Have you ever walked through a poetry book? This is what you will experience visiting Leiden. More than a hundred murals with poems compose the literary route, created in 1992 by the Tegenbeeld (Contrast) foundation, in response to traditional urban art centered on pictures. Each piece has been written in the author’s language, including Japanese, Russian, Arabic, Hebrew, Mandarin, Catalan, and 33 others.  

In 2018, Ivo van Vulpen and Sense Jan de Molen suggested adding another language to the Leiden wall poems project: math; the language of the Universe, as Galileo claimed

Loving the idea, Tegenbeeld Foundation designed and painted eight breakthrough physics formulas on the city walls, whose authors worked or studied at the University of Leiden. Each of these equations tells a story of nature. Let’s hike through them.  

A living work of art

Between 1901 and 1913, five Dutch scientists received the Nobel Prize. One of them was Hendrik Lorentz (1902), a leading contributor to special relativity. Lorentz is so famous among scientists that the University of Leiden created an institute in his name. Two of his findings are shown on Leiden walls.

Albert Einstein and Hendrik Antoon Lorentz, photographed by Paul Ehrenfest, 1921. Credit: Museum Boerhaave, Leiden.

Diverting particles

Lorentz force. Credit: muurformules.nl.  
At De Hooykist Café, below the formula, you can order a Lorentz Steak or an Einstein E=mc^2 Burger.  

The formula located at the corner of the Groenesteeg and the Hooigracht describes how electric and magnetic fields affect electric-charged particles. While the electric field (E) accelerates the particle in a straight line, the magnetic field (B) deflects it. This is called: Lorentz force.    

These are real particle tracks detected at CERN. Charged particles create those curls when a magnetic field is present.  

The shrinking effect in special relativity

Lorentz length contraction. Credit: own work.

The second Lorentz’s formula on Leiden walls describes the length contraction of a stationary object observed by a traveler. And vice versa: a stationary observer will see a traveling object shrink. 

To illustrate, picture a soccer ball on a pillar. A traveler passing by will see it deformed like a rugby ball in the direction of its movement, as it was sketched under the equation.  

Why don’t we see anything deformed while going in our cars? The effect is noticeable when the traveler moves close to the speed of light. For this reason, it is imperceptible in our daily lives. Still, it is frequently observed in big particle accelerators and cosmic ray detectors. 

This formula was crucial for the development of Einstein’s special relativity.  

Lorentz was a professor at the University of Leiden and a member of the Academy of Sciences and Arts of the Kingdom of the Netherlands. So brilliant he was that after a conference in 1927, Einstein called him: “a living work of art.”

Measuring time 

Huygens’ pendulum clock formula. Old church of St. Peters bell tower at Plaatsteeg 20, which in 1936 turned into a firefighter training center. Huygens studied law and mathematics at the University of Leiden. Credit: muurformules.nl.  

Christiaan Huygens (1629 – 1695) was one of the most important scientists in history: He came up with a breakthrough description of light as waves, explained Saturn’s rings, spotted its moon Titan for the first time, and observed features of Mars’ surface, among other discoveries.

In 1637, Huygens showed that to calculate the period of a pendulum—how many swings it makes per minute— we do not need to know its mass; it is enough to know how high it is pending. This discovery boosted the knowledge to build clocks. 

Galaxies rotating in a strange way

Credit: muurformules.nl.

In the 70s, American astronomer Vera Rubin observed, astonished, that the Andromeda galaxy was rotating wrongly, challenging both Newton’s and Einstein’s gravity theories. The stars at the spiral’s arms moved as fast as the stars near the center of the galaxy. According to the physics known, the stuff far from the center of a galaxy should be moving much slower —unless astronomers add invisible mass to the calculations: dark matter.    

Jan Oort’s formula on Witte Singel 58 C helps astronomers calculate how stars rotate with respect to the galaxy center since no accepted gravity theory works. Jan Oort was an astronomy professor in Leiden. 

Rotation curve of spiral galaxy Messier 33. The discrepancy between the two curves can be accounted for by adding a dark matter halo surrounding the galaxy. Credit: Dr. Mario De Leo, astronomer.

