Pythagoras was many wonderful things. A delirious mystic. A benevolent cult leader. A bean-hating vegetarian. A real person (maybe).

One thing he was not: the guy who gave us the Pythagorean Theorem.

So why does he get his name on it? I cry foul. I cry “no more.” I cry, “Let us band together and vote on a better name for this ancient theorem! Not because it will actually result in a name change, but because it’s a fun debate!”

Who’s with me?!

I submit for your consideration the following names, from Hambrecht and the other clever folks in his thread:

**The Three Squares Theorem**. Although we perceive it as a claim about numbers, for most mathematical cultures, this was a claim about*shapes*. To wit: if you affix squares to the sides of a right triangle, the two smaller areas add up to equal the largest.**The Babylonian Formula**. Give credit where credit is due! As Hambrecht says, this name “has the hint of far-away times and places… Through millenia and continents, this piece of math connects us to strange, alien people, yet so much our equals.” He calls it “fuel for children’s imaginations.” Even more important, as an astute observer points out: it can be abbreviated as “Baby Formula.”**The Distance Theorem**. The theorem’s most ubiquitous use is in finding distances, especially in higher dimensions.**The Huey Lewis Theorem**. Proposed (or, pun-posed) by Susan Burns, because, and I quote: “It’s hyp to be square…or is that b-squared?”**The Adrakhonic Theorem**, because that’s what it’s called in Neal Stephenson’s novel Anathem (which I just added to my reading list).**Squaring the Triangle.**Olaf Doschke’s suggestion, with a ring of the famously impossible “squaring the circle.”**The Sum of Squares Theorem**. Descriptive, clear.**Garfield’s Theorem**. Because if we’re just naming it after a random dude who supplied a proof, why not pick an assassinated U.S. president?**Theorem 3-4-5.**After the most famous Pythagorean triple.**Euclid, Book I, Proposition 47.**“Like chapter and verse in the mathematical bible,” explains George Jelliss.**The Hypotenuse Theorem**. Because it’s all about that longest side.**The Right Theorem**. Because it’s all about that right angle. (Also, because it’s right.)**The Distance/Area Theorem**. Because it’s all about multiple things at once.**The Benjamin Watson Theorem**. Because of this heroic, historic tackle, brought to my attention by Fawn Nguyen in her appearance on My Favorite Theorem.

Now, we could leave it there. We could say, “This has been a fruitful discussion. Let’s call it a day!” We could say, “Obviously a random blog post isn’t going to succeed in renaming the most famous theorem in mathematics, so let’s go home and eat raisins and watch sitcom reruns like the human mediocrities we are.”

But I say no! I say it’s time for a referendum!

What say *you*, good people of the internet? What is the best name for this fundamental theorem of geometry?

Other ideas are, of course, welcome in the comments below.

]]>**Rhymes with Orange, by Hilary Price.**

Where *Doonesbury* bites, *Dilbert* groans, and *Peanuts* waxes philosophical, *Rhymes with Orange* has a lighter touch. The gags come out of nowhere, like tickling fingers.

I wasn’t thinking about math when I picked up a copy of the first collected volume (published back in 1997). But then I ran into this pretty great surface area joke:

That got me seeking more. And, with the help of Price’s admirably searchable website, I found the gallery that follows – it turns out that math is a topic she has circled back to time and again over the years.

From December 17, 1997:

(This contradicts the famous “half plus seven” rule, whereby the youngest partner a 34-year-old can consider is half 34, plus 7. In other words, 24. Price’s model looks more complicated, but may have more empirical validity.)

From May 9, 2000:

(Sigh.)

From February 17, 2013:

(I don’t know why I find this so funny, but I find it SO FUNNY.)

From August 22, 2011:

(Moore’s Law for stones?)

From January 17, 2015:

(I hope to make this order in a restaurant someday.)

From October 26, 2017:

(I assume that’s division rather than a cube root. Anyway, it’s a great gag.)

From October 17, 2013:

(True facts! Well, not about Rome’s fall – as I understand it lead pipes and bureaucratic sprawl were involved – but about the disadvantages of Roman numerals, and the efficiencies that led to the adoption of our own system!)

