He's just preparing the stage, possibly.
He's in the process of "developing", so to speak.
https://www.space.com/30608-mysterious-saturn-hexagon-explained.html
Graphene is just a fancy name for (single-layered) Graphite, which in turn is carbon. So yes, hexagonal structure is what I'd expect to see.
On the pic though one can see rotation which is kind of a giveaway for the saturn option in the anon's "quiz" from above.
you can easily read that up. The O doesn't change structure, but strengthens bonds and changes some other properties.
https://en.wikipedia.org/wiki/Graphene#Graphene_oxide
As for the "greater significance" of hexagonal structure in nature, all I know is that it's the most space-efficient "packing" and is among the best re stability/material ratio. I also remember hexagons to be favoured by nature (in many cases) due to energy constraints (that's why probably the hex structure in that Saturn storm above)
Many other crystalline substances form hex structure (e.g. quartz, h2o-ice), but others prefer diff structures, as e.g. NaCl.
There's 7 systems of crystalline structure, iirc
https://en.wikipedia.org/wiki/Crystal_structure#Lattice_systems