Joseph Fourier and the Sinusoids

Joseph Fourier, linocut, 11" x 14", by Ele Willoughby, 2024

Sometimes people ask to commission a scientist portrait of someone I had already considered portraying. I don't think you can study physics for as long as I did and not think about Joseph Fourier. So when asked to make his portrait, I had a plan in mind.

French mathematician and physicist Jean-Baptiste Joseph Fourier (1768-1830) is best remembered for his work Fourier series, Fourier transforms, Fourier’s law of conduction, Fourier analysis and harmonic analysis and their use to solve heat transfer problems. He is also often credited with proposing the greenhouse effect as early as 1824.

Born in Auxerre, a son of a tailor, he was orphaned at age nine. Recommended to the Bishop of Auxerre, he was educated by the Benedictine Order of the Convent of St. Mark. Being a commoner he could not seek a commission in the scientific corps of the army but took a military lectureship in mathematics. He took a prominent part in promoting the French Revolution in his district. Nonetheless he was briefly imprisoned by the Terror. In 1795 he was appointed to the École Normale and later succeeded Lagrange at the École Polytechnique. After accompanying Napoleon as a scientific adviser on his Egyptian expedition in 1798 he was appointed secretary of the Institut d’Egypte, where he organized workshops for the French to make munitions, wrote several papers for the Egyptian Institute (now Cairo Institute). He returned to France in 1801 after the British victories, to resume work at the École Polytechnique but Napoleon appointed him Prefect of the Department of Isère in Grenoble. In Grenoble he began to experiment on the propagation of heat, and contributed to the comprehensive catalogue Description de l’Égype.

In 1820 he published his theorem on polynomial real roots, that a polynomial with real coefficients has a real root between any two consecutive zeros of its derivative. (This is technically a corollary of a theorem published by Budan in 1807 and 1811, also now known as Fourier’s theorem). In 1822 he succeeded Delambre as Permanent Secretary of the French Academy of Sciences, and he published his heat flow research in his Théorie analytique de la chaleur (The Analytical Theory of Heat) which included his important claim that any function of a variable, continuous or discontinuous, can be expanded in a series of sines of multiples of the variable. The idea is true (with some conditions, later discovered by Dirichlet) and a breakthrough. This is the foundation for what we call the Fourier transform, an invaluable tool in math, physics and engineering. During the 1820s he realized that the Earth, at its distance from the Sun should be colder is it is only heated by solar radiation. He wrongly thought Earth has must then receive significant radiation from interstellar space but he did consider the correct answer that the atmosphere acts as an insulator. He referred to experiments by Saussure who measured temperature in a vase with several inset panes of glass, who noted that the inner chambers (under more glass) become hotter. Fourier noted that if the atmosphere makes a layer like the glass in the vase we would get the same effect on Earth (though he noted that Saussure’s experiment did not account for the convection we get in the atmosphere). This is now recognized as the first conception of what we know as the greenhouse effect.

He became a foreign member of the Royal Swedish Academy of Science in 1830. He experienced heart aneurysms in both Egypt and Grenoble and died 16 May 1830. He was buried in Père Lachaise Cemetary in Paris in a tomb with an Egyptian motif and his name is one of 72 inscribed on the Eiffel Tower.

My portrait (inspired by the portrait by Boilly) alludes to Fourier decomposition: the sine waves sum to the violet waveform framing him. With an infinite series of sines, his very outline, as a function of x (the horizontal axis across the page) could be made. Likewise the circles in the background are another visualization of building up an image of Fourier himself from sine waves. If you trace the perimeter of a circle on a continuously advancing sheet of paper you produce a sine wave. (Mathematicians and physicists often use this equivalency to think of a real sinusoid as the real part of a complex function where a vector formed by by its real and imaginary parts traces a circle around the origin). So a simplified function tracing the main lines in Bouilly’s portrait can be produced by the sum of sine waves produced by tracing the series of these epicycle circles. 

There are a lot of great tutorials about this online, but a particular shout out to 3Blue1Brown on YouTube who not only has an excellent series of videos (it's everything you ever wanted to know about Fourier series but were afraid to ask) but he's also specifically made animations of producing a path that approximates Bouilly's portrait with rotating vectors or Fourier epicycles here.