How Fourier’s transformational thinking still makes a difference


When you listen to digital music, the harmonies and chords that you hear have probably been reconstructed from a file that stored them as components of different frequencies, broken down by a process known as Fourier analysis. As you listen, the cochleae in your ears repeat the process — separating the sounds into those same sinusoidal components before sending electrical signals to the brain, which puts the components together again.


Fourier analysis allows complex waveforms to be understood and analysed by breaking them down into simpler signals. And it’s a shining example of the power and value of intellectual boldness.




As a troubled orphan in France, Fourier was transformed by his first encounter with mathematics. Thanks to a local bishop who recognized his talent, Fourier received an education through Benedictine monks. As a college student, he so loved math that he collected discarded candle stumps so he could continue his studies after others had gone to bed.


As a young man, Fourier was soon swept up by the French Revolution. However, he became disenchanted by its excessive brutality, and his protests landed him in prison for part of 1794. After his release, he was appointed to the faculty of an engineering school. There he proved his genius by substituting for ill colleagues, teaching subjects ranging from physics to classics.


Before he inspired a revolution in science, Fourier helped to trigger one in his native France. He came of age in the ferment of the 1790s and signed up as a committed révolutionnaire français — a decision that almost led to him losing his head to the guillotine during the Reign of Terror that followed the establishment of the First Republic. He joined the army of Napoleon Bonaparte on his invasion of North Africa, alongside dozens of other experts in science, medicine and engineering. With colonial zeal, Napoleon claimed that these intellectuals would help to spread the civilizing values of the Enlightenment.


Fourier worked in Egypt as an administrator, where his efficiency and smart ideas prompted Napoleon to earmark him for a similar position home in France. Back in gloomy northern Europe, Fourier became obsessed with heat and started to apply his mathematical skills to understanding how heat was transferred. He is widely credited as the first scientist to discuss how the greenhouse effect could warm the planet.


Traveling with Napoleon to Egypt in 1978, Fourier was appointed secretary of the Egyptian Institute, which Napoleon modeled on the Institute of France.


One of the most important fruits of Fourier’s studies concerns heat.


Fourier’s law states that heat transfers through a material at a rate proportional to both the difference in temperature between different areas and to the area across which the transfer takes place. For example, people who are overheated can cool off quickly by getting to a cool place and exposing as much of their body to it as possible.


Fourier’s work enables scientists to predict the future distribution of heat. Heat is transferred through different materials at different rates. For example, brass has a high thermal conductivity. Air is poorly conductive, which is why it’s frequently used in insulation.


Remarkably, Fourier’s equation applies widely to matter, whether in the form of solid, liquid or gas. It powerfully shaped scientists’ understanding of both electricity and the process of diffusion. It also transformed scientists’ understanding of flow in nature generally – from water’s passage through porous rocks to the movement of blood through capillaries.


Today, when helping to care for patients, radiologists rely on another mathematical discovery of Fourier’s, now referred to as the “Fourier transform.”


In CT scans, doctors send X-ray beams through a patient from multiple different directions. Some X-rays emerge from the other side, where they can be measured, while others are blocked by structures within the body.


With many such measurements taken at many different angles, it becomes possible to determine the degree to which each tiny block of tissue blocked the beam. For example, bone blocks most of the X-rays, while the lungs block very little. Through a complex series of computations, it’s possible to reconstruct the measurements into two-dimensional images of a patient’s internal anatomy.


Thanks to Fourier and today’s powerful computers, doctors can create almost instantaneous images of the brain, the pulmonary arteries, the appendix and other parts of the body. This in turn makes it possible to confirm or rule out the presence of issues such as blood clots in the pulmonary arteries or inflammation of the appendix. It’s difficult to imagine practicing medicine today without such CT images.


Commemorative plaque at the birthplace of Jean Baptiste Joseph Fourier in Auxerre (France)


Fourier received many honors during his lifetime, including election to the French Academy of Science, but he would surely be delighted that his ideas have endured. Writing to a friend 229 years ago, he lamented his lack of achievement up to that point: “Yesterday was my 21st birthday; at that age Newton and Pascal had already acquired many claims to immortality.”

He succeeded in his fifties. 

In May 1830, he died of an aneurysm at the age of 63.


Today, Fourier’s name is inscribed on the Eiffel Tower. But more importantly, it is immortalized in Fourier’s law and the Fourier transform, enduring emblems of his belief that mathematics holds the key to the universe.



Source: nature.com, theconversation.com



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