**Q0:** The mathematical principle that prefers *hexagonal* planar networks is Euler's theorem,
$$V-E+F=0,$$
with $V$ the number of vertices, $E$ the number of edges, and $F$ the number of cells. Because every vertex is incident with three edges (a higher coordination number would be mechanically unstable) and every edge is incident with two vertices, we have $2E = 3V$, hence $E/F=3$. Since every edge is adjacent to two cells, the average number of sides per cell is 6 --- hence a uniform network must be hexagonal.

**Q1:** Euler's theorem still allows for curved rather than straight walls of the cells. The optimality condition that prefers straight walls is the minimization of surface area. 

[![enter image description here][1]][1]  
<sub> source: <A HREF="https://doi.org/10.1098/rsif.2013.0299">Honeybee combs: how the circular cells transform into rounded hexagons</A></sub>   

An experiment that appears to be directly relevant for honeybee combs is the transformation of a closed-packed bundle of circular plastic straws into a hexagonal pattern on heating by conduction until the melting point of the plastic. Likewise, the honeybee combs start out as such a closed-packed bundle of circular cells (panel a). The wax walls of the cells are heated to the melting point by the bees and then become straight to reduce the surface energy (panel b).  

  [1]: https://i.sstatic.net/ImiXa.jpg