Everything Bagel Notes & Sources

Last updated: August 20, 2026

These are notes and sources for the EVERYTHING BAGEL deck. Please note that this page will make minimal sense without the cards. But I am continually updating it (chaotically and out of order) so feel free to check in!

If you have the deck, just hit "Ctrl+F" (or ⌘+F on a Mac) and type “Card 6” (or whatever you need) to jump to the relevant notes! The full bibliography is at the bottom.

A little of this, a little of that. Coming very very soon!

Card Notes

Intro Card

“roofing and siding” (Quinn 2013)

This anecdote is in a piece by Frank Quinn in The Best Writing on Mathematics 2013 called “A Revolution in Mathematics? What really happened a century ago and why it matters today.”

There is so much more I wish I could write about the wonderful, colorful mathematician Paul Halmos. “Roofing and siding” is just the tip of the iceberg. On writing (and probably the first time “Euclid” and “vermiform appendix” have been used in the same sentence):

The English language can be a beautiful and powerful instrument for interesting, clear, and completely precise information, and I have faith that the same is true for French or Japanese or Russian. It is just as important for an expositor to familiarize himself with that instrument as for a surgeon to know his tools. Euclid can be explained in bad grammar and bad diction, and a vermiform appendix can be removed with a rusty pocket knife, but the victim, even if he is unconscious of the reason for his discomfort, would surely prefer better treatment than that. (Halmos 1970)

And later:

The overworked period is no worse than the overworked comma. (Halmos 1970)

Since the end of proof symbol is called a halmos, I will use one in his honor here. ▪️

I mentioned Greeks and the square root of two but made sure not to use the phrase “ancient Greeks,” a tarp that blurs real distinctions between classical Socrates-era philosophizing and the far more rigorous and deductive Hellenistic-era science and math that began around 300 BC. Pythagoras and his crew were aware of the incommensurability of the side of a square and its diagonal but lacked the deductive reasoning (it simply wasn’t invented yet) to say much else about it. See Russo for more. This book blew my socks off. (Russo 2000)

Card 1

“Nathaniel Johnston” (Johnston 2009)

You can still read his 2009 blog post here! Johnston consulted the Online Encyclopedia of Integer Sequences (OEIS) and found that 11630 was the first number that didn’t appear anywhere in the database. How clever is that! (Note: He calls this number “uninteresting” and not boring.)

There’s a neat book called Those Fascinating Numbers by Jean-Marie De Koninck that goes through almost every number and lists something interesting about it. It’s a fun book to have around! I have a copy and the first number that doesn’t have an entry is 95.

Card 2

I never would’ve thought this completely elementary topic we learn about in 5th or 6th grade would turn out to be such a rabbit hole!

There is a quote I wanted to include by Roger Penrose. He writes: “What might it mean to say there are minus three cows in a field?” (Penrose 2005) We take so many math concepts for granted that we gloss over, or forget, how baffling these ideas once were. (And then wonder why people don’t understand that -4 x -2 = 8!) The idea that Augustus De Morgan, the great pioneer of logic and computer science, was still unsure about negative numbers in 1843 is, to me, a testament to this.

Information about De Morgan plus Liu Hui and Bhaskaracharya are revealed in an incredible paper called What’s so Baffling About Negative Numbers? – a Cross-Cultural Comparison by the genius David Mumford. Honestly I think it’s telling that after reading a hundred things, the most insightful, comprehensive and thoughtful work about this little sixth grade topic came from a Fields Medalist, MacArthur winner and Putnam fellow. See the paper here.

