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Not really.

The hexagons in beehives are what happens when bees stack circles as efficiently as possible and remove the excess material.

The angle in snowflakes is due to the 120 angle between the bonds in a water molecule.

Coincidentally, carbon atoms can also form bonds with each other at 120 degrees

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Carbon assumes a hybridized valence structure in molecular orbital theory where it forms four bonds, not the two you would assume from its atomic orbital structure. In the case of graphene those carbons are sp2 hybridized so it makes 4 bonds to 3 other carbons.

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So carbon was the bestagon all along?

Water has less than a 120° angle.

In the case of carbon it's sp² hybridized, other hybridizations like sp³ form angles of ~109.45°, while sp forms 180°.

Something like bor that's sp³ hybridized also can form hexagons due to the same reason. It's simply triangular shaped.

Also also: That's just the average bond angle. Due to temperature it fluctuates.

I would argue it is connected. SP2 hybridized carbon atoms form bonds at 120 degree angles because that is the most efficient packing for electrons around the atomic nucleus.

It isn’t exactly the same, but both situations arise due to a most-efficient-packing problem (and the hexagon happens to be the optimal packing in both cases).

The reason for ice molecules to form a hexagonal crystal structure is related to the bond angles in the molecule but not necessarily the way you think. Ice has a whole bunch of stable phases at different temperatures and pressures, at our common temperatures and pressures, hexagonal is dominant (but not exclusive). It's a gross simplification to say 120 degree bond angle = 120 degree crystals - many molecules have defined bond angles that doesn't really relate to their crystal structure.
i.e. silica in quartz which has a very defined 144 degree bond angle so forms these really clear tetrahedra on the molecular level, but the coarse crystal structure is a kind of skewed square, related to the hexagon. If you increase the temperature the tetrahedral are basically unaffected but the crystal structure changes significantly.

The hexagonal crystal structure is very very common because it is a "close-packed" structure, so it is thermodynamically favourable in a lot of situations, including in some carbon allotropes.

My field isn't really minerals like ice(more of a metallurgist) so sorry if this isnt super clear, maybe a geologist can weigh in a bit more. Metals are a good example of how bond angles aren't too important in crystal structure, they don't form "molecules" as such, so there isn't any define angles between atoms exactly(e.g. In liquid form), but they still have very defined crystal structures as solids

Your question is hard to answer for several reasons, mostly because you do not define what unique traits you mean. Also, it is hard to assess your level of knowledge in chemistry and physics.

I would argue that mostly it is not due to any packaging effects.

As a biochemist, I consider carbon to be unique mostly because of its nuclear makeup, meaning its position in the periodic table. As first member of main-group 4 it is a relatively small, abundant atom that is capable of forming a variety of different bonds. Meaning sp1-hybridized single bonds, sp2-hybridized double bonds and sp3-hybridized triple bonds. On the other hand, its mediocre electronegativity of 2,5 makes it possible for C to bond with a variety of different other elements more or less stably.

These factors allow for carbon to be perfect as base for organic chemistry under terrestrial conditions - meaning the diverse chemistry of life.

"Perfect Stacking" geometries are very common in metals, because metal atoms like to get close to one another and don't care too much about the angles of interaction. Nonmetals like carbon and ice care a lot more about the angles of bonds between atoms, so perfect stacking is less common. This is one reason that metals tend to be denser than nonmetals.

Carbon can be thought of as hybridized sp, as in acetylene where the bond angles are 180, sp2, as in benzene (120) or sp3, as in methane, (109.5).

The oxygen in water has 2 lone pairs and two protons around it so it’s best to think about it as sp3 hybridized. The lone pairs repel a bit more than the protons do, squeezing the bond angle a little bit below 109.5, to like 104.

But yes it’s always flexing and such at room temperature. Also, the protons are constantly exchanging, so any one distinct water molecule has a pretty short “lifetime.”