The π-ful Tower

Katie Steckles

On a recent trip to Paris, alongside the other obvious tourist traps, I found myself by the Eiffel Tower – a globally recognisable landmark, and lasting monument to engineering ingenuity. While sitting in front of the tower, I was intrigued to notice something I had not seen before when looking at photos of the tower: Around the sides, just below the first balcony level of the tower, were engraved a list of names – and some of them were familiar ones.

In case you had never noticed this before, the tower has a list of 72 French names on it – engraved into the tower and highlighted with gold paint, in an unending line running around all four sides. It turns out that Gustav Eiffel, when building the tower in 1887, was keen to make sure it represented not just a frivolous use of resources to create a striking landmark, but that it was also a celebration of the scientific achievements of the people of France from the preceding several decades.

“At the first level, Eiffel decided to paint the names of 72 French scientists who had distinguished themselves since the Revolution in 1789. This ‘invocation of science’, as he called it, reflected his worry over accusations that the tower was useless and wasteful. The names of Ampère, Daguerre, Gay-Lussac and more than 60 others in gilded letters two feet high all around the tower would, he hoped, convince critics that the thing had a nobler purpose than to titillate the masses.”
Joseph Harriss – from the book The Tallest Tower: Eiffel and the Belle Epoque 

While France was obviously not the only country in the world to be enjoying mathematical and scientific advancements during this time, it was definitely a productive period – and the list of names includes many recognisable and distinguished scientists, including engineers, chemists, physicists, geologists and geographers. It contains several names so high-profile they are also units of measurement – including the coulomb, becquerel and ampere.

And as you might expect, the list includes a large number of mathematicians – which is why my interest was piqued. The names I was peering up at were familiar to me from my old mathematics textbooks, many with theorems and objects named for their creators.

So, since it is π day this week (3/14, or March 14th), I thought I would go on a π hunt: to explore the mathematical and scientific discoveries owed to this group that also somehow involve the number π. With such a list, we are limited specifically to French mathematicians (and further limited by the choice of Eiffel to only include men: Sophie Germain would have been a perfectly valid addition). But here I present my discoveries, some of which I found interesting to uncover, and hopefully you will find something new to you as well.

The Tenuous Ones

I will admit that some of my connections to π are more tenuous, so I will get those out of the way first. Since π is the circle constant, but also connected to the measurement of angles in radians, I considered anyone whose work involved angles or rotation – or any other types of calculations which would undoubtedly involve the use of π at some point – to be legitimate inclusions on the list. For example, astronomers like Le Verrier (who discovered the planet Neptune), Delambre and Delaunay would definitely have used π in their work.

There are also scientists like Étienne-Louis Malus, who was a pioneer in the field of optics, which certainly involves angles and trigonometry. Lazare Carnot gave us Carnot’s theorem, which is specifically related to circles; Michel Chasles worked on projective geometry and in particular studied cones and conic sections, so would certainly have come across π at some point. Then there is Gabriel Lamé, who laid the foundations of study for curvilinear coordinates (which include polar and spherical coordinates) and superellipses (a class of shapes based on ellipses).

Louis Poinsot gave us Poinsot’s construction, which is used for studying the motion of rotating rigid bodies (definitely some angles involved there). There is also François Arago, whose eponymous rotations are a phenomenon observed in a rotating disc interacting with a magnetic needle.

More tenuous connections come from Gaspard Monge, who was widely hailed as the inventor of descriptive geometry – the mathematical basis of technical drawing – and the father of differential geometry. Surely you cannot complete a technical illustration without angles coming in somewhere? And there is Jean-Baptiste Bélanger, who was a hydraulic and hydrodynamic engineer, who delivered influential teaching on the subject of mechanics.

Joseph-Louis Lagrange established the field of Lagrangian mechanics, which involves many calculations with angles. Lagrange also studied the three-body problem, and various other astronomical calculations, which would definitely have involved using plenty of π.

I will also throw in the fact that Jean-Charles de Borda worked with angles; although the connection to π is broken slightly, as he actually wanted to decimalise angles – so there would be 100 degrees in a right angle (and time, so there would be 10 hours of 100 minutes per day). I am not sure how that would work with conversion to radians!

There is of course one French mathematician you might most associate with π – Georges-Louis Leclerc, Comte de Buffon gave us the Buffon’s Needle technique for approximating π using probability, which I described in a post back in 2022. Sadly, Buffon was too early for the period Eiffel chose to celebrate, so he did not get a spot on the tower.

The Proper Ones

There are some incredibly famous mathematicians included on the list – notably Augustin-Louis Cauchy. According to Hans Freudenthal, “more concepts and theorems have been named for Cauchy than for any other mathematician.” And he definitely used π: Cauchy’s integral formula has a π in it, as does the formula for the Cauchy distribution.

Then we move on to Joseph Fourier, whose eponymous transforms and analysis permeate so much of maths and science. And there is plenty of π there too – the constant π appears naturally in Fourier series of periodic functions, and the formula for continuous Fourier transforms has a π in it as well. There are even connections via other mathematicians too – the Schönhage-Strassen algorithm is based on Fourier transforms, and can be used to compute the value of π.

Another mathematician with a generous slice of π is Adrien-Marie Legendre – as well as the Gauss-Legendre algorithm for calculating π, there are also Legendre polynomials which can be expressed using formulae that involve π. Legendre also invented the gamma function, and Euler’s reflection formula for the gamma function involves π.

Gaspard de Prony was a prolific inventor – he invented the Prony brake, which has a formula involving π for calculating the power produced by an engine. He also proposed (but was not the first to implement) a device which became known as Kater’s pendulum, and uses a formula including π to measure gravity.

While on the subject of the pendulum, we cannot forget Léon Foucault – as well as inventing the gyroscope, he is most famous for Foucault’s pendulum – and the formula to approximate the period of a pendulum involves π. Another big name on the tower is Charles-Augustin de Coulomb – his work in physics led to Coulomb’s law… which includes π. 

Many of the other mathematicians also have things named after them which directly include the ubiquitous circle constant – Augustin-Jean Fresnel has the Fresnel integral, the formula for which includes π; there is also a π in Ampére’s force law, named for André-Marie Ampère. Claude-Louis Navier is at least partly responsible for the Navier-Stokes equations, which often involve π, especially when talking about periodic solutions. And thanks to Siméon Denis Poisson, we have Poisson’s equation – a partial differential equation used in theoretical physics, which – surprise! – contains π.

Pierre-Simon Laplace is regarded as one of the greatest scientists of all time (sometimes called ‘the French Newton’) – contributions to celestial mechanics, calculus and statistics, among other things. The Wikipedia page for Laplace’s equation mentions π 16 times.

And another name you might recognise is Gaspard-Gustave de Coriolis – as well as being the person who established the correct expression for kinetic energy, ½mv², his famous Coriolis effect takes place in a rotating frame of reference, so angles are definitely involved. And there is another π connection: he published a book on the physics of billiards, which you may not be aware has an intriguing connection to π.

In fact, of the 72 scientists listed on the tower, and of the subset of them who are mathematicians, there is only one for whom I cannot find a specific or tenuous connection to π from: Jacques Sturm’s main legacy was Sturm’s theorem, which concerns how many real roots a polynomial will have – but nothing specifically π-related that I can see. If you can find a connection – to Sturm, or anyone else I’ve missed, let me know in the comments!

The post The π-ful Tower originally appeared on the HLFF SciLogs blog.