Bees and their hives: engineering genius and optimization
The bees build their combs with cells in the shape of a hexagonal prism to place their larvae and store honey. This had already attracted the attention of Pappus of Alexandria who mentions him in an essay titled “the sagacity of the bees.” Charles Darwin studied in his now famous book “The Origin of Species” and describes it as a masterpiece of engineering in minimizing labor and material and a demonstration of the principle of natural selection.
Beyond its undeniable beauty what makes you so special? If it’s economy to maximize perimeter area is well known that the solution is a circle, and to save area to maximize volume a sphere. But in this construction the pieces fit together perfectly without wasting space or Overlap that cannot be achieved with circles and spheres. This is known in mathematics as a tiling and the periodic case is the same geometric shape which is repeated over and over again to generate it.
Front projection of this structure is the regular hexagonal tiling known to all. This in itself is remarkable, because it is only possible to achieve periodic tiling of the plane with squares, equilateral triangles and hexagons and those who accomplish more with less perimeter area to be the case of the regular hexagon, which concluded Pappus motivated by the bees.
Even more complex
But if we go deep into the honeycomb structure we find a more complex and wonderful. The lid is supported by six hexagonal trapezoids and the lower base we have a base of 3 equal rhombuses that allow fit to leave no gaps.
In the study of optimum dimensions for this structure, the mathematical disagreed with bees by a difference of two minutes of arc for many years. Maraldi, who first measured the angles of the rhombuses of the base of the diaper. Another naturalist René Reamur asked the mathematician Samuel Koenig that calculated with the new methods of differential calculus which were the optimum angles, giving that little difference. Finally a maritime tragedy highlighted an error in logarithmic tables that had been used by Koenig and gave it to be corrected right to bees and impeccable solution.
In the wise words of the great Edgar Allan Poe:
“Leaving this land, we come at once to another, in which the bees and birds are both mathematical genius and erudition that daily give scientific lessons of geometry to the scholars of empire”