3. Infinite double layer structures

To go one step further towards the Leonardo grid space frames I first tried to find a way to construct infinite double layer structures. Space frames can be built by connecting polyhedra in systematic way. Cubes stack so as to fill space completely. And when you look at the graph that represents the cube you will notice that all vertices have degree 3, which was a condition for the Leonardo grids. A cubic frame can be made as a Leonardo grid construction in three different ways (Figures 12, 13 and 14; only in Figure 14 the ends point to the X, Y and Z-direction).
Figure 12: Cubic frame - A
Figure 13: Cubic frame - B
Figure 14: Cubic frame - C
A way to make a double layer structure is to connect these kinds of cubes as in Figure 15. This will result in a non-planar infinite construction that has also dynamic properties. The elements can slide between certain boundaries and the total construction can be pressed together or stretched.
Figure 15: Double layer structure
Figure 16: Pressed together
Figure 17: Stretched
4. Rings and strings

In the double layer structures the basic elements, the rods with the 4 connecting points, are linked together to form bigger units. We can distinguish two kinds of these bigger units: rings, a closed concatenation of a finite number of basic elements, and strings a (open) concatenation of an infinite number of basic elements. Some examples of both categories are shown in Figures 18, 19 and 20 (rings) and Figures 21, 22 and 23 (strings).
Figure 18: Base for double layer structure - rings
Figure 19: Double layer structure - rings
Figure 20: Double layer structure - rings
Figure 21: Base for double layer structure - strings
Figure 22: Double layer structure - strings
Figure 23: Double layer structure - strings
And the constructions don't have to be limited to 2 layers as can be seen in Figures 24, 25 and 26. Here an infinite 3D construction is built with one type of string, which is a concatenation of basic Leonardo grid elements.
The question still is whether it is possible to make space frame constructions out of basic elements which are not linked.
Figure 24: Strings 3D
Figure 25: String structure 3D
Figure 26: String structure 3D