2022-06-14 17:46:39
Today we are going to start a new modality of Kilo of Science that I like to title: the Kilo of my teacher. I explain. As some of you know, I maintain a friendly relationship with my Biochemistry professor during my university studies and during my thesis, Miguel Pocoví Mieras. Yes, yes, dear students who can listen to me, it is possible to maintain friendships with former teachers. I know that this is difficult to imagine today, but I assure you that it is possible, and I realize it.
In addition to being an excellent teacher, Miguel is also an excellent disseminator. He has written dozens of very interesting articles on a variety of scientific topics. The articles have been so interesting to me that I have told myself that it is a pity that I do not share them with you and thus also make you part of the teachings of my professor.
The first installment of “Kilo de mi profe”, Miguel Pocoví:
The Sagacity of Bees: The Mathematical Conjecture of the Honeycomb, Thomas Hales and the James Webb Space Observatory.
Miguel Pocoví (17-03-2021)
Constructions made by bees have always attracted the attention of scientists, writers and artists. If we want to fill a plane with identical geometric shapes, that is, to put tiles, doing so with hexagons is the most efficient way and offers the smallest perimeter.
The bees choose this hexagonal shape to build the cells of the combs and in this way they use the least amount of wax possible. The bees store their honey in cells to form a surface without gaps, optimizing the use of space.
Wax making is an expensive process that requires time and high consumption of calories. It is estimated that to make 1 kg of wax, bees, depending on the temperature, must ingest an average of between 4 and 12 kg of honey.
The honeycomb is built by worker bees and is used to deposit honey and pollen. At the same time, the cells are the habitat for the breeding of workers and drones.
Around the year 36 BC, Marco Terentius Varro, in his book on agriculture, wrote about the hexagonal shape of the honeycomb. There were two competing theories to explain this hexagonal structure. One theory held that the hexagons were better suited to the bee’s six legs.
The other theory, supported by mathematicians of Varro’s time, was that the structure was explained by an isoperimetric property of the hexagonal honeycomb and constitutes what we know as the “Honeycomb Conjecture.”
Honeycomb conjecture and its proof by Hales.
The honeycomb conjecture was a conjecture until it was proved, as we will see later, and became a mathematical theorem. The conjecture states: “Any partition of the plane into regions of equal area has a perimeter of at least that of the hexagonal honeycomb mosaic.” That is, the honeycomb is the best way to divide a surface into regions of equal area and with the minimum total perimeter.
There are only three regular polygons that tile the plane: squares, equilateral triangles, and regular hexagons. If we have a square, an equilateral triangle, and a regular hexagon with the same perimeter, the hexagon is the one that contains the most area. Therefore if I want to save material, hexagons are the best.
For the purposes of saving material, two adjacent hexagonal cells are already cheaper than two triangular or square ones. On the other hand, it is impossible to tessellate the entire plane with regular pentagons, heptagons and octagons.
Partly due to the isoperimetric property of the honeycomb, there is a vast literature through the centuries that mentions the bee as a geometer.
over the century XVIII, the mathematical architecture of the honeycomb was seen as evidence of a large teleological tendency of the universe. All this shows that our ancestors were often interested in matters that were not trivial.
The honeycomb problem had never been solved, except under special hypotheses, such as convexity. In 1999 the American mathematician Thomas Callister Hales (1958) submitted for publication an article on this conjecture entitled: “The Honeycomb Conjecture”. After review by experts
the article was published in 2001 in the journal Discrete & Computational Geometry. Hales’s paper provides a proof without the assumption of convexity so the conjecture becomes a theorem.
We have just seen that the hexagonal structure of the cells of the bees in a plane is the ideal one to spend less wax and accumulate more honey. However, if we look at a honeycomb from the front, the lattice of hexagons are only the entrances in a plane, while the study of the bottom of the cells is also very important both to save material and to save wax, as well as to fit into two plans.
The most logical thing was to assume that the cells are simply hexagonal prisms with a closed bottom. But if the cells are observed, the bottom of the honeycomb is not flat, but rather forms a pyramid with three dihedral-shaped diamonds.
These cells fit perfectly into a two-layer system: when three cells are placed together in the same orientation they leave a gap where a fourth cell placed in the opposite orientation fits perfectly.
Therefore, the cells, following this double-layer system, perfectly fill the space between two parallel planes without leaving gaps and forming a highly rigid structure.
Practical applications.
Next we are going to see that mathematics is used for development and does not remain only in numbers, conjectures and theorems. Human beings have known how to use them to convert their applications into research and exploration.
As an example, to illustrate what we have just said, we are going to see an application of this conjecture, and theorem, in the construction of a telescope, the James Webb Space Telescope (JWST), in honor of the one who was administrator of the NASA during Project Apollo. This telescope is expected to be the subject of news this year. The NASA After many delays due to the coronavirus and some technical problems, it is scheduled to launch from French Guiana on October 31, 2021.
The capacity of a telescope is conditioned by the size of the mirror it has, because the objective is to collect the greatest possible amount of light from the cosmos over a surface, a total area without inactive areas. He JWST it is included in a space observatory that will help to discover the mysteries of the cosmos. This telescope requires a primary mirror, which in its case is a beryllium reflector with a diameter of 6.5 meters and an area of 25 square meters. It is so large that it would not fit in any existing rocket.
He JWST It has been built with a set of 18 pieces of hexagonal mirrors that can be folded to fit in the rocket, plus its
honeycomb shape makes it possible for each mirror to fit perfectly on its edges (see figure)
The observatory will detect light from the first generation of galaxies that formed in the early universe after the Big Bang and will study the atmospheres of nearby planets for possible signs of habitability.
Sources consulted.
1. Hales, T. The Honeycomb Conjecture. Discrete Comput Geom 25, 1–22 (2001). https://doi.org/10.1007/s004540010071
2. González Llorente J. Prodigious architects, heroes, billiards and problems of maximums and minimums. The Gazette of the RSME, (2013), Vol. 16 No. 2, Pages 241–269. https://gaceta.rsme.es/abrir.php?id=1143
3. James Webb Space Telescope https://www.jwst.nasa.gov/
4. Räz, T. On the Application of the Honeycomb Conjecture to the Bee’s Honeycomb. Philosophia Mathematica 2013; 21(3): 351–60.
http://philsci-archive.pitt.edu/10918/1/honeycomb_final.pdf
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