TITLE: Why do bees create hexagonal cells ? (Mathematical reasons)
QUESTION [52 upvotes]: Question 0 Are there any mathematical phenomena which are related to the form of honeycomb cells? 
Question 1 Maybe hexagonal lattices satisfy certain optimality condition(s) which are related to it? 
The reason to ask - some considerations with the famous "K-means" clustering algorithm on the plane. It also tends to produce something similar to hexagons, moreover, maybe, ruling out technicalities, hexagonal lattice is optimal for K-means functional, that is  MO362135 question. 
Question 2 Can it also be related to bee's construction? 

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,
"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.

REPLY [11 votes]: Isn't it just the 2d sphere packing? If one assumes that the larvae needs a disc of fixed radius to grow up to an adult form and that the bees want to have as many cells as possible then the hexagonal lattice is the optimal one.