PPT Slide
Many of the regular tilings we have been discussing arise naturally. Most people are aware of the hexagonal tiling of a honeycomb and would believe that this is an optimally efficient way to store something. But how did the bees arrive at this optimal choice ? Was it the result of eons of trial and error guided by natural selection or is it the consequence of some fundamental physical principle ? One may never know the answer for certain, but there is a simple physical principle that should serve:
All physical systems prefer to be in their most energetically stable state, called equilibrium.
In fact, though it may seem optimal to most people, a mathematical proof that honeycomb hexagonal tiling has walls with shortest lengths per unit area (as opposed to, say, an irregular tiling) was only given last month !