Taste in Math: A love letter to the Atlas
The best story about math is:
In the time of the July Revolution in post-Napoleonic France, there lived a boy called Évariste Galois who saw something beautiful and fundamental no one else had yet articulated. As he came of age, Évariste lost his father to suicide and was expelled from the École Normale for political insubordination. He was arrested on Bastille Day in 1830 and sent to prison, where he continued to work on math through depression and desperation. Évariste was released in the spring of 1831. One month later, he was killed in a duel with an unknown opponent. He was only twenty.
The night before his duel, Évariste wrote sixty feverish pages of mathematical notes to his friend Auguste Chevalier. His parting words pleaded that the trusted mathematicians of his day would “give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess”.
Évariste’s letter worked. A grand, intergenerational project began to untangle the ideas of a boy who died young and unheard. It pushed the limits of mathematical convention–people chased mere coincidences until they resolved into patterns that resolved into answers. In 1985, a group of five mathematicians completed the project. They archived the work of the previous two centuries in a torso-sized, fire engine-red book, and decided to call it an Atlas.
Évariste’s idea was that symmetry is a language. He saw that symmetry is a tool to describe and study other patterns. The field of mathematics he originated and named is called group theory. The project that produced the Atlas is called the classification of finite groups.
In group theory, the operative question is “when is doing something complicated the same as doing nothing?” It feels like: turning a hexagonal tile six times until it aligns again, making four left turns on a city grid, collapsing military time into two cycles of an analog clock, braiding someone’s hair to create a memory of a permutation, or learning the orthogonal roll, pitch, and yaw to control a camera gimbal or pilot a helicopter. Group theory is about patterns like this in pure math. Évariste first saw them in the roots of polynomials, but they exist almost everywhere.
The goal of the Atlas project was to find and classify all of these patterns. To do this, they sought the “atomic” ones–indivisible systems from which all others can be constructed. But there’s more nuance than enumerating the periodic table. Many symmetrical patterns work in any dimension, like scaling up a recipe for more dinner guests. The Atlas is more a collection of recipes than of individuals. A recipe for the six rotations of a hexagon doesn’t appear, as they can be constructed from smaller, simpler systems. But you can find orthogonal transformations (the camera gimbal or helicopter controls).
The classification project shaped the life’s work of countless people. The Atlas authors describe how they “culled material from many places, sometimes barely remembered, which might have been transformed beyond recognition by our own investigations or transliterations.” In May 1977, mathematician Daniel Gorenstein told The New York Times that he had been working on the classification problem five hours a day, seven days a week, 52 weeks a year since 1959.
The hardest part was finding the symmetry groups that don’t fit a recipe–misfit creatures that required special attention. The first five were found thirty years after Évariste Galois’ death by the French mathematician Emile Mathieu. Despite steady effort, a century slipped by before the next was found by German mathematician Zvonimir Janko in 1965. Then, three more were found within three years by American Mathematician John Conway, who would lead the creation of the Atlas. All 26 were found by 1985.
Many of these misfit creatures were conjectured and constructed by different people, without overlapping careers, or even overlapping lives. One person proved a group must exist, someone else found out how it works. The Atlas dutifully includes a record of both roles.
In 1975, whilst the Atlas was underway, Jorge Luis Borges’ published There are More Things in memory of horror author HP Lovecraft. In the story, a young man returns to his family estate that has been sold and remodeled in secret. The man sneaks inside, and discovers strange furniture that implies something of the inhuman form of the inhuman creature that now inhabits his childhood home. He flees, but curiosity overwhelms his fear and he cannot help but glimpse at the creature as he runs. The story ends before the reader can learn what he saw. Group theory is like that. From morbid curiosity about strange creatures came the Atlas.
The Atlas’ origin is a beloved proof of why taste matters in math. Theoretical math is prose–all reading and writing. Chalkboard, paper, and pencil. There are rarely calculators or computers. As in all writing, mathematicians make choices under constraints. The constraints of axioms, logical correctness, and proof-writing convention are strict. But the author of a structured sonnet still makes choices. Mathematicians still express a consistent principle or emotion underlying a problem. There are sparse moments to make these tasteful choices in math, so they matter greatly.