From December 5, 2011:

(I’ve heard tell of a problem like this: so many variables that they exhausted the English *and* Greek alphabets, and had to summon emergency back-ups from Hebrew.)

From October 2, 2010:

(Sounds persuasive to me.)

From April 3, 2008:

(Totally using this next time I teach inverse relationships.)

And finally, from February 17, 2010, by guest cartoonist Mo Willems:

**

To recap: how does Price find humor in math? In any number of ways. I find their variety instructive:

- Satirizing its inscrutability (as in the “girl years” and “run out of letters” gags);
- Playfully misapplying its concepts (“only $0.79 an ounce”; “60/40 split pea”);
- Drawing unexpected parallels (“more power in this little abacus”);
- Lamenting the experience of math education (“now we are going to learn percentages”; “no relying on the wand”);
- Playing the logical strictures of math against the illogic of emotion (“isosceles”; “couch-to-door ratio”).

Some of these take the stance of an outsider, to whom math is a blur. Others feel like jokes any math insider might make.

What I appreciate most is the coexistence, the blending, of those two perspectives. It suggests that this thing called mathematics – this exalted, despised, exoticized subject – is perhaps a human activity like any other. It’s part of our common inheritance, along with language and color and humor.

Of course, I’m sure Price intends no such grand statement through her mathematical cartoons – which is exactly why I love ’em.

]]>I was wary of getting too political in a volume of math jokes, but I find it the E.C. such a peculiar and compelling mathematical object that I couldn’t help myself.

A highly abridged version of the chapter follows, and then some questions. (For more, check out the book!)

**What is the Electoral College?**

It’s how the President becomes President!

Each state gets a certain number of Electors (the bigger the state, the more Electors). It’s these Electors, officially, who pick the president.

**How does it work in practice?**

States can choose Electors however they like: roulette wheel, softball match, arm-wrestling tournament. Possibilities abound!

But as it is, almost all follow this all-or-nothing electoral approach:

- Hold a statewide popular vote.
- Choose Electors who all pledge to vote for the statewide winner.

**Why was the Electoral College created?**

Three reasons, more or less.

First, in a big country, long before high-speed communication, it seemed unlikely that far-flung voters could knowledgeably judge national candidates.

Second, the president needed independence. Making him accountable to the House could undercut the balance of powers; but direct election could promote demagoguery. Hence, a kind of one-time CEO search committee.

Third, there had already been delicate negotiations over legislative representation. Slave states got the 3/5 clause. Small states got the Senate. The Electoral College imported these compromises into the selection of presidents.

It’s arguable whether any of these reasons still apply today. Slavery is abolished; the all-or-nothing approach has eliminated small states’ advantage (see below); and mass communication has nationalized and democratized our politics (for better or worse).

**So why do we still have the Electoral College?**

Replacing it never felt urgent. From 1892 to 1996, it had no discernible effect on election results. And amending the Constitution is a long, hard road.

**Who benefits under the Electoral College?**

One tempting answer, given 2000 and 2016 (when the Democrat won the national popular vote, but lost the Electoral College), is “the Republican Party.”

But a closer look suggests that n = 2 is too small a sample.

Who does benefit, then? That’s a tricky (and rather delicious) math question!

In general, voters in “close” states (such as Nevada, Wisconsin, and Florida) wield extra power, at the expense of voters in states with a dominant party (such as Kansas, Massachusetts, Oklahoma, and Hawaii).

Arguments that the E.C. benefits a specific, stable group – e.g., rural voters, or voters from the Midwest – don’t tend to withstand scrutiny. Such an advantage may endure for a couple of elections, but in the long term, randomness seems to reign.

What about state size? It’s complicated. A Wyoming voter has a better chance of swinging her state’s outcome than a Texas voter does. But, thanks to all-or-nothing Elector apportionment, her state has a much worse chance of swinging the election has a whole. The result is a bit of a wash.

**Why do people want to replace it?**

You’d have to ask them – I’m just a math teacher!

Mathematically, it seems the Electoral College gives very similar results to a popular vote, with some deviations (caused by the all-or-nothing approach) that are effectively random.

