The Shape of Letters: From Leonardo da Vinci to Donald Knuth: Étienne Ghys on 500 Years of Typography
✒️ The Shape of Letters: from Leonardo da Vinci to Donald Knuth · Étienne Ghys · MAA-AMS-SIAM Porter Lecture (2026)
Étienne Ghys is a French mathematician and permanent secretary of the Academy of Sciences. He studies dynamical systems, geometry, and topology. He also makes beautiful films about mathematics. This lecture is about letters.
📺 Watch here
Not reading letters. Drawing letters. The shape of letters as geometric objects that we spend our lives looking at and almost never examining carefully.
The Central Idea
Ghys opens with a confession. More than 20 years ago, he was correcting proofs of a paper for a prestigious journal and found a letter printed upside down: the dot below the line. He asked the editor-in-chief, Jacques Tits, how this was possible with modern computers. Tits was delighted. He explained that the journal was the last in the world still using Gutenberg-style lead type, and sometimes the machine inverts a character. Then he told Ghys to close his eyes and run his fingers over the paper to feel the embossing.
Ghys was hooked. Tits asked him to take over as editor-in-chief the next week.
That transmission of tradition (the weight of a metal form, the feel of embossed paper, the materiality of letters), is the thread running through the entire talk.
Four Heroes, 500 Years
Ghys picks four moments.
Gutenberg (1445). Not a mathematician, not an artist. A businessman who wanted to print Bibles that looked like handwritten manuscripts. His real contribution was transforming abstract signs into concrete metal objects. A punch carved in steel, one week per letter per size. A matrix struck from it. Molten lead poured in to create type. Upper case and lower case; from the physical position of the boxes. Gutenberg used 290 different sorts for his Bible. Try reading it. You will not succeed.
Leonardo da Vinci and Luca Pacioli (1509). Pacioli wrote The Divine Proportion, a math book about the golden ratio. At the end, an alphabet drawn by Leonardo. Each letter constructed with compass and ruler; circles, lines, geometric prescriptions. The letter A is a diagram. The letter S is already the hardest. The illustrations sit alongside Pacioli’s translation of Euclid, and Ghys notes that in these old books it is unclear whether the pictures explain the text or the text explains the pictures.
Dürer (1525). Artist and mathematician. He wrote The Just Shaping of Letters and drew two alphabets. But he noticed something: Euclid’s geometry is not enough. A ruler and compass produce letters that are correct but not beautiful. The bowls of the B should be fuller at the bottom than the top, as they are when written with a pen. You must sometimes complete the drawing by hand.
Louis XIV and the Romain du Roi (1690s). Politics enters typography. Louis XIV wanted a font that embodied centralized power. A commission of five or six scientists and artists invented the concept of the pixel (a grid of 2,304 squares), and spent ten years designing letters. The result was beautiful but rigid. Every official printer in the kingdom had to use it. The lowercase L and capital I were identical, so they added a small spur to distinguish them. You can still spot it today.
Then Ghys skips a few centuries and arrives at his fourth hero.
Donald Knuth. He wrote The Art of Computer Programming and when he saw the proofs of the second edition, he was devastated by the quality of the printing. Mathematical books did not look as beautiful as they used to. So he did what mathematicians do: he decided mathematics could solve the problem. He created TeX for typesetting and Metafont for drawing letters.
Metafont is a program that draws letters as parameterized curves. Knuth’s fonts depend on 72 parameters. You can change the thickness, the width, the shape continuously. He produced a version of Psalm 23 where the font morphs from Roman to modern across the page. He never succeeded in convincing professional designers, who told him mathematicians know nothing about art.
The most difficult letter was S. Knuth struggled so much that he once joked he should rewrite his book without using the letter S. He published a paper called “The Letter S”, which Ghys recommends to everyone, student or not.
The Mathematics
The technical core is Bézier curves. Pierre Bézier was an engineer at Renault designing car bodies. He needed a way to draw smooth curves using four control points. The formula is a single line: a barycentric combination of four points where the coefficients come from expanding (1-t + t)^3. The result is a curve that starts at A in the direction of AB and ends at D in the direction of CD, with curvature changing continuously; like the transition curves on train tracks, called clothoids.
Every letter on your screen is drawn this way. There is no such thing as a circular O. Every O is a sophisticated composite of Bézier segments.
Variable Fonts
Knuth’s parameterized approach was years ahead of its time. Adobe PostScript, Apple TrueType, Microsoft OpenType; they all converged on the same idea. In 2016, Google joined them to create Variable OpenType, which is exactly what Knuth had done decades earlier. A continuous space of fonts rather than discrete files.
Today you can open a Google Fonts variable font and slide parameters to change weight, width, slant, optical size; continuously. Knuth’s version from the 1980s already did this, and he put the whole system in the public domain.
Political Letters
The talk ends with national stereotypes in type. Bodoni is Italian: look at me. Didot is French: we like rules. Bauhaus letters must be useful; no capitals needed. Caslon is British tradition. Gotham speaks to everyone; used by Obama and Trump in successive campaigns.
Macron commissioned a font called Marianne in 2020. The US government recently switched from Calibri to Times New Roman. A font carries culture. You do not read poetry, history, and mathematics in the same typeface.
Ghys’ closing line: “Letters are the guardians of history. Without letters, we would be nothing.”
The full transcript runs nearly 800 lines and covers much more than I have summarized here: the neuroscience of letter recognition, the pixel grid of the Romain du Roi, the diffeomorphism exhibition in Paris. It is a proper lecture by a mathematician who cares about beauty.
Worth watching. Worth reading the paper on the letter S. Worth remembering next time you type an S without thinking about it.
Crepi il lupo! 🐺