“as early as 1000 BC” (Mumford 2010)

“as late as 1843!” (Mumford 2010)

Mentioned in the footnotes of this card are two additional sources: Roger Penrose’s The Road to Reality and Jiuzhang Suanshu (The Nine Chapters on the Mathematical Art). On the latter, Harvard mathematician Oliver Knill provides these additional papers (both linked as PDFs) as teaching materials:

A Classic From China: The Nine Chapters, by Professor Randy Schwartz

The Suàn shù shū ┝ᆚए ‘Writings on reckoning’: A translation of a Chinese mathematical collection of the second century BC, with explanatory commentary by Christopher Cullen

Card 3

“definitely true or definitely false”

The study of logic begins with statements. A statement is a sentence or mathematical expression that is either definitely true or definitely false. (Hammock 2018)

Card 4

"The ancient Greeks ran into” there’s a rumor that irrational numbers were so upsetting someone was drowned over this! (Newman 1956)

“Eudoxus of Cnidus” (Newman 1956) Almost everything we know about Eudoxus comes by way of Euclid. Book V of The Elements, which concerns the theory of ratios (what I called Eudoxus’s workaround to irrationals), is almost entirely inspired by (or written by?) Eudoxus.

“Richard Dedekind” (Newman 1956)

This is one of those topics that’s so big it felt impossible to write something meaningful in 90 words or less. Many cool things were left out, like the Babylonian tablet YBC 7289 which contains an estimate of the square root of 2 to six decimal places! There is so much to say about Richard Dedekind, the legendary mathematician who came up with his theory of irrationals while teaching at a technical high school. (Newman 1956)

Also, it’s hard to overstate how unsettling irrational numbers were — and for how long. Here’s E.T. Bell in 1937:

As not one of these three roots can be extracted exactly, no matter to how many decimal places the computation is carried, it is clear that the verification by multiplication as just described will never be complete. The whole human race toiling incessantly through all of its existence could never prove in this way that √3 x √2 = √6. (Newman 1956)

But people! Let’s not forget cows.There was a recent article about burping, methane, and cows and it included startling gems like :

Ruminant livestock, including cows, goats, and sheep, have burped out about a third of all methane gas in the atmosphere. (Starling 2026)

So the gut of the cow is a fascinating place ruled by these rumen microbes that are making cows burp up and/or fart out all this methane and — well maybe that’s the square root of cow?

Back to math (sorry cows), here’s an anecdote about Nobel Prize-winning phycisist Paul Dirac and his path to the relativistic wave equation for the electron:

As the story goes, in 1927 Niels Bohr asked Paul Dirac “What are you working on Mr. Dirac?” to which Dirac replied “I’m trying to take the square root of something.” (Arkinstall 2017)

Card 5

DISCLAIMER: Card 5 has a very long unfocused and rambling note because this topic is enormous and very interesting!

A postulate is a statement we can assume is true without proof. Euclid starts The Elements with 23 definitions of things like circles, points and lines. He then writes five postulates — statements he can safely and logically gather from these definitions without proving them. Here are the first four postulates (Euclid/Thomas L. Heath, The thirteen books of Euclid’s Elements, 135):

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance.

4. That all right angles are equal to one another.

Here is the fifth:

5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side which are the angles less than the two right angles.

There is an easier-to-understand restatement of the fifth postulate called Playfair’s Postulate:

Through a given point only one parallel can be drawn to a give straight line. (Heath 1956)

Back to Euclid. Right away, something is different. This postulate is long and not at all obvious. But Euclid included it as if it was! Was it obvious to him? Did he add it on a wing and a prayer?

It’s a very strange statement. It’s a blot. Because it’s a leap of faith unlike all the other postulates. (Gray 2009)

When we consider the countless successive attempts made through more than twenty centuries to prove the Postulate, many of them by geometers of ability, we cannot but admire the genius of the man who concluded that such a hypothesis, which he found necessary to the validity of his whole system of geometry, was really indemonstrable. (Heath 1956)

It turns out the parallel postulate is true in Euclidean geometry (the flat paper geometry we learn in school) but false in other geometries. These other geometries were discovered by Gauss, Bolyai, Riemann and Lobachevsky, not in that order. I have to add a note here by Roger Penrose:

It is, however, the conventional standpoint (somewhat unfair, in my opinion) to deny [Heinrich] Lambert the honor of having first constructed non-Euclidean geometry, and to consider that (about half a century later) the first person to have come to a clear acceptance of a fully consistent geometry, distinct from that of Euclid, in which the parallel postulate is false, was the great mathematician Carl Friedrich Gauss. (Penrose 2005)

