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still haven't seen a satisfactory explantion for the pointed tetrahedral apex in the honeycomb where the displacement is approximately 35% of the length of the side of the hexagon (this results in a local minimum on the area). Bees doing calculus just doesn't cut it for me.

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.

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.