The most important choice is which threads to pull on and when to stop. In a field where creating new abstractions and generalizations is productive, there’s a surprising amount of creativity involved in deciding what’s worth your time. It is a matter of taste to care about the classification of groups at all. A group is a particular type of symmetry. Other constructs emerge from a stricter or a more relaxed approach (with similarly bland names like algebras, rings, and fields). There’s no reason that groups are the most important, only that many people find them the most beautiful.
It’s not taboo to admit that math is made for pursuit of beauty and satisfaction. American mathematician Martin Gardner argued that the “great elegance” of symmetry groups was plain motivation to pursue them. In its opening pages, John Conway tells us that he “originally conceived this Atlas to convey every interesting fact about every interesting finite group.” He makes no overwrought effort to define interesting. Instead, he muses that they just kept going “until the groups became either too big or too boring.” And in doubtful cases, “our rule was to think how far the reasonable person would go, and then go a step further.”
It was a matter of taste that the Atlas itself was the right capstone for the classification project. It fills 252 pages published in English in 1985. The works that support it and prove it is complete fill tens of thousands more, written over decades in English, German, Japanese, Polish, and French. The authors knew it was not enough to drip their final findings through conventional journals. Instead, they created something that demands attention.
The Atlas is a huge book. It is bright red and spiral bound. You cannot get a smaller or duller copy. Almost every page is covered with miniscule, pale gray numbers arranged in careful grids. They form hazy patterns from far away–some like a checkerboard, others a soft ombre. Designing the pages was so complicated the Atlas team invented new mathematical systems and new ways of typesetting them.
You’re right that this sounds impractical–the Atlas is not a practical work. I’ve never met someone who needed one for utilitarian reasons. It isn’t just a reference document, its an epitaph.
We know the Atlas’ authors cared about more than practicality because it was an ordeal to publish. The Atlas team kept discovering new things, noticing errors, and changing their mind about layout throughout the typesetting process, much to the displeasure of publishers. The Oxford University Press Production Department continually pasted in small edits.
After seeing the first copies with flimsy paper covers, the Atlas team insisted on something stronger. The Production Department obliged, but neglected to upgrade the ring binding to support a heavier cover. Atlas author Robert Wilson admits that “the result has been a source of embarrassment to us all ever since.” They planned to publish a reprint with corrections in the late 1990s, but Robert Wilson also admitted that “the Production Department still has nightmares about this book, so it may take some time to persuade them.” After 1985, the Atlas was never printed again.
In the summer of 1995, a hundred of the living mathematicians whose work touched the Atlas gathered in Birmingham. They celebrated the Atlas’ 10th birthday with a bright red, Atlas-shaped cake, and ten candles. Perhaps with lingering resentment, the proceedings of this conference to celebrate the big, red book with a big, red book-cake were published in an awful dark teal.
The Atlas authors made captivating choices about which ideas to pursue and how memorialize them. They also had great taste in names. Names are an olive branch to the mathematical reader–an incentive. If you write a methodical proof about something that alights your imagination, then the right name preserves that feeling.
Group theory does this the best of all branches of math. Its nomenclature is a vernacular architecture. No ivory tower of Greek and Roman roots. Instead, there are braids and knots, tilings and twistings, socks and shoes, wreaths and latices, bird-tracks, lamplighters, caterpillars, orbits, chains, and kaleidoscopes.
Consider, the real name for groups that don’t fit an orderly recipe is “sporadic.” But I called them misfit creatures. Mathematician Daniel Gorenstein called them pathologies. Mathematician Robert Griess called some of them pariahs. This should give you the right feeling, no details required. And of all the sporadic groups (the misfit creatures, pathologies, and pariahs of the Atlas), the most eldritch is called the Monster.
The Monster is the crown of the classification project–it comes last in the Atlas. It sits atop the ten thousand of pages of proofs from two centuries of effort. The Monster was first predicted to exist Bernd Fischer and Robert Griess in 1973, but over a decade was required to fully understand it. In 1980, mathematician Martin Gardner wrote that people all over the world “have been struggling to capture the monster.” Gardner’s article was bisected by an ad for French brandy called “the brandy of Napoleon,” which is in poor taste considering the Évariste Galois’ anti-monarchy motivation.