“over 1000 books” is from page 59 of a terrific book called The Poincaré Conjecture by Donal O’Shea. This book also mentions the second most-read book and Lincoln. (O’Shea 2007)

“bouts of insanity” people the world over really went insane over the parallel postulate and I wish I had more room on the card to elaborate. János Bolyai was one of them. János’s dad, Farkas Bolyai, was a genius. (He was friends with freakin’ Gauss.) Farkas decided hey — let me make my son János an even bigger genius! (János knew calculus, analytical mechanics and several languages by 13. (O’Shea 2007)

Farkas, the dad, worked on the parallel postulate with Gauss but it was János who got really obsessed. Here’s what Farkas wrote to his son:

I implore you to make no attempt to master the theory of parallels; you will spend all your time on it…Do not try…either by the means you mentioned or any other means…I passed all through the cheerless blackness of this night and buried in it every ray of light, every joy in life. For God’s sake, I beseech you, give it up. Fear it no less than sensual passions, because it too may take all your time, deprive you of your health, peace of mind and happiness in life. (O’Shea 2007)

A quick note about Euclid: It’s true that very little is known about him. Lucio Russo, in (the mind-blowing book) The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had To Be Reborn writes:

Euclidean geometry has remained throughout the centuries the framework for basic mathematical teaching. But Euclid himself has been taken out of history. In his case the mechanism is opposite the one used for Archimedes: instead of being depicted in legend and in anecdotes, he is offered to us without any historical context, laying down “Euclidean geometry” as if it were something that had always been there at mankind’s disposal. If you are not convinced of this, try asking your friends what century Euclid lived in. Very few will answer correctly in spite of having studied Euclidean geometry for several years. (Russo 2000)

Lastly, If you want a masterpiece treatment of the parallel postulate in the broader context of logic, read Logicomix, An Epic Search For Truth. The parallel postulate is on page 70.

Card 6

Not to pile on trapezoids more, but triangles are even mentioned in Plato’s Timaeus in relation to the human body:

[Consider] the young constitution of the whole animal, which has the triangles of the elements new…Since the triangles coming in from the outside, which make up food and drink, are older and weaker than its own triangles, it overpowers them and cuts them up with its new triangles, making the animal grow by nourishing it with many similar elements. (Russo 2000)

Also, when Archimedes was trying to find the area of a parabola bounded by a line (Quadrature of the Parabola), he did so by adding up the areas of infinitely many triangles:

This example makes it clear why Hellenistic mathematicians laid out with great care such simple theories as that of triangles, presented in the Elements: they were useful tools for tackling even problems whose original statements had no connection whatsoever with the auxiliary theory. (Russo 2000)

Triangle tidbit: One way to find the area of a triangle is through its base. The first use of “base” in Euclid’s Elements was in Book 1 Proposition 4:

Here we have the word base used for the first time in the Elements. Proclus explains it as meaning (1), when no side of a triangle has been mentioned before, the side “which is on a level with the sight” and (2), when two sides have already have already been mentioned, the third side. (Heath 1956)

Proclus wrote the commentary for Euclid’s Elements.

Card 7

“a closed loop in space that doesn’t intersect itself”

A knot is just such a knotted loop of string, except that we think of the string as having no thickness, its cross-section being a single point. The knot is then a closed curve in space that does not intersect itself anywhere. (Adams 2)

There are so many cool things in this book. I wanted to include a bit on Reidemeister moves, but feel free to explore them if you’re curious!

For so much more, including a periodic table of knots, dive into this brilliant column by David S. Richeson in Quanta Magazine called Why Mathematicians Study Knots.

Card 8

“in one study” (Boaler 2016)

“packing problems” The fact that hexagons are the best possible shape to fill a space was proved by Thomas Hales in 1999! (Hales 1999)

Card 9

“Flip a coin” This has been done several times and in several places but here is a nice example of the coin flip exercise from MIT’s Course 8.334, Statistical Physics of Fields. For further dive into stochastic processes and Brownian motion (including an aside about pollen grains) here is Dr. Peter Kempthorne, Lecture 14, Stochastic Processes II.

