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.

Heart Space

Once I started understanding the slightly irregular geometry of the Chestahedron, I grew fond of it. Two variations of it now decorate the Neurophysics Functional Movement Centre in Queensland, as well as being used to demonstrate complex processes.

Even without anything suspended from the top, the Chestahedron favours an upright position in relation to gravity. 'Chiming Heart' was sized to fit into a 50cm cube, made of non-elastic string and Tasmanian Oak. It has a Chinese Bell as attachment, introducing more moveable elements to a relative sturdy structure.

'Fiery Heart' needed to be more moveable, even the small weight of a bell (ca. one ounce) put too much stress on a similar structure with elastic strings. I decided to use non-elastic string for the lateral 'corners' to maintain the overall shape after distortion, and to make the non-elastic attachment point part of the tension network.

While I used whatever I could find as attachments so far, I needed now something of decent size, low weight and potentially bell-shaped. A small 6-strut tensegrity tetrahedron fits the bill perfectly, creating pleasing size relations. Some dashes of colour high light this centre, combining the shape for Fire with its colour red.

Meanwhile, 'Mottled Heart' gets taken over a by tomato, rosemary and mustard. The dodecahedron on the roof suffered a bit with the stormy weather lately, but 'Mottled Heart' shows only slight fading of the strings. I wonder whether its attachment will interfere with the mustard underneath, it won't take long before I find out.

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.