And finally, optimal networks in 3 dimensions.
The DG network
This network is in three dimensions are is composed of two types of beams. It can be shown that it reaches isotropy and optimality and the cross-sections reach a given ratio. The mesh is included in a Kevin cell, which is known to tile the space. No optimal mechanical network have been found before, to the best of our knowledge. More meshes are represented below.
Bonus — A nice 3d network which is only optimal for conductivity
This network has three types of beams and can be viewed as an extension of the Star Network presented earlier. It has the benefit to have beams which are not collinear by pairs are every junctions (except for the central one). The ratios can be tuned so that the network is optimal from the conductivity point of view, but there are many equations for the mechanical optimality. I show more meshes in the next figure, because I think that it’s nice :p.
That’s all folks!