Card 10

“Kepler dedicated” (Newman 1956)

“took a voyage”

“My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto Astronomy, viz. the Logarithms.” — Henry Briggs (Newman 1956)

“was wishing he’d hurry up”

The Danish astronomer looked for an early publication of the logarithmic tables; but it was long before they were completed. Napier, in fact, was slow but sure. (Newman 1956)

And here is the link to the James Tanton video in the footnote! Tanton also has an 18-page logarithm PDF bonanza here.

Card 11

We return to David Mumford’s paper (everyone should read this and he deserves all of his awards) for an example of Girolamo Cardano trying to figure out a real-world example of negative square roots:

The dowry of Francis’ wife is 100 aurei more than Francis’ own property, and the square (?) of the dowry is 400 more than the square of his property. Find the dowry and the property (Mumford)

Mathematicians thought, then, that imaginaries, though apparently uninterpretable and even self-contradictory,  must have a logic. So they were used with a faith that was almost firm and was only justified much later. Mathematicians indicated their growing security in the use of √-1 by writing “i” instead of “√-1” and calling it “the complex unity,” thus denying, by implication, that there is anything really imaginary of impossible or absurd about it. (Newman 1956)

On Ψ and the necessity of complex numbers, I refer you to MIT professor Dr. Barton Zwiebach, who goes on to show how Ψ (the wavefunction) being a real instead of complex results in a contradiction in quantum mechanics!

Card 15

This card mentions bona fide math genius Terrance Tao. He keeps a blog here that everyone should read.

Euclid proved that primes are infinite in Book IX Proposition 20 of The Elements. Note that he does this “without ever dealing directly with infinity by reducing the problem to the study of finite numbers.” (Russo 2000)

Card 17

There is a quote I wish I had room to add on this card and it’s by Tom Sherrington, author of the great teacherhead blog and also of the book The Learning Rainforest: Great Teaching in Real Classrooms. On one of his Great Lessons posts, he writes:

I’d suggest that the most important diagram in the universe is the number line.  The key to good numeracy is a strong mental model of numbers in sequence and scale. I’ve often found that people with weak numeracy skills have a poor foundation at this basic level.  Before we get into complex operations, just having a really good feel for number is vital.  Having an intuition that 0.6 is less than 2/3 or that 3/4 is bigger than 0.7 – and so on come – from a good visual map of numbers in scale and sequence. (Tom Sherrington, Great Lessons 6: Explaining, 2013

I agree with him and really wish I had space on the card to include that quote!

Card 19

Frege published his Begriffsschrift in 1879, “perhaps the most important single work ever written in logic.” (Gray 2009)

Card 24

“the night before he was killed in a stupid duel”

The term group was first used in a technical sense by the French mathematician Évariste Galois in 1830. He wrote his brilliant paper on the subject at the age of twenty, the night before ht was killed in a stupid duel. (Newman 1956)

“a big part of quantum mechanics”

The mathematics of groups consisting of a finite number of elements goes back to the French mathematician Évariste Galois who…used them to show the impossibility of finding a formula for the solutions of polynomial equations of degree five or higher. [Note — see Card 30.] These finite groups and their representations were studied during the late nineteenth century, a period that also saw the formulation in 1873 by Sophus Lie of the definition of a Lie group. (Woit 2006)

Now Hermann Weyl, a physicist who considered himself a mathematician first, was working on something called the representation theory of Lie groups. I mention that he was a mathematician first because his book Group Theory and Quantum Mechanics was so math-heavy that most physicists couldn’t follow it. Wolfgang Pauli, winner of the Nobel Prize in 1945, called it die Gruppenpest— the plague of group theory:

For all his brilliance and good intentions, Hermann Weyl had discouraged a fair number of physicists from studying group theory. (Woit 2006)

“the monster is a very big group” it is technically the largest of all sporadic finite simple groups. (Gowers 2008)

Card 28

A fun bit about large numbers: Immanual Kant argued that all you need for proofs is your intuition. Frege said something like how do we know that 123,456,789 + 987,654,321 = 1,111,111,110? Not by counting dots! You can’t use your intuition for things like really big numbers. You need rules. (Gray 2009).