What is the Monster? John Conway’s student Richard Borcherds lamented how hard it is to give an intuitive answer. First, the Monster is an enormous. For groups, we count actions, like how the six rotations and six reflections of a hexagon make twelve. The ~8.08E53 actions that make the Monster are more than the number of atoms in the sun.
Second, the Monster is unwieldy. With our hexagon, its easy to trace what happens when you combine a thirty-degree rotation and a horizontal reflection. In the Monster, similar calculations are labyrinthine. This isn’t just because its large. There are “atomic” groups much larger than the Monster with structures so simple that calculations can be done easily by hand.
Mapping elements of the Monster can be done with computers. But in the early 2000s, a single calculation took 45 minutes. In 2024, mathematician Martin Seysen published a simple and fast version in python. It is terribly bittersweet that so many people who studied the Monster lived almost until the age in which computation became powerful enough to make a difference in their work. Only one of the original five authors is still alive.
To build an intuition for the Monster, it’s best to learn where they found it, which is that it dwells in the symmetries of spheres. It operates on a conceptual structure called the Leech Lattice, discovered by John Leech in 1967. The Leech Lattice is a dense packing of hyperspheres. It is the the imaginary way to efficiently stack 24-dimensional oranges or tennis balls. (In our 3D world, an orange touches at most 12 others. In 24-dimensions, it touches 196,560–called the kissing number).
The Leech Lattice lives beyond the realm of human visualization, but not imagination. It shares some affect with early ambient musician Mike Oldfield’s The Songs of Distant Earth, where a well-dressed man stacks glowing spheres on the cyan salt flat of an alien world. Or Borges again. In his Book of Imaginary Beings, Borges includes a record of Animals in the Forms of Spheres. He traces a lineage of Plato, Kepler, and others who personified the planets and stars of ancient cosmology as living beings with warmth and reason, a celestial zoology where vast creatures “cast aspersions on those slow-witted astronomers who failed to understand that the circular course of heavenly bodies was voluntary.”
The Monster and its fellow pariahs and pathologies created a great tradition of nicknaming. John Conway later named the second-largest sporadic group the Baby Monster, which then became nickname for his toddler son, who would play with his grad students. In 1982, Robert Griess tried to get people to start calling the Monster “The Friendly Giant” instead, but the Monster had already stuck. Conway diplomatically included both in the Atlas.
Without this love for idiosyncrasies, we would never have solved a great mystery of the Monster. In the final decade of the classification project (although no one knew it was so), mathematician John McKay noticed a coincidence. John was working on a complicated problem, but the coincidence was simple. It was merely that two important numbers were one apart: 196,883 is important to the Monster and 196,884 is important in the seemingly-unrelated study of modular functions.
John McKay told John Conway what he noticed, and Conway called it Moonshine. He meant it in the sense of being a crazy or foolish idea. Conway harkened to a A Midsummer Night’s Dream, when Bottom proclaims “A calendar, a calendar! Look in the almanack; find out moonshine, find out moonshine!” to see whether the moon will light their dreamlike play.
“Moonshine” also works because coincidences are a bootleg way of doing math. After all, a few important numbers are going to be close by chance. John McKay was told that his observation was as useful as tea-leaves. Scottish mathematician Eric Temple Bell called the whole endeavor “little advanced beyond mathematical illiteracy.”
Despite this resistance, John Conway formally published a paper on the mysterious connection in 1979. He called it “Monstrous Moonshine” and wrote about the “moonshine properties” and “immaterial moonshine” that he saw. His Moonshine nickname became the conventional way to describe the weird relationships between sporadic groups and modular functions. More people grew curious, and in time, Moonshine became canonized as a legitimate field of math in its own right.
In 1998, John Conway’s student Richard Borcherds finally solved the mystery behind the Moonshine coincidence and won the Fields Medal, the highest honor in math. In November of that year, venture capitalist Wayt Gibbs visited Richard in his Cambridge University office. He wrote that talking to Richard was “unnerving” and that “looking through his eyes, through his work, you can get a glimpse of a whole alternative universe, full of wondrous objects that are real but not physical.”
