# August 2024 (with bad drawings)

As I weep at the sight of orange leaves, and queue up my favorite end-of-summer songs, let me also queue up this end-of-summer blog post, full of things that (1) happened, and (2) did not yet happen to appear on this blog, raising the question of (3) did they happen at all?

Yes, yes, they did. (Not all questions raised are good questions.)

Anyway, highlights from the month:

**1. A favorite puzzle style: “Which is bigger?”**

To help promote the forthcoming *Math for English Majors, *the fabulous Alex Bellos featured some puzzles of mine in his *Guardian* column. All are variations on a simple question: *Which is bigger?* The fraction one and the root one previously appeared on this blog, but I was especially pleased with two new ones.

The first, approached from a certain angle, is quick to solve. But a surprise lies in just *how *quick.

Which is bigger: the sum of all perfect squares from 1 to 100, or the sum of all perfect cubes from 100 to 200?

The second is not so quick to solve. The surprise is that it can be managed by hand, with very little computation.

Which is bigger: 2

^{100}or 5^{45}?

You can check out solutions here. Many thanks to Alex and the folks at the *Guardian*!

**2**. **Beyond “hard” and “easy” math.**

Earlier this year, I mused on whether math class should be hard or easy. (My vague answer: it depends, in a paradoxical way, on your goals.)

This month, I crystallized those thoughts into a more definite answer. Hard or easy? Neither.

Who really cares about “easy” and “difficult”? They are only proxies for two higher virtues, the actual qualities of successful instruction.

First, math class should be

welcoming….Second, math class should be

challenging….Unlike easy vs. difficult, welcoming and challenging aren’t opposites. We don’t need to choose between them. The best math instruction braids the two together, in puzzles that are clear yet subtle. A good math lesson, like a good sudoku, can welcome and challenge students simultaneously — welcome them

bychallenging them.

Thanks to the Hechinger Report for running this piece, and thanks to the thoughtful folks on social media who ventured to share and discuss it (and no thanks to Human Nature and the Infernal Design of Social Media which conspired to reduce those conversations into the most meaningless and conflictual forms possible).

**3. Math for English Majors and Everyone Else**

Having long enjoyed Jim Propp’s monthly essays, I was honored that my new book (really, you should buy it!) provides the launching point for his latest. He ends with wise words on math popularization (which, conveniently for me, double as a nice advertisement for my book):

The pool of people whose lives could be enriched by a deeper appreciation of what math is about is a diverse one, so the math popularizer community needs to adopt what game theorists would call a “mixed strategy” for getting out the word that math is about more than formulas and instructions.

I think a big selling point of Orlin’s books is that they’re #NotLikeOtherMathBooks, in large part because of the prominently displayed silly drawings. The very crudeness of Orlin’s drawing skills is part of what draws a crowd. Here’s a bad-at-art guy who does it anyway because he loves doing it. This invites readers who identify as being bad-at-math to stop worrying about being bad at it and have some fun under the guidance of an author who knows how to give readers a good time.

Propp talks about math enriching people’s lives. There is also the converse possibility: that if we can grow the community of mathematical thinking, then those new people can enrich math.

Either way, to reach a variety of people, we need a variety of reachers, reaching in a variety of ways — from Kyne’s drag to Eugenia’s baking to my cartoons to Jim’s lovely essays. Each of us is a distinct creature pursuing our own little strategy. Together we create a little ecosystem of ideas, full of mutuality and symbiosis.

**4. Math in its natural form: audio recordings!**

I had great fun on the lovely Manuscript Academy podcast, touching on such issues as (1) how to turn your writing project into a parameter-setting math problem, and (2) whether there might be a standardized unit of snark.

And I loved chatting with Sarah Kesty on the Executive Function podcast, with a focus on the state of being stuck, but with all kinds of fun detours.

**5. A Sharp Inequality**

Okay, not actually a sharp inequality in the technical sense, but it cuts like one:

I wrote this cartoon ages ago (note the handwritten text, instead of my snazzy Chank-designed font). So I had to solve the problem again when re-posting it this month.

I mean, I guess I didn’t *have* to solve it, but I did, and guess what? I like this style of problem! I recommend you tackle it yourself, if you enjoy doing a few lines of algebra, and then saying to yourself, “Ah yes, that all checks out.”

**6. Jokes I Seem to Have Come Up with While Reading a Cosmology Book**

As I noted: never a good idea to estimate a muffin’s mass from its optical properties alone. (Muffin dark matter interacts with bathroom scales, but not with retinas. Very tricky.)

Except it’s not really constant, is it?

When I was a kid, in 1990s New England, three children seemed a totally ordinary number. Five was big but not wild.