**How do reformers propose to replace it?**

Twelve state legislatures have passed laws to this effect:

- We pledge to give all our Electors to the winner of the national popular vote.
- This law will only take effect when a critical mass of other states pass it, too.

If the critical mass of 270 electors is reached, then the country will have a de facto popular vote, without the need for a Constitutional amendment. This is known as the National Popular Vote Interstate Compact.

**You’re a math teacher. Do you have some exercises for math students?**

Why, sure! Here are two strings of problems.

**ELECTORS PER CAPITA: WHAT DOES IT TELL US?**

The formula for your state’s number of Electors is roughly this: Population/700,000 + 2, rounded to the nearest whole number. Assume that the winner within each state gets all of its Electors.

1. Compute the number of electors for Alaska (737,000 people), South Dakota (882,000 people), Mississippi (2,986,000 people), and Alabama (4,887,000 people).

2. Now, compute the number of electors *per capita* for Alaska, South Dakota, Mississippi, and Alabama.

3. Under this system, which sorts of states will have the most Electors per capita?

4. Which will have the fewest?

5. Who do you think is more powerful – voters in small states, voters in big ones, or does it not matter? Explain.

Let’s imagine the country were made of 2 states: *Megastate*, with a population of 1.4 million, and the *State of Moe*, with a single resident named Moe.

6. How many Electors does each state receive?

7. How many Electors per capita does each state have?

8. Whose vote has a better chance of swinging the election: Moe’s, or a voter’s in Megastate? Think carefully, and explain!

9. What does this two-state scenario tell us about the usefulness of “Electors per capita” as a measure of power?

10. What is another way we could measure a voter’s power?

**SYSTEMS OF APPORTIONMENT: DO THEY MATTER?**

Currently, 48 out of 50 states apportion their Electors on an *all-or-nothing* basis: the winner of the statewide vote gets all of the Electors.

Imagine if states switched to a *proportional* system, whereby if you win X% of the vote in a state, you get X% of the Electors (rounded to the nearest whole number).

1. Suppose that Minnesota votes 68% for A, 30% for B, and 2% for C. How should it apportion its 10 electors? Explain.

2. Suppose that Minnesota votes 53% for A, 44% for B, and 3% for C. How should it apportion its 10 electors? Explain.

3. How would the effect of this change be different for big states like California (with 55 electors) than for small states like Vermont (with 3 electors)?

4. Imagine going to Hawaii (which usually votes Democrat) and asking a Democrat and a Republican whether they support this change. What do you think they would say, and why?

Imagine if states switched to a *district-by-district* system. For example, if a state has 5 electors, it breaks its voters into 5 districts, and assigns an elector to the winner of each.

5. Suppose that in Massachusetts, this has no effect on the electors. What does that tell us about Massachusetts? Be specific.

6. Suppose that in New Hampshire, this has a big effect: instead of winning all 4 electors, the Democrat now wins only 2. What does this tell us about New Hampshire? Be specific.

7. Suppose a Republican and a Democrat in New Hampshire are each asked to divide the state into districts. Do you think they’d make similar divisions? Why or why not?

]]>There’s a time when I’d have heard this Galileo praise, and nodded along.

But lately I’ve been listening to Opinionated History of Mathematics, a podcast by historian of mathematics Viktor Blåsjö. The first season is a startling anti-Galilean polemic, with arguments range from the speculative to the devastating, and from the sassy to the hilariously sassy.

This is no frivolous hit job – there are real historical questions at stake here. How did science go from Ptolemy to Newton? What distinguished empiricism as an approach? Just how much did ancient Greeks know? You’ll find some of Blåsjö’s answers more persuasive than others, but he’s got the receipts, and if you embrace even a fraction of what he says, you’ll find your Galilean faith shaken.

I suggest you listen to the whole thing, but here’s an episode-by-episode teaser. (My apologies to Viktor for eschewing the more subtly argued passages in favor of juicy attacks. I couldn’t help myself.)

