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.

Hanging around

Guerilla tensegrities have an unpredictable lifespan. Not necessarily by decomposition, that happened only twice as far as I can say, but rather due to their accessibility. If it's easy for me to install them, it's easy for anyone to take them, whether for their own enjoyment or as part of their paid jobs.

I released 13 objects into the wild so far, with only two or three still in place. I still wonder if my motivation reflects more narcissism or exercises in letting go. However, I cannot regret the experiences I made, the bits of adrenaline rush while doing it, and about how to increase the positioning to appeal at least to me when I cruise through the neighbourhood to check whether they're still in place.

The first object I released took the most effort and excitement, and it lasted for nearly a year. A three strut minimal tensegrity made of pencils, attached to a power wire with an elaborate mounting mechanism. It survived many periods of wild weather, and I found it on the ground after the attachment broke, still intact.

Yet I wonder if it was ever noticed, being tiny and in a place the usual gaze would wonder about to find something unexpected. Two objects of similar size in plain eyesight were gone quite fast, and lately I went to much bigger sculptures for outdoor installations - I ran out of space, and I hoped to increase their visibility.

The first sculpture put up in a place for its splendid visibility didn't even last 24 hours, installed in a stealthy night action it was gone before I could take any photo during daylight. Another one, which I placed in the trunk of cut-off tree, survived some flooding from the creek next to it, and I adjusted it several times for maximum viewing pleasure, is gone now as well.

My attempts to attached some medium sized octahedra on top of wooden poles might have gone with the wind - I found one of it next to the pole after a stormy day, yet I'm sure now that the way of attaching them is simply not viable.

One of the objects that is still happily hanging around, provided me a flood of synchronicities. I met someone living just about 50 metres away from where I hung it out, having one of the most amazing sunday afternoons, a day after it got into place.

Initially, it hung much closer to the branch it's suspended from than I intended. I wanted to give it visibility and freedom of movement, and instead pulled it very tightly to the branch. I still have no idea how it lowered itself down, the elements might have helped me in giving it a more prominent position. One day, instead of being snuggly close to the branch, it had at least the little bit of clearance to rotate in the wind, became more visible, and is now affected by the slightest breeze.

It's now about seven weeks that it's hanging around, and I haven't got a name for it. Hold on, I just found one: Louise, please. As it's along one of my favorite bike paths, I will have ample opportunity to check on its longevity.

Louise, please (6 strut icosahedron)

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)

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.

Wrap up

I hope my thirst for novelty got quenched for a while - I still feel the itching to build something fantastic, yet I'm more than happy that I mastered some structures which posed lots of challenges. What started off with revisiting the x-module, and having some tetrahedral galore, ended with a four tensul (3 strut prism)  multi-coloured tetra with non-elastic string which balances in 20 constellation (four corners, four faces, six edges in two constellations).

Vier gewinnt (12 struts, outlining an tetrahedron)

Although I like the visual effect by using different colours, when I tried to make a digital 3d model out of the structure, the optical continuity of same coloured struts overwhelmed the software, and led to very blurry results. However, when shot against a suitable background the colours create many interesting perspectives, and even without being collapsible, the structure is very playable.

Vier gewinnt

I drafted the model with elastic string, using nylon for the 'base' of each tensul. It was tricky to balance, and untuned itself easily. Once I replaced all tendons, the model had movement as well as balance. I dread scaling the concept up, as the build was quite challenging with plenty of hick-ups on the way.

Clover (6 strut tetrahedron)

Clover isn't really a new structure for me - it's tetrahedron where the struts meet in close proximity at its center, instead creating more clearance and 'central space'. However, in this configuration more symmetry than usual can be seen, especially with the use of different colours. Nylon provides plenty of sturdiness, and with my corner configuration it looked a bit like a 4-leaved clover.

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.

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

Scaling

I used up my supply of recycled struts from my geodesic dome, well, the larger length. Fuller claimed that the ratio of strut to string length remains constant independent of scale - so far, my scaling attempts had quite some errors in it.

As I was curious whether I can use eight centrally joined tendon trinities to build a six strut icosa without any support than my body and dexterity, and furniture nearby. The symmetry of this structure made it so much more elegant (at least conceptually) than most other tensegrity builds. I prepared eight corner joints with three tendons each, based on calculating the string:strut ration from a smaller and larger model.

I even got fancy and used two different colours for the tendons, I hope the different qualities of the string used won't create problems later on. After only one minor hick-up the model came together nicely, with the length I measured being precisely what I wanted according to my calculations. I can imagine that even a bit less tension would provide a stable (not disassembling) model.

The tetrahedron survived some wild storms and still hangs in its slightly hidden space, while three other outdoor installations have gone.

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.

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

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.