But family sizes have shrunk. Now, one is the new two, two is the new three, and three is the new five. Based on this limited data, I conjecture that the Fibonacci sequence is somehow at work.

This is one of my best and least scrutable jokes. I stand by it.

**7. Are there physical forces too big (or too small) for us to perceive?**

In one of my favorite blog posts from this month, the philosopher Eric Schwitzgebel argues that the answer may be yes.

Start with the (plausible) premise that the universe is infinitely large. Then, consider:

At sufficiently small scales, the effects of gravity are virtually undetectable…. However, it accumulates over long distances, making its influence detectable at larger scales.

…if there are entities vastly larger than us, we might imagine them knowing of a force vastly weaker than gravity but which accumulates detectably over distances much larger than the mere tiny, minuscule, virtually negligible 93 billion light years that we can observe.

Of course, Occam’s Razor would counsel against believing in a force for which we have no evidence. But on the other hand…

Copernican or anti-specialness principles hold that we should default toward assuming that we aren’t in a special position in the universe (such as its exact center), pending contrary evidence. Considered in a certain way, we

wouldbe oddly special if we were just the right size to observe all the forces of nature….

If you feel a strange attraction to this argument, well, perhaps that’s just the tiny ripples of a force vaster than imagination.

One last joke that may be relevant:

**8. Separating the Math from the Mathematician**

In math popularization circles–the tiny, non-Euclidean circles in which I reside–there’s a certain celebrity and joy attached to the name Piet Hein. It’s the joy of hex, of the soma cube, of the superllipse (and its 3D kin, the superegg), and of the aphoristic poems known as grooks. Hein was (like M.C. Escher and John Conway) a favorite muse of Martin Gardner, who championed Hein’s work, and who titled his own autobiography (*Undiluted Hocus Pocus*) after a grook:

We glibly talk

of nature’s laws

but do things have

a natural cause?Black earth turned into

yellow crocus

is undiluted

hocus-pocus.

But–as recently revealed for the first time in English, in a well-researched essay by Michael Pershan–Piet Hein was violently abusive. (Pershan’s language is stronger; he calls him a “monster.”) Hein’s abuse spanned several decades, children, and marriages. It seems to have contributed to the suicide of his fourth wife.

I find Pershan’s essay important, all the more so because I’m not sure what to make of it.

One might expect mathematical creations–compared to, say, films or pop songs–to be the instances of art that are most easily separable from the artist. If new evidence revealed that Euclid was a terrible guy, would it matter? I don’t really think so. The geometry is no longer his; it’s ours, a collective possession. Mathematical ideas are bigger than their thinkers. The art o’erleaps the artist.

But Euclid is vast and ancient. What about someone more recent, more human-scale?

That’s where Hein comes in. His legacy isn’t just mathematical ideas. It’s their packaging and salesmanship. The superellipse, as Pershan points out, is kind of an obvious move:

I don’t mean to be a jerk—but couldn’t anyone have thought of this? Maybe not the equation. But who needs the math at all? Go ask literally any artist for something between a circle and a rectangle, I promise you’ll get it.

It is not that hard. Whatever. They asked Hein, so he gets the credit.

Grooks aren’t mathematics, either. They’re poems that happen to appeal to mathematicians. Poems that, in their precision and tightness and universality, adopt the aesthetic of mathematics. You can dislike them (Pershan dubs them “self-satisfied and cloying”) or you can like them (I’ve always admired the line “err, and err, and err again, but less, and less and less“), but either way, such poems aren’t timeless wisdom.

Their air of timelessness is a garment, placed there by the poems’ creator.

In short, I think Pershan makes a compelling case that Hein’s work cannot be divorced from Hein the person. But I differ from Pershan on this key point:

[K]nowing about his personal life makes Hein’s intellectual story

make more sense.Hein liked precision, structure, a kind of cleanliness…. That’s not what

Ilike in my art—I like big messy feelings and huge dramatic gestures. I don’t like aphorisms very much. But Hein sure did.In art and life, Hein thrived when the rules of the game conformed to his expectations… But step outside those lines, and Hein would fight to fence you back in.

I don’t see Hein’s grooks and his monstrosity as emerging from the same excessive love of tidiness and order. Pershan knows Hein better than I do, and I have no alternate theory to posit, but I tend to be wary of this kind of dimension collapse. Art does not necessarily encode the moral qualities of the artist.

If you want morality to be easy (and who doesn’t?!) then it’s the worst of both worlds. If the work was cleanly separable from the man, we could embrace it more easily; or if the work bore all the stamps of the man’s faults, we could reject it more easily; but neither is true. The work is neither separable from Hein nor coextensive with him. The overlap is muddy.

Or, as Hein himself put it, in a grook that reads more darkly now:

If virtue can’t

be mine alone

at least my faults

can be my own.