What Archimedes did trying to count the number of grains of sand that would fit in the universe is nothing short of jaw-dropping.

Card 29

“if a quantity is increased or decreased by an infinitesimal, [it] is neither increased or decreased” Johann Bernoulli (Thompson 1998)

“was pilloried far and wide for a long time” Bishop George Berkeley, 1734: “And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small nor yet nothing.” (Thompson 1998)

Bertrand Russell (1903) called them “mathematically useless.” Charles Pierce “strongly disagreed” but “was almost alone in his day in siding with Leibniz, who believed that infinitesimals were as real and as legitimate as imaginary numbers.” (Thompson 1998)

See also this recent preliminary paper submitted to arXiv called A philosophical history of infinitesimals:

Indivisibles were controversial in the 17th century, not least in the eyes of the jesuits who issued numerous bans against them. (Kanovei 2026)

There’s evidence Zeno of Elia was aware of infinitesimals in 500 BC! Let’s say you’re crossing a street. You walk halfway across the street and pause. Then you walk half the distance that’s left and pause. Etc. Etc. The size of your steps is getting really really small right? Also: Will you ever cross the street? That’s Zeno’s Paradox.

Card 31

What a joy this card was to draw and write! I am so happy to be able to pay homage to this legendary critic in my own small way.

“an eye in action” (Webb 2020)

The idea of beauty shaping the trajectory of math and science is an interesting one. Theoretical physicist Sabine Hossenfelder, who also has a great and hugely popular YouTube channel, wrote a book called Lost in Math: How Beauty Leads Physics Astray. In it, she wonders if physicists are relying a little too much on mathematical beauty to guide their research.

Let’s backtrack. Paul Dirac, one of the architects of quantum mechanics, cared about beauty (the math kind) a lot. His first mention of it is even pointed out in his biography:

[Henry Baker’s fascination with ancient Greece and beauty] may be one reason why Dirac drew attention to the aesthetic appeal of Einstein’s theory of gravity in a talk he gave at one of Baker’s gatherings, having pointed out that its predecessor, Newton’s law of gravity, ‘is of no more interest — (beauty?) — to the pure mathematician than any other inverse power of distance.’ This is Dirac’s first recorded mention of ‘beauty.’ (Farmelo 2009)

That was 1924. Here is Dirac 57 years later:

A good deal of my research work in physics has consisted in not setting out to solve some particular kind of problem, but simply examining mathematical quantities of a kind that physicists use and trying to fit them together in an interesting way regardless of any application that the work may have. It is simply a search for pretty mathematics. It may turn out later that the work does have an application. Then one has had good luck. (Dirac 1982)

What he is essentially saying (in a paper called “Pretty Mathematics,” no less) is that he stumbled into his Nobel Prize-winning work on the electron by concerning himself with math he found beautiful. Right? Is anyone else reading it that way? Let me leave speculation to smarter people:

The historian Helge Kragh concluded his biography of Dirac with the observation that “after 1935 [Dirac] largely failed to produce physics of lasting value. It is not irrelevant to pint out that the principle of mathematical beauty governed his thinking only during the latter period. (Hossenfelder 2018)

Here’s another excerpt from Lost in Math:

“We cannot give exact mathematical rules that define if a theory is attractive or not,” says Gian Francesco Giudice. “However, it is surprising how the beauty and elegance of a theory are universally recognized by people from different cultures. When I tell you ‘Look, I have a new paper and my theory is beautiful,’ I don’t have to tell you the details of my theory; you will get why I’m excited. Right?”

I don’t get it. That’s why I’m talking to him. Why should the laws of nature care what I find beautiful? Such a connection between me and the universe seems very mystical, very romantic, very not me. (Hossenfelder 2018)

Also, does the math have to be difficult and/or profound to be pretty? Can 1 + 3 = 4 be pretty?

