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!

https://vimeo.com/64244306

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