# Bee-Cell Structure.mw

Nature, in general, affords many examples of economy of space and time for anyone who is curious enough to stop for a while and think.
Pythagoras (6th Century BC) was reported to have known the fact that the circle is the figure that has the greatest surface area among all plane figures having the same perimeter which is obviously an example of economy of space.
Heron (2d Century BC) deduced the fact that light after reflection follows the shortest path, hence an example of economy of space.
Fermat (1601-1665) was seeking a way to put the law of light refraction under a form similar to that given by Heron for the light reflection but this time he was looking for an economy of time rather than space.
Huygens (1629-1695), building on Fermat finding, considered the path of light not as a straight line but rather as a curve when passing through mediums where its velocity is variable from one point to an other. Hence economy of time.
Jean Bernoulli (1667-1748) he too was building on Huygens concept when he solved the problem of the brachystochron which is based on economy of time.
This is not to say that nature economy is concerned with only physical phenomena but examples taken from living creatures abound around us. To take one example that many researches have examined very carefully and in many details in the past is that of the honeycomb building in a bee hive.
It turns out, as we shall soon prove, that the bottom of any bee-cell has the form of a trihedron with 3 equal rhombi (rhombotrihedron) which, once added to the hexagonal right prism, will make the total surface area smaller resulting in economy on the precious wax that is secreted and used by worker bee in the construction of the entire cell.
Our plan in this article has a double purpose:
1- to prove the minimal surface we referred to above using a classical proof.
2- To start with no preconceived idea about the bee-cell then
A- to consider a trihedron having 90 degrees dihedral angle between all 3 planes.
B- To get an equation relating dihedral angle with the larger angle in a rhombus that, once
solved, gives exactly the dihedral angle of 120 degrees along with the larger angle in each
rhombus as 109.47122 degrees which are the exact data one can find at the bottom end of a
honey-cell.
This configuration which is that of a minimal surface of the cell is the only one that our
equation can give for all 3 planes to have in common these two angles . All others have
different dihedral and larger angles.
I believe that the neat and simple equation I arrived at is somehow original and so far I have no idea if anyone else has found it before me.