For that reason, this card has Roberta Smith staring at an elementary equation but also at some terribly difficult ones, including a few featured in The Metropolitan Museum of Art’s exhibit Where Beauty Meets Math: The Concinnitas Series.

There is a also a paper on ugly and pretty equations, and the winner (and loser!) of the study are on this card. (Zeki 2014)

Card 32

“terror” (Thompson 1998)

“When you…by the symbols that follow. That is all.” (Thompson 1998)

“invented by Gottfried Wilhelm von Leibniz” (Newman 1597)

Unfortunately I have no reference for the tort law-practicing integral in a top hat.

Card 34

“here, color is to do everything” (van Gogh 2000)

“in a word, looking at the picture” Van Gogh enclosed a lovely black and white drawing of his bedroom. The colored-in version would one day be world famous! (van Gogh 2000)

Card 36

There is so much more to add to this card about modernism. One of the thrilling ideas — to me — is that math lost touch with “reality” (i.e. basing things on representations of the real world) at about the same time that art did.

Card 38

The footnote is from a letter from André Weil to his sister Simone from Bonne-Nouvelle Prison — a military prison — in Rouen, March 1940:

One would be totally obstructed if there were not a bridge between the two.And just as God defeats the devil: this bridge exists; it is the theory of the field of algebraic functions over a finite field of constants (that is to say, a finite number of elements: also said to be a Galois field,, or earlier "Galois imaginaries" because Galois first defined them and studied them; they are the algebraic extensions of a field with p elements formed by the numbers 0, 1, 2, .. . , p- 1 where one calculates with, them modulo p, p = prime number). They appear already in Dedekind. (Krieger 2005)

He also said some pretty complimentary things about another person we know — Riemann!

I am surely one of the most knowledgeable persons about this subject; mainly because I had the good fortune (in 1923) to learn it directly from Riemann’s memoir, which is one of the greatest pieces of mathematics that has ever been written; there is not a single word in it that is not of consequence. (Krieger 2005)

Card 42

David Hilbert: “I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, we are a university, not a bathing establishment.” (Woit 2007)

“stop by later for more food, drinks and discussion” Noether would host events at her apartment where students and professors could come to relax and chat about math over food, desserts and wine:

Auguste Dick also writes about a lighter side of Emmy. She would host gatherings in her apartment where students and professors could come to discuss the mathematics of the day or to relax in eating sweets and drinking wine. (Radford 2016)

Card 43

“shrinks to a point” Note that the sphere is a “three-sphere” and both the torus and sphere are examples of “three-manifolds.”

Card 44

Cantor’s set theory was the first big move to strip intuition from logic and make it cold, consistent and formal. Set theory faced infinity head-on (Gauss did not want to face infinity head-on) and made math axiomatic from the ground up. (Goodbye intuition! This distressed Poincaré who viewed the new agenda as a move to math a soulless machine that spits out answers. In the based-on-reality but fictionalized Logicomix, Poincaré is depicted as saying “[Hilbert] wants a machine to feed it axioms and make theorems, like one where a pig enters the one side and the sausages come out from the other!”)

Russell’s paradox about sets that contain themselves turned this whole project upside down. Probably Poincaré was pleased. (A distraught Frege wrote an addendum in his Grundgesetze basically saying Russell collapsed one of his laws. See his letter to Russell here.) Cantor it seems took it somewhat well because his “set of all sets” was now impossible.

In modern ZFC set theory, there is no set that contains itself.

Here is a neat quote about Cantor’s proof by Hans Hahn:

The essence of Cantor’s proof is that no comprehensive counting procedure can be devise for the entire set of real numbers, nor even for one of its proper subsets, such as al the real numbers lying between 0 and 1. By various ingenious methods certain infinite sets such as all rational fractions ir a agebracic numbers can be paried off with the natural numbers; every attenmpt, however, to construct a formula for counting the all-inclusive set of real numbers is invaribaly frustrated. No matter what counting scheme is adopted it can be shown that some of the real numbers in the set so considered remain uncounted, which is to say that the scheme fails. It follows that an infinite set for which no counting method can be devised in noncountable, in other words nondenumerably infinite. (Newman 1956)

Card 45

“talk nobody cared about” Riemann’s talk at the University of Göttingen wasn’t meant to be a big ordeal because the habilitation is a ho-hum requirement for a German teaching position. But one person in attendance was paying close attention: Gauss.

