[Pete]

Please explain this in slightly more normal english. I think you are saying because of the point of the hexagon less volume is used so it (bee's wax) is a very efficient structure. However, I have been wrong before and I will be wrong again. So, please enlighten me.

That is essentially what I am saying.

Not only is it an efficient structure, it is the most efficient structure. It is straight forward to use calculus to calculate how "high" the point has to be in order to minimize the surface area of a pointed tetrahedral apex. It turns out that bees set the point height exactly to the value calculus tells us it needs to be to minimize the wax used.

[Right Wing Professor]

It turns out that bees set the point height exactly to the value calculus tells us it needs to be to minimize the wax used.

Individual bees can't do calculus - they just do what they're programmed to do. However, evolution can do calculus. If a more efficient structure causes more bee genes to be passed on, eventually evolution will select for the genes that cause the bees to build the most efficient structure - in effect, finding the point at which the slope of the efficiency with respect to each independent structural variable is zero.