Gravity according to Einstein

Einstein general relativity equation. Credit: own work

Albert Einstein was an extraordinary professor at the University of Leiden, visiting it frequently to collaborate with his colleagues. In honor of his genius and close friendship with Lorentz, the Lange Sint Agnietenstraat Street mural exhibits his famous General Relativity equation. 

This equation tells us that massive objects deform space and time around them. Consequently, any celestial body bends the light, and, if it is massive enough, for example, a galaxy, behaves like a lens, focusing on the light coming from a star behind it. This phenomenon is depicted in the drawing above the formula. 

The gravity of a luminous red galaxy has distorted the light from a much more distant blue galaxy nearly completing a ring. Credit: Hubble Space Telescope. 

After a hundred years of being published, Einstein’s theory of gravity continues to produce successes. Among the latest: the discovery of gravitational waves. We use general relativity every day, for example, when orient ourselves with a GPS. 

A quantum feature of matter

Goudsmit and Ulenbek’s spin formulas are shown in the Gerecht 13 mural. Credit: muurformules.nl.

Amidst the strange results in experiments that gave rise to the quantum revolution in the first half of the XX century, physicists struggled to describe atoms’ behavior in a magnetic field. 

Samuel Goudsmit and George Uhlenbeck solved the problem by introducing an electrons’ intrinsic property: the spin. The spin became crucial to explain the atomic structure and subatomic particle interactions. Although the spin can only be seen at a subatomic level, it explains many macroscopic phenomena like magnetism in iron, nickel, and cobalt, or the behavior of superfluid helium4. 

Samuel Goudsmit (1902-1928) studied physics at the University of Leiden. After that, emigrated to the USA, pushed by the nazi persecution. George Uhlenbeck (1900-1988) met Goulsmit when he was Ehrenfest’s assistant at the University of Leiden.

Atoms feeling each other

Van der Waals formula. Credit: own work.

In gases, as the pressure increases, so does the temperature, and their volume shrink. So says a handy formula. But that statement assumes gases are made out of sizeless dots that do not “see” each other. Despite its simplicity, it works well in many situations. However, considering that atoms and molecules do occupy space and “feel” each other may be crucial. 

Johannes Diderik van der Waals (1837-1923)  introduced these facts in his equation showcased in Fruinlaan 18. His findings helped explain water vapor condensation. Van der Waals was a student at Leiden University and proposed his equation in his Ph.D. thesis.

On how light bends  

Snell’s law. Credit: muurformules.nl.

When the light goes from a substance into another, it appears to be bent. You can see it looking at a straw in a glass of water. This phenomenon, called “refraction,” occurs because light travels at different speeds in different substances. The slower the light in a substance, the smaller the angle. 

In the XVII century, scientists didn’t know about the speed of light. Even so, Willebrord Snel van Royen (1580 –1626) came up with a general expression describing refraction: Snell’s law.

This formula is located at the intersection of the Sint Jorissteeg and the Hooigracht. Snel van Royen, also known as Snellius, was a mathematics professor at the University of Leiden. 

Light refracted by a plastic block.

Street art is not to be understood but to enjoy. Vulpen and de Molen showed that physics could also decorate the city even though non-scientists may not fully understand its meaning. 

No doubt, the formulas blend beautifully among Leiden’s poems. After all, quoting biologist Richard Dawkins: Science is the poetry of reality. 

I am a science and technology communicator and a bilingual freelance writer —Spanish and English— with a research scientist background (Ph.D. in Physics). I hold a specialization in communication and divulgation of science and technology from the University of Oviedo, Spain (2014). My fifteen years of experience as a scientist include a postdoc position at the Institute for Theoretical Physics at the Leibniz University of Hannover, Germany (DAAD fellow); Professor of physics, tenured, at Universidad Simón Bolívar, Caracas, Venezuela; and Research Associate at the University of Sydney, Australia. For the last three years, I have been the Leader of the non-profit organization Persea Foundation. I joined United Academics Magazine as Editor-in-chief in 2021. My passions are theoretical physics, particle physics, climate change, astronomy, and space exploration.

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