This is a story about a 1,800-year-old game that began with a Greek man living in the second century.
His name was Diophantus. Historical records do not have much information about this figure. Historians have spent a lot of ink debating his origins.
We only know that Diophantus loved to do math. He wrote many math books and some of them have survived to this day.
No one knows why, but Diophantus loved to solve equations with two unknowns. Perhaps Diophantus solved equations for fun, to relieve the boredom of everyday life. There was probably no TikTok at that time.
Who would have thought that 18 centuries later, the math problems Diophantus did for fun would form the mathematical basis for technologies that protect the global Internet. Every time you send an email, chat with friends, or shop online, your information is protected by cryptographic algorithms based on Elliptic Curve Cryptography (ECC). ECC was invented in the late 20th century, but there is a thread that shows everything originated from Diophantus.
Diophantus was interested in questions such as: Find pairs of rational numbers (x, y) such that x2 + y2 = 1. Diophantus' question is equivalent to finding rational points on a circle. In addition to circles, Diophantus wanted to find rational points on many types of curves.
Diophantus wrote a famous series of books called Arithmetica, consisting of 13 volumes, to discuss such questions. The fate of these books is linked to the history of world mathematics.
The Lost Books
The Arithmetica series was kept in the Library of Alexandria. Through many historical upheavals, the Library of Alexandria was burned down many times. Arithmetica was lost, gradually falling into oblivion.
In the next thousand years, Arithmetica only reappeared a few times. Six volumes of the 13-volume original Greek series reappeared in Europe during the Renaissance. Four volumes of an Arabic translation were discovered in 1968 in a temple in Iran. The remaining volumes have disappeared forever.
Arithmetica officially returned to the mainstream of world mathematics in 1572, when 143 of Diophantus' problems unexpectedly appeared in "Algebra" by Bombelli, a professor at the University of Bologna. A friend of Bombelli's had discovered six volumes of Arithmetica in the Vatican Library.
You can imagine that while the Library of Alexandria was burning fiercely, someone tried to save the most precious books. They hastily grabbed six volumes of Arithmetica from the bookshelf, put them on a boat, crossed the Mediterranean, and put them in the Vatican Library, where they lay dormant for the next thousand years.
What could have happened if humans had discovered these books sooner? Could we have developed a thousand years further than we are now? These haunting questions were from professor Dan Boneh at Stanford University, when discussing the preservation of human knowledge.
Fermat's Last Theorem
The six surviving volumes of Diophantus were first translated into Latin in the 16th century. After many different translations, in the 17th century, a translation by Bachet reached Fermat, and then the French lawyer who also did math for fun gave birth to number theory, the queen of mathematics.
It was in the margin of his Arithmetica that Fermat wrote the words that haunted the world for over three hundred years:
It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
This is the famous Fermat's Last Theorem. This theorem is dubbed the golden goose of mathematics, because the efforts to prove it have created many new types of mathematics. In other words, while playing Fermat's game, people came up with many other fun games.
Although it attracted the attention of the greatest mathematical minds, Fermat's theorem was only proved by Andrew Wiles and Richard Taylor in 1995. Interestingly, Wiles and Taylor's proof relies on elliptic curves, which originated from Diophantus. So we have come full circle, from Diophantus to Fermat, Wiles and Taylor and then back to Diophantus. The thread that connects them all is elliptic curves.
Curvy Curves
Arithmetica mainly discusses problems that we consider simple today. But if you read carefully you will see a problem very different from the rest: Problem number 24 in volume IV.
In this problem Diophantus wants to find the rational points (x, y) that satisfy the expression y2 = x3 - x + 9. Today we know this expression represents an elliptic curve. Call this curve E.
It is easy to see six points lying on E: (0, 3), (0, -3), (1, 3), (1, -3), (-1, 3), (-1, -3). Diophantus' question remains: are there any more points?
This is not a simple problem, even by today's standards. But, by a series of ingenious transformations, Diophantus found another point (-17/9, -55/27).
It took 1,500 years for Earthlings to understand the mathematical basis of the transformations Diophantus performed to solve problem 24. By the 17th century, Bachet and Fermat came up with an algebraic formula to "double" a point, while Newton pointed out the beautiful geometric basis of these transformations.