**Episode #1: Galileo Bad, Archimedes Good**

A classic math problem is to find the area of the cycloid. Blåsjö lays out what happened when Galileo attempted it: he floundered, failed, turned to crude trial-and-error, and *still* got the answer wrong. (Several of his contemporaries, meanwhile, calculated the precise answer.)

This, Blåsjö says, is typical of the fellow from Galilei:

He was not a pioneer of scientific method. He was not the father of modern science. He was not a heroic knight defeating dogmas and superstitions with the light of empirical truth…. Galileo was, first and foremost, a failed mathematician….

**Episode #2: Mathematics Versus Philosophy, Then and Now**

Galileo’s major works are, in effect, refutations of Aristotle. They’re dialogues between a foolish Aristotelian and a wise Galilean.

But according to Blåsjö, Aristotle was a straw man, whose shortcomings serious mathematicians had always known. Refuting Aristotle is shooting fish in a barrel, and impresses only those who don’t know any better.

Galileo’s books are “Science for Dummies”. He drones on and on about elementary principles of scientific method….

Galileo needs us to assume that… no one had ever heard of Archimedes. Only then do his so-called accomplishments come off looking any good.

**Episode #3: Galilean Science in Antiquity?**

Millennia before Galileo, were “Galilean” ideas already in circulation? Blåsjö’s says yes.

Either you are a cultural relativist and you think Galileo was a revolutionary… or you think mathematical thought is the same for you, me and everybody who ever lived, and then you think Galileo was just doing common-sense stuff.

(A possible counterargument: shouldn’t we credit Galileo for bringing these “common-sense” ideas to wider audiences? In my view, Blåsjö is too dismissive of popularization. But if the argument turns from “Galileo is the father of modern science” to “Galileo had a big impact as a popularizer,” then I think Blåsjö has already won.)

**Episode #4: The Case Against Galileo on the Law of Fall**

Among Galileo’s finest achievements: debunking Aristotle’s claim that heavier objects fall faster. (Legend holds, falsely, that he did this by dropping stones from the Tower of Pisa.)

But was the refutation really such an accomplishment? Aristotle makes the claim only once, in a paragraph-long aside; it does not seem central to his thinking. And isn’t the experimental verification a pretty straightforward idea?

Of course one can drop some rocks and see if it works…. In fact, Philoponus—an unoriginal commentator—had clearly and explicitly rejected Aristotle’s law of fall by precisely such an experiment more than a thousand years before Galileo…

**Episode #5: Galileo’s Errors on Projectile Motion and Inertia**

Galileo gets a lot of credit for articulating a law of inertia that’s halfway to Newton’s. But he made several big errors; for example, he applied the rule only to objects whose initial motion was horizontal, waffling on whether it applied more broadly.

Calling this “halfway to Newton” is too generous, Blåsjö argues. Even poor, benighted Aristotle articulated a similar idea!

So take your pick. Here are the [two] options:

Option 1. Galileo’s understanding of inertia was very poor.

Option 2. Galileo’s understanding of inertia was pretty good, but so was Aristotle’s….

**Episode #6: Why Galileo is Like Nostradamus**

Galileo made a lot of striking errors. His gravitational constant is way off, because he inexplicably used made-up data. His theory of planetary speeds (that the planets “fell” into the solar system from a tremendous distance) fails the most basic mathematical test. And his “proof” that objects could never fly off the spinning earth is totally wrong (because if the earth spun fast enough, they absolutely could). He even tries to pass off one of his miscalculations as a “joke”!

Why so many mistakes? Blåsjö pulls no punches:

Galileo is another Nostradamus. He too threw a thousand guesses out there and hoped that one or two would stick. Like Nostradamus, Galileo’s reputation rests on his admirers having selective amnesia, and remembering only the rare occasions when he got something right.

**Episode #7: Galileo’s Theory of Tides**

Galileo rejected the true explanation of tides (which his contemporaries embraced) as “childish” and “occult.” His alternative theory contradicted all the data, as well as Galileo’s own scientific principles. Blåsjö explains:

Galileo’s theory implies that high and low [tides] should be twelve hours apart rather than six… The fact that everyone could observe two high and two low tides per day Galileo thus wrote off as purely coincidental….

