Mathematical Optimization in Nature: Honeycomb Structures and Sea Turtle Navigation

on two sides of tessellated hexagonal cells where the base is formed by three rhombi. From an optimization point of view, we are in the domain of total wonder. Clearly hexagonal cells are more suitable than round cells since the latter arrangement wastes a lot of space between cells. Square or triangular cells would not have empty spaces, but since the larvae raised in the cells are neither square nor triangular in cross-section, space would be wasted anyway. The bottom of each hexagonal cell has the shape of a pyramid - again a more efficient solution than a square bottom -, and the two sides interface perfectly with each other through these pyramidal bases of the cells. Unlike the combs of some stingless bees, the honeybee comb must be vertical so that honey can be stored on both sides without dripping out, and the comb cells are slightly tilted downward from the opening to the base. In cavity species (Apis mellifera and A. cerana), multiple combs are built in parallel, leaving sufficient space for workers to move freely. This despite the fact that the cavities in which these bee species naturally nest are highly irregular in shape. Beyond intuitive arguments in favor of a two-sided hexagonal structure, it has been pointed out that the structure is indeed a mathematically optimal, or near-optimal, solution for saving on construction material while maximizing storage space. Analysis of the geometry of tessellated polyhedra has shown that the most economical cell construction comprised a hexagonal cell with a base formed by two squares and two hexagons. However, not only would the savings be less than $0.035\%$, at the expense of greater construction complexity, but a subsequent study demonstrated through the use of self-aligning soap bubbles, that at a certain wall thickness, the alternative solution would be unstable and the optimal one would become that favored by bees. In essence, we can say that the structure chosen by bees for their construction is effectively the theoretical-practical optimum. We can therefore ask ourselves who taught bees such a wonderful mathematical structure. We note that any bee, even one born and raised in isolation, as soon as possible, produces hexagonal structure cells while being, when necessary, capable of building pentagonal and heptagonal ones. The bee is therefore not obligated by Nature to structure hexagonal cells, but rather follows this natural inclination. Who then is it that suggests to the bee which mathematical structure to adopt and why do all bees adopt the same structure? The navigation of sea turtles. Knowledge of the extent and course of the routes followed by sea turtles has increased considerably in recent years, thanks to the spread of satellite telemetry techniques, which allow species that breathe air to be followed in great detail. As far as we are concerned, the exemplary case is constituted by the herbivorous green turtle, Chelonia mydas, which usually aims to reach specific objectives during its migrations and therefore, their journeys must be controlled by navigation mechanisms that allow correct orientation and overcome current drift. The perhaps most spectacular case is represented by the migratory habits of some of these turtles towards Ascension Island. Ascension Island is one of the most remote and desolate places on the face of the earth, it is a volcanic peak in the middle of the Atlantic, $8°$ south of...