Although Diophantus stopped when he found the first point, his method could help find infinitely many other points. The obvious next question is: does Diophantus' method help find all rational points?
It was not until the early 1900s, under the guidance of Euler, Jacobi, Abel, Gauss, Poincare, and Mordell, that humanity was able to answer this question.
After hundreds of years of research with many breakthrough ideas, today we know that the set of points on an elliptic curve forms a mathematical structure called an Abelian group. Only cubic curves have this property. The set of points on quadratic or higher-order curves does not form a group.
In the 20th century, elliptic curves flourished to become the main research topic of world mathematics. The main branches of mathematics such as number theory, algebra, geometry, and analysis meet at elliptic curves, intersecting to create brilliant results.
One of the highlights is the proof of Fermat's Last Theorem, as mentioned. In addition, the Birch and Swinnerton-Dyer (BSD) conjecture about the size of the group of rational points on elliptic curves is one of the seven Millennium Prize Problems. If you solve it, you will be awarded 1 million USD. I tried and found there might be easier ways to make money.
In addition to theoretical breakthroughs, the late 20th century also witnessed extremely useful applications of elliptic curves. What Diophantus and many mathematicians pursued for thousands of years out of intellectual curiosity, unexpectedly created secure communication methods, making the Internet much more useful and private. The security foundation of Bitcoin and many other cryptocurrencies is also built on elliptic curves.
Langlands' Garden
A mathematician once said, "One can write endlessly about elliptic curves." Although I have only just started playing Diophantus's game, I can also write forever. I was lucky to know this subject more than 10 years ago, initially for work, but now it has become a lifelong hobby.
My idea of a great weekend is lying on the sofa reading an old math book or listening to lectures on YouTube. There are books I have read 10 pages of, for 10 years continuously, fainting on page 11 because I can’t understand.
When I left Google, my colleague, a PhD in elliptic curve mathematics, gave me an introductory book on algebraic geometry. Actually, I already had this book, it is one of my "10 pages 10 years" books. I’ve tried to read it, forever.
Then recently I suddenly understood a little more. Oh my god, I felt elated. I was able to peek through the narrow door into a secret garden. That garden is called Langlands, named after the mathematician who has the vision to unify mathematics, where the Millennium Prize Problems are also just a special case. Few people may know Langlands, but the fundamental lemma that professor Ngo Bao Chau proved was also "grown" in this garden. He won a Fields Medal for this work.
I told a friend about this joy, he said that studying great works like this made him feel humble and calmer in everyday life. Sometimes stressfulness is no longer a problem if you just know where you are.
In addition to learning math for fun, looking back on my engineering career, I believe it's difficult to do a good job without math. I’ve worked at the largest technology companies in the world, and everywhere is full of math.
Recently my work has increased dramatically. In my limited free time, I just want to study math, really just for fun, but many customers want to hire our team, just because we know a little math.
I'm not surprised, because I've seen the silent but intense influence of mathematics in the past 18 centuries. So have you today! Welcome to the game.
If you haven't fallen asleep yet, the next part is a gift.
Why are elliptic curves named that way, when they don't look like ellipses? Interestingly, the answer is related to the godfather of Big Data and orbits of the planets.
The Godfather of Big Data
In addition to Arithmetica, the 16th century also witnessed the return of heliocentrism. Following ancient Greek and Indian astronomers, Copernicus argued that the sun, not the earth, was at the center of the universe.
Today, we take for granted that the Earth revolves around the sun, but Copernicus's astronomical model was not welcomed by his contemporaries. It was not until the early 18th century, with the great contributions of Tycho, Kepler, and Galileo, that heliocentrism was accepted. History has talked a lot about Galileo and Kepler, but few people know about Tycho.
Tycho Brahe was a brilliant Danish astronomer. Born into an aristocratic family, living a rebellious life and devoting his entire career to the stars, he could be considered the Elon Musk of the 16th century.
At the age of 13, Tycho witnessed a total lunar eclipse on August 21, 1560, and was delighted to learn that this phenomenon had been predicted, albeit a day late. Tycho realized that with better observational data, the prediction would be more accurate.
At the age of 16, while observing the conjunction of Jupiter and Saturn, Tycho once again saw many errors in the observational data at that time. He started collecting data himself.