Galileo even has some fake data to prove his erroneous point: namely that tides twelve hours apart are “daily observed in Lisbon,” he believes, even though that is completely false.

**Episode #8: Heliocentrism in Antiquity**

Galileo doesn’t just refute Aristotle. He also refutes the geocentric astronomy of Ptolemy.

But did Ptolemy really speak for all Greeks? Blåsjö argues otherwise, speculating that Archimedes’ pal Aristarchus had a well-reasoned heliocentric model.

Nowadays we are stuck with Ptolemy as the canonical source for Greek astronomy. But Ptolemy lived hundreds of years after the golden age of Greek science. It is likely that he was not the pinnacle of Greek astronomy, but rather a regressive later author who perhaps took astronomy backwards more than anything else.

**Episode #9: Heliocentrism Before the Telescope**

Galileo is remembered today as the greatest champion of Copernicus. But while other scientists filled the margins of Copernicus’s book with calculations and annotations, Galileo’s copy is bizarrely blank, as if he had not given the text a serious reading at all.

Blåsjö quotes another historian:

“…when I saw the copy in Florence, my reaction was one of scepticism that it was actually Galileo’s copy, since there were so few annotations in it. … This copy had no technical marginalia, in fact, no penned evidence that Galileo had actually read any substantial part of it. … Eventually, … I realized that my scepticism was unfounded and that it really was Galileo’s copy.”

There is no need for surprise, of course. Galileo was a poor mathematician.

***

More episodes are coming. (In an email, Blåsjö told me he’s barely halfway through his Galileo material!) I’m especially eager to hear the next one, which will tackle the discoveries Galileo made with his telescope.

Was Galileo a great scientist? A skilled popularizer? A talentless hack? I don’t honestly know. But the fact that we’re asking the question feels pretty darn radical to me.

*EDIT: The first version of this post ran a little too snarky. I’ve toned it down because I find the historical question here – to what degree was Galileo a popularizer instead of an innovator, and how should we value that work? – genuinely interesting. **And if you’re thinking, “Wait, this post was once even SNARKIER?” then you see why I changed it!*

In Newtonian mechanics, it’s not too hard to figure out how two celestial bodies (e.g., Earth and Moon) should behave. But three bodies (e.g., Earth, Moon, and Sun)? That’s a computational nightmare.

Anyway, speaking of chaos…

For my money, this is the most exciting research area in applied mathematics. Not computational biology, not dynamical systems, but Who Does the Dishes.

Just consider the open questions here:

*How does***total dish labor**(L) scale with**household size**(n)?*How does household size affect the***latency period**between dish-dirtying and dish-washing?*How does the*?**maximum**latency period relate to the**average**latency period*What is a***typical distribution of dish labor**between the various household members? Does*everyone**feel like they’re doing more than their share? Or do the freeloaders recognize themselves as such?*

I’m sure that some of these problems will prove computationally intractable. Still – and you heard it here first – I prophesy that the 21st century’s most breathtaking mathematical progress will come in understanding the domestic chaos theory of the shared living space.

]]>

Archimedes was born in Syracuse, in the early 3rd century BCE, on the island of Sicily. He was probably not fluent in Greek at birth. (Then again, who knows?)

**2.**

He assembled one of the wildest and most impressive careers in mathematics, ancient or otherwise. My favorite work of his: “Sand Reckoner,” an estimate of how many grains of sand it would take to fill the universe.

**3.**

Archimedes mastered the “method of exhaustion,” which smells an awful lot like what we today call “calculus.” He used this technique to deduce the circumference of circles, the volume of spheres, and the area enclosed by parabolas.

**4.**

I’ve heard several historians (namely Viktor Blasjo and Chris Rorres) hail Archimedes’ analysis of floating bodies as one of the great achievements in math. It was a systematic analysis of a chaotic system, more than 20 centuries before chaos theory.

**5.**

Legend has it that Archimedes refused to yield when a Roman soldier ordered him to stop drawing diagrams in the dirt. Classic mathematician.

