep the 4 struts together, but I wished I had more than two hands while I tried to give it more sturdiness.

I sealed the top of the struts with a glue gun once I was happy with the overall tension, and gave it another go.

New life (in a mist of breath in a cold night)

The Melbourne weather brought a bit of sunshine the next morning, so I could finally hope for a shot in daylight. After straightening the structure in its base a bit, I was quite happy with the result. The visibility isn't too great, but I guess the sun will bleach the struts which currently blend into the background a bit.

I can only hope that the council won't rip it out too soon, it's along my favorite unicycling route so I have a fair chance to keep an eye on it for some longer. It's very accessible, and I rather have it taken by a flooding Merri Creek than over-eager council worker or some destructive neighbours. Time will tell.

New life (in an old Willow tree)

Homage to Pars

When I came across the website of Marcelo Pars, a Dutch tensegrity artist, I knew that my explorations wouldn't find an end soon. The site looked a bit different at the time, but had already some fantastically inspiring pieces on display. I had the impression that he builds models on a larger scale than I do, and much more with 'redundant' struts that I explored so far.

Besides exploring basic geometry, he managed to give his objects a solidity in appearance (and potentially in mechanical behaviour) that eluded me. I totally enjoy the ethereal appearance of some of my own structures, and managed to resist the call for more solid struts.

Parsed Rastafarian (12 strut tetrahedron)

While I consider the tetrahedron (in some educated believe in Fuller that the tetrahedron comprises the smallest unit in universe) a very appealing shape, most people exposed to my work prefer the icosahedron when it comes to 6 strut structures.

Parsed Rastafarian (sitting on a tetrahedral face)

While playing with the Java tensegrity viewer by Bob Burkhardt, a pioneer of constructing tensegrity structures, I realised have easy it should be to build a 12 strut tetrahedron like Pars did. I had a small colourful tetrahedron lying around, and decided to transform it. 

Parsed Rastafarian

The tetrahedron 'proves' that one and one add up to four, and it hides 3 pairs of orthogonal edges in it. Three colours suffice to show this twisted pair of edges. While I love many visual aspects of the small version, it's tricky to balance it on all corners, and the relative large diameter in relation to the length of the struts brings the struts nearly into contact.

Homage to Marcelo Pars (12 strut tetrahedron)

The proximity of the struts could easily be changed by upsizing. The corners are held together in a single spot, I'm tempted to connect the corners to the center. Well, as the first emanation with 60 cm struts already has quite a large prestress, I might delay this idea until the next build. 

Homage to Marcelo Pars

The larger build unveiled the space in between the intricate weaving patterns in the center of the sculpture. I'm still hesitant to play it hard, I had some accidents during the build and it feels so taut that I fear to break some of the flimsy struts used.

Genesis revisited (4 strut x-modul)

My attempts to scale the classic 4 strut x-module up didn't succeed. The 30 cm strut version, very taut with only nylon string, can easily be held in balance by slotting into a relatively heavy base. I hope my lungs didn't take damage from drilling the fibreglass base...

Double plus good

Using two colours for the tendons allows to show the 'centre' and 'periphery' of the x-module. Using two colours for the four struts will enhance the polarity of the struts, could be fun be find out how a chain of x-modules behaves. I did this before, very early in my tensegrity exploration, certainly worth doing again.

Pented up

The small 30 strut dodecahedron I had lying around didn't really invite for playing, so I thought in combination with a basis its wobbly qualities come into good use.

Next big thing

I think I got sufficiently mad during the last week to call myself an artist. I upscaled to a degree that my calculations were quite off the target, and a lot of re-adjustment was required. Three structures with 95 cm struts emerged. The first one, attached to its current destination in a stormy night before the rain hit, still survives the wild weather. The second structure didn't stay for 24 hours in the wild, the latest one still needs some painting and still blocks some of my lounge.

While the tetrahedron exposes some efficiency in the use of materials (roughly a 1:1 ratio of string and tendon), the six-strut icosahedron seems wasteful. The initial string/tendon ratio was about 2.7:1, and as I was running low on nylon strings, I wanted more bang for my bucks. I can't really prove in any fashion that in order to create a tension triangle you can equally to from the corner along its sides or to its center, and the central joints introduced another additional node in the network of strings, but in practise it works just splendidly.

The six-strut icosahedron is a strange tensegrity, connecting the 12 corners of the icosa in a neatly way highly symmetrical through its center. In a way, the minimal 3 strut tensegrity can be understood as minimal octahedron. I better test this experimentally :) Being able to reduce to string/ratio to 1.5:1 made the idea of larger icosas much more viable, and I can't stop experimenting with the result.

I hit a sweet spot with the tendon lengths. Any three struts will balance (all 20 faces), drop and squeezing tests showed a lot of robustness, and the wigglyness can be hypnotising. I won't test it to breaking strength without camera, but I love the options offered by this design in a larger scale. It's simple to suspend small scale objects in the center, some sort of generic 'picture frame' to showcase more complex models.

I wonder whether I can use a set of centrally joined tendons to build from the scratch.

Do or die

2012 started suboptimal when it comes to market appearances. Without the more complex sculptures, there's less eye catchers on the table. Who cares, I'm basically trying to sell art, so can't really analyse my success in business terms.

On the other hand, the lack of sales means I can come up with new projects what to do with tiny supply of rods I still have at my disposal. The source has dried out, just after I got a better idea how to work with the slightly heavier material than usual.

However, I experimented with a different topology for the tendons in the corner. Instead of connecting the tendons between neighbouring struts, I used a little ring to join them in the center of the corner. The struts have now an individual degree of lateral movement, and the cube happily balances on each corner. I also build an octahedron and the Vector Equilibrium in star formation instead of using loops for the corner.

As I enjoyed the different movement patterns with centrally joined corners, I went a bit bigger, attempting a hanging installation for outdoors. It survived already some strong gusts in its test location, it behaves quite nicely in windy conditions. The structure tends to rotate slightly out of the wind pressure, and doesn't swing itself up easily.

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.

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.

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.

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.