Question 0 Are there any 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 the famous "K-means" 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 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.

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.

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" 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 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.

Question 0 Are there any 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 the famous "K-means" 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 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.