(By the way, I recommend the Twitter thread in which historians Alberto Martinez and Viktor Blasjo debate the plausibility of this tale.)

**6.**

The soldier, not knowing who Archimedes was, attacked him.

(And by the way, he really did view the sphere volume proof – that a sphere, fit snugly inside a cylinder, will fill 2/3 of its container – as his masterwork. Cicero tells us that its image appeared on his tombstone.)

**7.**

I imagine Archimedes departed our mortal plane beneath a fitting piece of geometry: a parabolic arch.

**8.**

(Not depicted: sphere volume proof.)

Rest in peace, Archimedes!

]]>It’s hard to say what inspired these cartoons. Well… actually, it’s easy:

- I love Dr. Seuss.
- I have a math blog.

Maybe that’s not much. But with Dr. Seuss’s 114th birthday approaching on March 2nd (along with my sister’s 28th – happy birthday, Caroline!) it seems as good a time as any to put a mathematical spin on Seuss’s style.

**Green Eggs and Ham and Diminishing Returns**

**The Lorax
**

**The Grinch:
**

**Yertle the Turtle,
**

In the Homeric epic that is mathematics, *e *gets a choice epithet. It is “the natural base.”

I inventoried the reasons last year (for the historic once-in-a-century *e* Day). And I later stumbled across an even more delicious “naturalness” argument for *e*.

But that’s hardly exhaustive. Here’s another thing I dig about *e*, an easy fact for any first-year calculus student to verify:

Also, an addendum: a fun observation from Twitter…

**Addendum #2, on March 4, 2019**: the inimitable Sam Shah asks how we can introduce *e* to algebra students, and then delivers a delightful post rounding up more lovely *e* facts.

But it captures a metaphorical truth, which is that becoming a teacher has estranged me from my friends.

I don’t mean in the present. We still drink milkshakes together, trade gifs, etc. I mean a kind of retroactive estrangement, an expulsion from our shared past. For my non-teacher friends, school is a place visited only in memory; you empathize with the kid you were, and regard the teacher as a distant authority (even if your age now exceeds theirs then).

But for teachers, school is a place visited daily. Our perspective shifts. I now identify less with my past self than with my own teacherly antagonists.

*Remember when the teacher took two months to pass back that paper? *Well, he was probably drowning in grading—I don’t blame him.

*Remember how she never noticed us passing notes?* Yeah, that was so stupid of us; we should have paid better attention.

*Remember when the teacher snapped at us even though *everyone* was talking?* You see, the thing about enforcing rules is that it’s impossible to be perfectly evenhanded…

I wish someone had warned me that becoming a teacher rewrites your past. Today, when I look back at my student experiences, I find my empathy turned inside out. Retrained by years behind the big desk, my first instinct is to identify not with myself, but with my onetime opposition, the teacher.

Only with effort can I flip the switch, and see through my student eyes again.

]]>In math, silliness happens. Slips of the pen, or the tongue, or the LaTeX-typing-fingers. We all make foolish mistakes now and again.

But I have on occasion noticed among students (and their parents) a peculiar tendency to attribute *all* mistakes to silliness.

“Look at his test!” a parent might say. “He lost all his points on silly errors. He *knows* how to do it.” Then we’d go through the test together, and in each question where he’d lost points, I’d see topics to address, room to grow. Not just hiccups and typos.

Why the divergent views?

To some, mathematics feels like the successful performance of prescribed steps. In that case, the whole subject is mechanical and straightforward. *All* mistakes, in this light, are “silly” misfires. Multiplied when I should have added? Hey, just silliness! Obviously I know *how* to add! I just happened to zig when I should have zagged!

I take a different view of math. Some aspects – say, employing rich mental models for key concepts – may be hard to assess, but are crucial. Multiplied when you should have added? It’s possible your mental models of “addition” and “multiplication” (or “area” and “perimeter,” etc.) are leading you astray, or could use further development.

It’s comforting, of course, to view all one’s mistake’s as “silly.” Easier to admit occasional clumsiness than real confusion. But it’s the deep mistakes that signal the greatest opportunities to learn.

Missing out on those chances – that’s the *really* silly mistake.