Mathematicians let a little magical realism in on purpose. Great taste in mathematical imagery matters because it is an act of meaning-making. What cannot be visualized can be visceralized by the right language. It can be personified in a way that shares beauty or mystery or uncanniness or morbid curiosity across generations. The classification of finite groups reflects rigorous, rational work. But the intergenerational persistence and canonized nicknames and absurd physicality of the Atlas are to make good on the last wish of a tortured boy.
I met my best friend because we sat in the front row of the seminar where I first learned about the Atlas. My notes were color-coded drawings and doodles. Hers were margin-to-margin prose in black pen. Not equations–monochromatic transcriptions of the lecture. She doesn’t visualize at all.
For a semester, we worked late into the night together, sometimes for many days on a single problem. We learned how to translate our perspectives, how to convince each other. To chase coincidences together until they resolved into patterns.
Our styles match our mathematical taste. She wears strictly black and all her things are black. I wore colorful, thrifted outfits. We once swapped all our clothes and school supplies for class. She wore my overalls. We tried to predictively ask each others’ questions. Didn’t get anything done because we were in tears laughing.
Our friendship makes a real thing out of an abstraction. Like how Baby Monster, here in the concrete world of physical beings, is John Conway's son playing with grad students in the Princeton math department.
On a drunken evening walk after our final exam, we stood at an empty intersection under the endless Iowa sky and she confided that math was the first thing in her life that felt big enough. Years later, I realized that I had never learnt her eyes are green, which I attribute to our time staring at chalkboards together, looking always in a shared direction, never at each other.
References
John Conway, Robert Curtis, Simon Norton, Richard Parker, and Robert Wilson. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford University Press, 1985, X
Robert Wilson and Robert Curtis. The Atlas of Finite Groups: Ten Years On, Cambridge University Press, 1995, X
John Conway, and Simon Norton. Monstrous Moonshine. Bulletin of the London Mathematical Society, 1979, X
Brie Wolfson. Notes on “Taste.” 2022.X
Lee Dembart. Theory of Groups: A Key To the Mysteries of Math. New York Times, 1977, X
Martin Gardner. The Capture of the Monster: A Mathematical Group with a Ridiculous Number of Elements. Scientific American, 1980, X
Michael Quinion. Moonshine. World Wide Words, 1996, X
Wayt Gibbs. Monstrous Moonshine Is True. Scientific American, 1998, X
Richard Borcherds. What is the Monster? Notices of the American Mathematical Society, 2002, X
Mark Ronan. Symmetry and the Monster. Oxford University Press, 2006, X
Adam Goucher. An attempt to understand the Monster group. Complex Projective 4-Space, 2020. X
Martin Seysen. The Monster has Been Tamed: A Fast Implementation of the Monster Group. Journal of Computational Algebra, 2024, X
Matt Clay and Dan Margalit. Office Hours with a Geometric Group Theorist. Princeton University Press, 2017. X
Jorge Luis Borges. The Book of Sand. Emecé Editores, 1975, X
Jorge Luis Borges. Book of Imaginary Beings. Fondo de Cultura Económica, 1969, X
Math notes
The Monster is not exactly the symmetry group (automorphisms) of the Leech Lattice itself. Instead it is the symmetry group of a vertex algebra associated with the Leech Lattice. Even Richard Borcherds admits that there’s no good shorthand for vertex algebras, other than “rings that don’t work everywhere.” Sadly, my understanding doesn’t go much further than this.
I took a few liberties with my list of good imagery in group theory. Namely from braid theory, knot theory, tessellations of Weyl groups, Twisted Chevelley groups, the socks and shoes theorem, wreath products, spherical lattices, lamplighter groups, birdtrack diagrams, caterpillars from hypergraph theory, Emmy Noether’s chains of ideal rings, and Coxeter’s kaleidoscopic diagramming system.
My favorite group theoretic phrase is the empty word, a formalization of the idea that many things can be the same as doing nothing. Rotating a hexagonal tile six times, or twelve times, or eighteen times, or eighteen forward and six backwards, are all the same as not doing anything at all. All these increasingly lengthy instructions belong to (the equivalence class of) the empty word.
If you insist on being curious about practical applications, check out how group theory is used to benchmark quantum computers