“one of the greatest moments in the history of science” (O’Shea 2007)

More praise:

The speech completely recast three thousand years of geometry, and did so in plain German with almost no mathematical notation. (O’Shea 2007)

Riemann’s geometry was the key to solving the puzzle Einstein had been wrestling with all those years. (Yau 2010)

…Bernhard Riemann, who was widely recognized as the most original mathematician of the mid-nineteenth century... (Gray 2009)

“I still can’t see how he thought of it” (Livio 2006)

“a terrible mess” this was said by Einstein’s friend, the geometer Marcel Grossman. The full quote: “a terrible mess which physicists should not be involved with.” (Yau 2010)

“general relativity is born” note that special relativity is another geometric framework: Minkowskian.

Card 46

"modern-day Euclid” (Roberts 2024)

“what shape is that?" (Roberts 2024)

“It seems worth while…” (Coxeter 1998)

Card 47

On hard problems fostering creativity and joy:

When students are invited to ask a harder question, they often light up, totally engaged by the opportunity to use their own thinking and creativity. (Boaler 2016)

Card 48

You might have a fun time seeing the Bourbaki group mentioned in the storied 294-page Grothendieck-Serre Correspondence, a torrent of letters of sent between Alexander Grothendieck and Jean-Pierre Serre. Grothendieck is considered one of the greatest geniuses in the history of math. (Colmez 2004)

Before leaving for a Bourbaki congress, I will try to answer the torrent of questions you asked in your last letter. — Serre

In no6, I have marked two passages with a “?” sign in the margin, to indicate that if you feel that such unhatched chickens have no place in a Bourbaki talk, then you can simply delete them. — Grothendieck

You must think me a terrible correspondent for not having answered your letter sooner, but I have just come back from the Bourbaki meeting, and I had loads of things to do.

We would really like you to come to the Bourbaki meeting in October, if possible (and ditto for the others, of course! I don’t remember exactly what the program is to be (in any case, there will be a reading of my draft on filtered rings etc.), and I don’t think you will find it particularly interesting. But one is not in Bourbaki for fun, as Dieudonn´e never stops repeating… — Serre

etc.

Card 50

“delivered his talk” The anecdote about Grigori Perelman is from Donal O’Shea’s amazing book The Poincaré Conjecture:

As at MIT, everyone in the room, young and old, except the reporters, realized that what they were hearing was the culmination of over a century of the greatest flowering of matheiacical thought in our species’ history. The lecture demanded close attention, leaving little space for stray throughts. (O’Shea 2007)

How did Perelman feel about the possibility of winning that kind of money? As it dawned on them that he did not care, they changed their approach and wrote stories about a reclusive Russian making a big math discovery, and speculated that he would reject the prize. (O’Shea 2007)

Card 51

This didn’t fit on the card but here are some very nice words about Arthur Cayley courtesy of James R. Newman:

Cayley brought mathematical glory to Cambridge, second only to that of Newton, and the fertility of his suggestions, in geometry and algebra, continues to influence the whole range that is now studied at home and abroad. To this versatility Cayley added a Gauss-like care and industry. (Newman 1956)

The technique for multiplying two matrices seems completely baffling until you learn that matrices describe linear transformations between vector spaces:

The definition [of matrix multiplication] becomes crystal clear once we look at a matrix as a tool to describe linear transformations between vector spaces, as the elements of the matrix describe the images of basis vectors. (Danka 2025)

“just what he needed” Ralph Fowler, Paul Dirac’s advisor at Cambridge, knew that Werner Heisenberg’s paper was something special and mailed it straight to 6 Julius Road, where his 23-year old student lived:

