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29 June 2018

The Unpredictable Power of Mathematics

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Many mathematicians understand their discipline as an art, as a way of expressing the beauty of correct and coherent reasoning, and that is the only end they pursue by studying it. Others have as their main motivation the applications of mathematics. But independently of their objectives, the power of numbers and formulas is so strong that, sooner or later, applications appear that their authors could never have glimpsed. This was the case for Fourier, a mathematician who worked under the command of Napoleon Bonaparte and who, without the slightest intention, left behind him the mathematical tools that many years later would revolutionize medicine and communication technologies.

Portrait of Jean-Baptiste Joseph Fourier. Author: Jules Boilly
Portrait of Jean-Baptiste Joseph Fourier. Author: Jules Boilly

2018 marks the 250th anniversary of the birth of Jean-Baptiste Joseph Fourier (March 21, 1768 – May 16, 1830), who studied at the École normale supérieure with great mathematicians such as Lagrange and Laplace and was a professor at the Polytechnic University of Paris. He also actively participated in politics, convinced of the principles of the French Revolution and even came to participate in The Terror, for which he was later arrested. “Actually, I fell in love with this cause, in my opinion the greatest and most beautiful that any nation has ever undertaken,” declared Fourier about his revolutionary ideals

In 1798, he travelled with Napoleon in his campaign in Egypt. He had a close relationship with the emperor, who appointed him Prefect (Governor) of the Department of Isère in 1802 on his return from Cairo. Those were his most scientifically productive years. He considered that “nature is the most important source of mathematical discoveries” and in 1807 he published his first report, On the Propagation of Heat in Solid Bodies, in which he formulated the heat equation.

 Fourier travelled with Napoleon in his campaign in Egypt. Credit: Art Renewal Center
Fourier travelled with Napoleon in his campaign in Egypt. Credit: Art Renewal Center

Mathematics that changed the world

But his fame is due, above all, to the method he developed to solve this equation (now called the Fourier method), which has proved to be very useful, powerful and applicable in many other mathematical and physics theories, so much so that it is now ubiquitous in science and technology and its applications decisively affect our daily lives.

Nowadays, these ideas are used in the algorithms that permit the efficient sending and storage of data (such as images or music) from mobile devices. They are also used to process the information obtained in medical tests such as tomographies (CT or CAT scans), in which an attenuation of the intensity of X-rays is recorded, which thanks to the Fourier method reveal clear images of the tissues they traverse. In addition, his great mathematical achievement is used in sound filters or musical equalizers, and in the interpretation of seismic waves (to obtain information about tectonic faults) and in the spectra of the diffraction of rays (to discern the internal structure of a crystal).

Fourier's mathematical ideas are used in musical equalizers. Credit: Kai Stachowiak
Fourier’s mathematical ideas are used in musical equalizers. Credit: Kai Stachowiak

However, when Fourier developed his ideas, he did it “only” to understand a basic question of the science of that time. He considered the following problem: in a thermally insulated enclosure, what is the temperature at each of its points at each moment of time? The law of conservation of energy (in this case of heat), together with the concepts of specific heat and thermal conductivity, allowed Fourier, with the help of differential calculus, to write the equation that governs this evolution of temperature.

The dreams of Fourier

The resulting equation has the important property of being linear, i.e. if we know several solutions we can automatically generate many others, without doing more than multiplying each one by a number and adding the results (linear combinations) thus obtained. Therefore, Fourier believed that a good strategy to solve the equation was to find a sufficient number of solutions, so that his linear combinations generated all the others. Starting from this idea, he formulated the hypothesis that all function can be expressed as a sum, or integral, of trigonometric functions.

Although in 1812 he received the prize of the French Academy of Sciences for that work, the academics did not fully believe his proposal. And they did not lack reasons for doing so because to rigorously prove the dreams of Fourier, it was necessary to create, now in the twentieth century, the Lebesgue Measure Theory and to improve the differential calculus known at the time.

At the same time, the field of its applications was extended, from number theory (calculation of the number of points of the reticule in a circle) to quantum mechanics (Heisenberg’s uncertainty principle), two examples in which Fourier analysis, also called harmonic analysis, plays a fundamental role. Beyond pure science, its impact extended to multiple technological fields. Although Fourier could not foresee it, his mathematics, in addition to being inspired by nature, have also become a powerful instrument to understand it and to transform it.

Antonio Córdoba and Ágata Timón

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