Do chemical bonds abide by widely accepted geometric principles? To elaborate, it is mathematically proven that triangles are the strongest shape. I don't know what strong materials there are in the world but I have heard of carbon nanotubes. In the case of nanotubes, as well as other carbon based compounds that I have seen, they seem to form hexagonal patterns as seen in this picture:

So when I ask if chemical bonds abide by widely accepted geometric principles, I am asking if I would be right to assume some of the strongest materials would be those that form triangle patterns? Sorry if this question seems dumb and simplistic, depending on the complexity of the answer I may take a larger interest in the subject. ALso I didn't know what tag to use so I listed as soft question but I still desire the nitty gritty if you don't mind (>~<)
 A: Triangles are the "strongest shape" only other things being equal, such as rigidity of sides. To the extent that the analogy works rigidity corresponds to the strength of forces between atoms in a lattice, and that depends on their type. So no, the strength of the lattice is not determined by geometry alone. For example, a triangular pattern would require six bonds meeting at one point. But forces between atoms with this valence may not be as strong as between atoms with valence three which can form a hexagonal pattern.
A: 
Above is a picture of a section of diamond crystal, the hardest naturally occuring material we know and to me that least, the triangle shape is not as evident in this section as it is in any man made structure. The other point that may be worth mentioning is that, as you probably know, a diamond is one giant molecule, whereas I don't think graphene nanotubes can be classed as clearly in their structure.
I am no expert, but I don't think the macro world "building" rules can be extended  to the quantum level. In support of this, molecules with definite triangular "sides", are not classed at the top end of the hardness scale. 
The gas methane for example, has the closest to triangular sides that I know of, but at STP, no structural strength worth considering .
 