The new theory looked completely different from its classical counterpart. For example, there there is no mention in the quantum theory of single numbers to represent the electron’s position; instead, position is replaced by numbers in a square array, an example of what mathematicians call a matrix. (Farmelo 2009)

A note on AxB is not equal to BxA: the non-commutation of matrices that Heisenberg wrote about (the fact that 3×4 is not 4×3, essentially) was so shocking that Dirac, who never talked about his work with anyone, excitedly told his parents. What Heisenberg discovered, and was almost embarrassed by, was the fact that position times momentum was not the same as momentum times position. But Dirac was not phased by this:

Unlike Heisenberg, who had never come across non-commuting quantities before, Dirac was well-acquainted with them — from his studies of quaternions, from the Grassman algebra he had heard about at Baker’s tea parties, and from his extensive studies of protective geometry, which also features such relationships. (Farmelo 2009)

Einstein was excited too. He wrote to Paul Ehrenfest that “Heisenberg has laid a big quantum egg.”

A fun note: Arthur Cayley was Henry Baker’s doctoral advisor at Cambridge.

Card 54

Apollonius of Perga discovered all of this in around 200 BC (!). See his eight-volume treatise Conics.

Card 55

“A screaming comes across the sky” (Pynchon 1973)

The Poisson distribution is just a special case of the binomial distribution. Will I get some confusion and pushback for introducing a card about statistics with a writer? Probably. But Thomas Pynchon is no slouch! He casually and correctly used the words “Poisson equation” and “Poisson distribution” in what has been called the greatest American post-World War II novel. Don’t you love that?

The Poisson is Siméon-Denis Poisson. Poisson brackets would lead physicist Paul Dirac to a groundbreaking flash of insight in quantum mechanics.

Card 58

“cutting and glueing”

There is a problem in The Knot Book that demonstrates the syntax of math surgery:

Cut S open along C, obtaining two copies of C in the cut open S. Glue discs to each of the new curves, where each disk is parallel to the disk bounded by C in F. (Adams 2004)

“to classify different knots” This is mentioned in an absolutely lovely paper by Jonathan Marty:

Despite the rather mechanical and unintuitive nature of the surgery operation, it has a wide variety of applications. Physicists use it to study topology change under events that “cut” spacetime. Knot theorists use it to classify knots by breaking down their the Seifert surfaces associated with them. (Marty 2021)

“Dehn surgery”

In 1910, Dehn and Heegaard published a famous paper that used Dehn surgery to produce an infinite series of three-manifolds that were homology spheres. (O’Shea 2007)

The 1910 paper is interesting for a number of other reasons. It showed that there was a connection between homology spheres and non-Euclidean geometry. It also investigated some connections between the theory of knots and three-manifolds. (O’Shea 2007)

“every 3-manifold”

It has been known for over 30 years that every closed connected orientable 3- manifold is obtained by surgery on a link in S^3. (Lackenby 1997)

“A Seifert surface is”

Given a knot K, a Seifert surface for K is an orientable surface with one boundary component such that the boundary component of the surface of the knot is K. We have just described one way to obtain a Seifert surface for a knot. However, there may be other Seifert surfaces for the same knot. (Adams 2004)

“Ricci flow with surgery”

Ricci flows typically develop singularities within finite time. Perelman’s groundbreaking work allowed for the classification of these singularities and introduced the concept of Ricci flow with surgery. (Yike 2024)

As Hamilton’s colleague Shing-Tung Yau of Harvard University pointed out, these necks mark the spots where mathematicians should perform the “surgery” Thurston’s conjecture requires. (Mackenzie 2003)

Now consider the Ricci flow and let the manifold evolve in accordance with it. If the manifold is simply connected (that is, if it is such that every loop can be shrunk to a point) , then Perelman proves that the Ricci flow, after perhaps some harmless surgeries, will eventually smooth out the extremes of curvature, giving a manifold with constant positive curvature homeomorphic to the original manifold. (O’Shea 2007)

“all the possible shapes”

Poincaré was particularly concerned with three-dimensional manifolds. These modeled possible shapes that our universe might have. (O’Shea 2007)

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