polygon composition with hinges

hello everyone,

i am having a very hard time building a simple structure from equilateral triangles, that have all their sides used as hinges. the idea is to create a composition from these triangular polygons , where i can pick a point and by draging it around, to change the angle between them. i watched and did many many tutorials with grasshopper kangaroo, but the problem in all of them is that they make the polygon become elastic and change the side length, the side length has to remain the same. after so much reading and watching tutorials i start thinking , that simple operation i impossible in grasshopper and kagaroo.

i hope i have described my idea crearly. i posted s structure i build, that should be like an example for the 3d.

i would be glad if someone could help

thank you very much in advance

Replies to This Discussion

Permalink Reply by Ted Fails on May 13, 2015 at 2:28pm

Well, depending on exactly what you are doing, it may not just be impossible in GH, it may be mathematically impossible! Try researching the simplest non-planar case: the tiling of the sphere.

Permalink Reply by Svetoslav Zdravkov on May 14, 2015 at 3:46am

well i don´t know if you saw the picture i posted.. it certainly works with the real model made of matches. the length of each match is not extending for sure. that makes me think, that if it works in reality, it should work using software. first i thought it would be realy easy, couse it´s actually a simple thing...

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P1050747.JPG, 2 MB

Permalink Reply by martyn hogg on May 14, 2015 at 3:56am

The matches might be the same length but the joints are not... it looks like you have about +/-2mm with the way you have joined them at the corners and some are not joined at all.

Permalink Reply by Svetoslav Zdravkov on May 14, 2015 at 4:22am

http://www.archipanic.com/wp-content/uploads/2013/06/woodskin-02.png

Permalink Reply by martyn hogg on May 13, 2015 at 3:54pm

I think what you are trying to do is impossible.

If you take just 6 equilateral triangles joined together to form a hexagon on the XY plane, like a small section from your image above, and tried to pull the centre point upwards in the Z direction then the only thing that can happen is that the 6 triangles fold in half along one of the lines that bisect the hexagon.

Otherwise the outer lines of the hexagon must change length!

If you had a big XY plane covered in equilateral triangles and pulled one point upwards in Z direction then the triangular mesh can fold along a row of edges but to create anything other than a linear fold the lengths of the edges would have to change I think.

You can form non-planar meshes from equilateral triangles but they have sharp kinks in them.

I think your image above are not actually equilateral triangles... there are small differences in the lengths of the sides due to the way you have glued them together. Probably a tolerance of +/-2mm which is enough to give you some curvature. There are also at least 2 edges that are not joined to anything.

Take a look at an icosohedron - a 20 sided shape made up of equilateral triangles. All the points of the icosohedron lie on a sphere. If you split all the triangles of an icosohedron into 4 (i.e. divide each side by 2 and join the dots) and then project these new points out onto a sphere you have a a basic geodesic dome which looks like it is panelled with equilateral triangles but they are actually not equilateral, like your triangles above.

There's been a few posts on this forum where people are trying to panel a surface with equilateral triangles but it's often impossible.

Hopefully someone on here might show you some mathematical proof!

Permalink Reply by Svetoslav Zdravkov on May 14, 2015 at 4:04am

hello,

i understand what you are saying. the glued off matches and the not exact point where they meet do not play a big role. please take a look at this link

http://www.mammafotogramma.it/woodskin/BroschureENG.pdf

the last 2 images show exactly what i would like to do.those are all equilateral triangles

Permalink Reply by martyn hogg on May 14, 2015 at 9:13am

Ah, I get what you mean now. You don't need full freedom of movement on each point. I don't think you can just pull a point because each point does not have freedom to move in any direction.

I 3d printed something to see what would happen... the white triangles are rigid and the dark lines are rubber so it's not quite a true representation as the rubber can stretch but it behaves similar to hinged panels.

As Daniel explains below, I imagine this is very hard to model as the vertices do not have full freedom of movement. The surface is awkward and it's hard to make it go into the shape you want.

Permalink Reply by Daniel Piker on May 14, 2015 at 10:17am

Nice multi-material print.

It's true that it can be hard to make it go into a particular shape - but for more of an exploratory process, pulling it around to see what it can do is quite easy to simulate with the definition below.

Permalink Reply by martyn hogg on May 14, 2015 at 2:08pm

it was done on an Objet500 Connex3 printer.

How do you drag vertices around? I can see the Grab component and it's input is set to true and it says hold CTRL down and drag particles but I'm just drawing rectangles in select mode... What am I missing?

Permalink Reply by martyn hogg on May 14, 2015 at 4:50pm

no worries... i'm on Rhino32bit :(

Permalink Reply by Svetoslav Zdravkov on May 14, 2015 at 4:24am

https://vimeo.com/64244306

Permalink Reply by Daniel Piker on May 14, 2015 at 6:38am

Hi all.

Surfaces composed entirely of precisely equilateral triangles are actually possible mathematically and in reality (without relying on construction tolerances). However, these surfaces will always be a bit, well, crinkly, so to speak (which is often something people want to avoid, but in this case it sounds like it is the desired result).

Also, they can indeed be created using Kangaroo, by simply setting a length goal for every edge:

(This was possible in the old Kangaroo, but such hard constraints work much better with the new solver in version 2)

Such meshes are sometimes referred to as Lobel frames after the French architect Alain Lobel.

I think some of the confusion arises because there have been numerous conversations on this forum where people were asking how to create a triangular mesh with precisely equal edge lengths on a given doubly curved surface, which also in some sense approximates smoothness, and this is generally impossible.

Of course no non-flat triangular mesh with a finite number of faces is ever actually truly smooth, since the individual faces are flat, while the curvature is concentrated at the vertices instead of distributed across the surface. However, by allowing slight variation in the edge lengths these kinks can be made small, and they get smaller as the mesh is refined (as is done for subdivision surfaces), approaching smoothness in the limit.

This isn't possible while keeping edge lengths equal, but interestingly Lobel frames can in some cases approximate slightly doubly curved surfaces, it's just that they have to take on a sort of up-and-down folding pattern between adjacent faces, like origami, instead of the faces lying tangent to the smooth surface like subdivision meshes do.

Also, such equilateral meshes inevitably form spikes at the extraordinary vertices (those surrounded by some number other than 6 faces), and unless the surface you are approximating is close to developable you usually need some extraordinary vertices.

Bearing in mind all these limitations, I still think equilateral meshes have some interesting possibilities and are relatively under-explored digitally, due to a former lack of tools for working with them.

They are also closely related to an interesting class of hexagonal beam structures, as described here:

http://www.geometrie.tuwien.ac.at/pottmann/2014/honeycomb/index.html

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