For nearly the next 40 years, Tycho diligently observed and recorded in detail the movements of the planets. He also spent a lot of time improving tools and building observatories.
As a result, Tycho collected a set of astronomical observational data that was many times more accurate than other datasets of the same time. Until the invention of the telescope in the early 17th century, Tycho's data was the most accurate.
It is no exaggeration to say that Tycho was the first data scientist in history, the godfather of Big Data. Today, his name is given to many projects related to data collection and analysis.
Orbits of the planets
Although Tycho's dataset was very valuable, it took another genius to understand what the data was saying: Kepler.
Kepler's life can be summarized by Einstein's remark, "This guy is very smart." Terence Tao commented that if Einstein thought you were very smart, then you were probably doing okay.
Kepler worked as an assistant to Tycho. Tycho saw that the orbit of Mars deviated from the predictions of the models at that time, so he assigned Kepler the task of analyzing the Mars data.
Kepler's work faced many difficulties because Tycho strictly protected the data, not allowing Kepler to copy it to "work from home." Kepler also supported Copernicus' heliocentric system, while Tycho wanted to develop a different model.
It was not until Tycho's sudden death in 1601 that Kepler gained access to the valuable data source. There is an anecdote that Kepler poisoned Tycho with mercury to steal data, but the results of Tycho's autopsy in 2012 showed that Tycho was not poisoned.
From the data Tycho left behind, Kepler discovered three famous laws named after him about the motion of the planets. The first law connects Kepler with Diophantus. Kepler discovered that the orbits of the planets are not circles, but ellipses.
From Ellipses to Elliptic Curves
Ellipses are deformed circles, flattened at both ends, like an oval. If a circle is a set of points equidistant from the center of the circle, an ellipse is a set of points with the same total distance to two foci.
Ellipses are represented by second-degree expressions, while elliptic curves are third-degree expressions. When projected into three-dimensional space, ellipses are equivalent to a ball, while elliptic curves look like doughnuts.
However, elliptic curves are named after ellipses because elliptic curves arise in the problem of calculating the circumference of ellipses.
My first mathematical memory is being taught by my elementary school teacher the formula for the circumference of a circle 2 * π * r, where r is the radius. But I can't remember the formula for the circumference of ellipses.
Do you remember? I guess not, because that formula doesn't exist! To calculate the circumference of ellipses, you have to calculate an integral called an elliptic integral of the second kind, but it’s been proven that there’s no closed form formula for this kind of integral.
Although elliptic integrals cannot be computed, in the 18th century, Fagnano and Euler discovered how to add elliptic integrals. What does that mean? Let the elliptic integral be the function f. No one can calculate f(x) and f(y), but there is a way to calculate z, such that f(z) = f(x) + f(y). How strange!
By the 19th century, Jacobi showed that Euler's elliptic integral addition algorithm was the same method that Diophantus used to solve the equation y2 = x3 - x + 9. From there, Jacobi, Abel, and Gauss proposed to invert the elliptic integral, focusing on elliptic functions, which are complex functions with two periods.
And then Clebsch, Eisenstein, and Weierstrass pointed out that the real problems to be studied were not even the elliptic functions, but the cubic curves that appear naturally when studying the properties of elliptic functions. They called them elliptic curves.
In short: Tycho → Kepler → ellipses → elliptic integrals → elliptic functions → elliptic curves.
Thanks to M and VH for reading drafts of this. Thanks to Dan for sharing this story with me and his endless love for math and cryptography.
References
Dan Boneh: Cryptography: From Mathematical Magic to Secure Communication
Isabella G. Bashmakova: Diophantus and Diophantine equations
Jose Barrios: A Brief History of Elliptic Integral Addition Theorems
Ezra Brown, Bruce Myers: Elliptic Curves from Mordell to Diophantus and Back
Ezra Brown, Adrian Rice: Why Ellipses Are Not Elliptic Curves
Happy to have run across this by accident as I was researching Diophantus! I also enjoy reading math books and am not very good at it. I worked through my college Calculus textbook over the course of 30 years, ramming my head into the wall and making tiny dents, again and again, until I finally finished it.
Amazing post. I learned a lot and thoroughly enjoyed reading it. What is the intro book on algebraic geometry you mentioned?