**Question 0** Are there some mathematical phenomena which are related to the form of honeycomb cells  ? 

**Question 1** May be hexagonal lattice satisfy certain optimality condition(s) which are related to it ? 

The reason to ask - some considerations with famous ["K-means"][1] clustering algorithm on the plane. It also tends to produce something similar to hexagons, moreover, may be, ruling out technicalities, hexagonal lattice is optimal for K-means functional, that is  [MO362135][2] question. 

**Question 2** Can it also be related to bee's construction ? 

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Googling gives lots of sources on the question. But many of them are focused on non-mathematical sides of the question - how are bees being able to produce such quite precise forms of hexagons ? Why  is it useful for them ? Etc. 

Let me quote the relatively recent  [paper from Nature 2016][3],
"The hexagonal shape of the honeycomb cells depends on the construction behavior of bees",
Francesco Nazzi:


> Abstract. The hexagonal shape of the honey bee cells has attracted the
> attention of humans for centuries. It is now accepted that bees build
> cylindrical cells that later transform into hexagonal prisms through a
> process that it is still debated. The early explanations involving the
> geometers’ skills of bees have been abandoned in favor of new
> hypotheses involving the action of physical forces, but recent data
> suggest that mechanical shaping by bees plays a role. However, the
> observed geometry can arise only if isodiametric cells are previously
> arranged in a way that each one is surrounded by six other similar
> cells; here I suggest that this is a consequence of the building
> program adopted by bees and propose a possible behavioral rule
> ultimately accounting for the hexagonal shape of bee cells.

  [1]: https://en.wikipedia.org/wiki/K-means_clustering
  [2]: https://mathoverflow.net/q/362135/10446
  [3]: https://www.nature.com/articles/srep28341