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Nature's cheap, cheaper in mathematics

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Nature's cheap, cheaper in mathematics

By Edgard Pimentel

Nature always seems to pursue energy savings. But is this intentional?

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Why is a soap bubble round? Is there a reason why the hive combs are hexagonal? The short answer is: because it is cheaper. As curious as a cubic soap bubble was, its cost would be very high. In the case of the hive, the hexagonal shape minimizes the amount of wax needed to build the structure. These are two examples of facts of nature in which the cheapest alternative prevails.

Another example. Have you ever realized the challenge of applying an adhesive film to a book cover? I have tried several strategies to avoid air bubbles that form. And I failed them all. In fact, the problem results from an agreement between those involved: the book and the adhesive film. The book cover and the film are made of different materials and have different properties. The elasticity, for example, is not the same: each material stretches in a different way. How can two things that stretch differently coexist together? The solution this pair found was to minimize their joint wear and tear. The appearance of bubbles along the contact surface is the configuration that minimizes such wear. In other words, the solution seems to be a combination of more efficient and cheaper.

Now let's think about the primate gestation process. It is considered long. From the moment of fixation of an embryo, it is able to affect the functioning of the pregnant woman's body. At the end of the corridor, to the left of whoever enters, is the evolution and the idea of ​​reproduction in the service of the perpetuation of the gene. Together, these facts suggest the need to select an embryo before it develops. The endometrium comes on the scene: an adverse structure for the ambitious embryo. And what happens? Only the fittest – whatever that means – overcomes these conditions and develops. When no candidate passes the test, the body gets rid of this material. The menstrual cycle is perhaps the cheapest option that nature has found to balance costs associated with pregnancy.

But does nature do it on purpose? It has been said that nature minimizes energy. In other words, that these manifestations result from an intentional process. It is clear that such an intense and beautiful statement is not unanimous. The mathematician Luc Tartar is one of those who refute this idea. He argues that conservation laws would travel through the phenomena of nature. Minimizing energy would be a consequence of these laws. Equally beautiful. Intentional or not, the settings that minimize energy are present in nature. In addition, they appear in mathematics. I'm glad.

Let's look at an area of ​​mathematics, the analysis of differential equations. With many applications in several areas, these equations help us to understand some day-to-day phenomena. The shape of an ice cube in a gin and tonic, for example, or as a population of voters votes, or even when investing in the Stock Exchange. For each of these problems, there is a differential equation.

And if this equation is connected to a configuration that minimizes energy – like the soap bubble – then its study is very simplified and the answers we find can be very fun, as in the case of the ice cube in the drink. Here, it is possible to understand the heat transmission between the drink and the ice, the shape of the “border” of the cube as it melts, and the temperature along this interface.

It is true that we do not know whether nature chooses to save or is minimizing itself on purpose. But we know that these configurations exist. When they appear in mathematics, the cheap is almost free.

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Edgard Pimentel is a mathematician and professor at PUC-Rio

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