The question arose whether it would be possible to make real 3D constructions based on the Leonardo grids. Is it possible to make multi layer objects using the Leonardo grids? Or even can we make 3D space frames built out of simple elements just like in the domes? So mathematically speaking instead of planar graphs I had to look for non-planar graphs now.
A first example of a real 3D construction is the double sphere in Figure 6. In this object you can see a sphere within a sphere. Each of the 24 elements is half in the inner sphere and half in the outer sphere. The design was made by starting with two layers of Leonardo grids. The layers were placed above each other in such a way that after cutting each element in two parts, half elements from the upper layer could be connected to half elements of the under layer. The resulting structure again has all the properties needed for a Leonardo grid: each element is connected to 4 other elements in the right way: all endpoint connected to midpoints and all midpoints connected to endpoints. Half of the connecting points are on the surface of the inner sphere, the other half on the surface of the outer sphere. A first non-planar or real 3D result. And surprisingly enough the total structure appeared to be stable.
Figure 7: Three interwoven patterns
Figure 8: Icosahedron - plan
Another approach is to start with interwoven patterns (Figure 7). A simple way to transform a flat pattern into a spherical construction is to use a polyhedron. When the pattern is hexagonal, the net of a icosahedron can be used. In the special case of the three interwoven patterns of Figure 8 the cut off elements of pattern A will be connected to the cut off elements of pattern B when folding the net to an icosahedron. And so with the cut off elements of B and C and of C and A.
The result is real 3D Leonardo grid construction. And now all the connecting points are laying in the same spherical surface. In the pictures (Figure 9, 10 and 11) you can see some variations.
Figure 9: Interwoven sphere - A
Figure 10: Interwoven sphere - B
Figure 11: Interwoven sphere - C
Although this is a good approach it is hard to find a good set of interwoven patterns that can be used for this method.