Tricks of the Trade

In the 1950s, a Scottish mathematician named C. Dudley Langford looked at a stack of six blocks his young son had assembled (see Endnote #1) and noticed something interesting that would lead him to the mathematical discovery he’s remembered for today. 

Langford noticed that between the two red blocks was one block, between the two blue blocks were two blocks, and between the two yellow blocks were three blocks. Being a mathematician, Langford immediately wondered “Could we do this with more than three colors?”

Can you figure out how to add two green blocks and arrange the eight blocks so that there will be one block between the red blocks, two blocks between the blue blocks, three blocks between the yellow blocks, and four blocks between the green blocks?

And, having succeeded with four colors, can you do it with five?

Breaking Pi

I love working with others to discover new mathematics, but there’s a kind of research I’d love even more: helping decode a Message from an extraterrestrial civilization. The chance to do that would make me drop all my mathematical projects — though in a way it wouldn’t, since decoding the Message would almost certainly involve a lot of math.

As a teenager I was captivated by a 1973 book called Communication with Extraterrestrial Intelligence. It was edited by a not-yet-world-famous astronomer named Carl Sagan who was interested both in sending messages to the stars and in seeking messages from the stars to us. He went on to host the incredibly popular TV program “Cosmos” and to write several best-selling books, including the novel Contact which I’ll have a lot to say about later.

The reason I’m writing this particular essay this month is because almost exactly two centuries ago, the mathematician and astronomer Carl Friedrich Gauss proposed sending a message to the moon. (Gauss’ ideas about life on other worlds had a respectable pedigree in European thought; see the excellent articles by Aldersey-Williams and Dillard listed in the References.) Gauss had invented a kind of signaling device he called the heliotrope, and on March 25, 1822, he wrote a letter to the astronomer Heinrich Olbers, saying “With 100 separate mirrors, each of 16 square feet, used conjointly, one would be able to send good heliotrope-light to the moon. … This would be a discovery even greater than that of America, if we could get in touch with our neighbors on the moon.”

The Clatter of the Primes

The trouble began, as trouble often does, with a rivalry between friends. It took place during the Big Before, when numbers and operations were new and still figuring themselves out, and none of them had any idea what a universe was or whether having one would be a good idea.

Plus said to Times “No offense, friend, but I’m just better at building numbers than you are. Starting from 1, the smallest number, I can build lots of new numbers: 1+1 is 2, 1+1+1 is 3, and so on. But look at you! 1×1 is just 1. 1×1×1? 1 again. And so on. Boring!”

Times naturally became defensive. “Now that’s just not fair. You’re using the wrong building block. Instead of 1, try 2.” And the number 2 began to twinkle. “2×2 is 4. 2×2×2 is 8. And so on. See, I get new numbers, just like you, and mine are bigger than yours!”

Plus said “I can get all those numbers, and more; it just takes me longer. But I get some numbers you can’t get. 3 is 1+1+1, but you’ll never get 3 by multiplying 2’s.”

Times, thinking quickly, retorted, “I never said I could get everything from 2’s. I also use 3 as a building block.” Then the number 3 began to twinkle. “For instance, with 2 and 3, I can get 2, 4, 8, and so on, and 3, 9, and so on. And I mix 2’s and 3’s, so I get 6 and lots of other numbers too.”

Plus said “What about 5? How do you get 5 by multiplying 2’s and 3’s?”

Times airily answered “Oh, I never said 2 and 3 would be enough! 5 is another one of my building blocks.” And the number 5 began to twinkle.

Plus asked “How many of these building blocks do you have?”

Bad Soup

My father once told a story about going to a restaurant with some friends, one of whom ordered some soup and was very unhappy with what he got. When the waiter came by and asked what the problem was, my father’s friend said “The soup is bad.”

“I’m sorry the soup did not meet with your approval,” the waiter replied. But this only made my father’s friend angrier.

“It’s not that the soup didn’t meet with my approval!” he shouted. “This is BAD SOUP!”

Like my father’s best stories, this is one that has meant different things to me at different times. Originally I found the story funny because of the patron’s insistence on being not just entitled to his opinion but objectively right, in a matter of taste where there’s no such thing as right or wrong. Later, I realized (okay, my wife pointed out to me) that it is possible for soup to be bad; for instance, it can be teeming with mycotoxins that are as poisonous as they are distasteful. And what’s poignant for me today is that it’s now impossible, all these years later, to determine whether the soup was bad or my father’s friend was fussy. I’m not even sure how I’d find out what the friend’s name was, or where they ate.

But the topic I want to treat ever-so-superficially (before getting back to preparing for the first day of classes tomorrow; I do have a real job) is this desire we often feel to be right not just in our own minds but objectively. Sometimes this is secondary to the petty desire that someone else be wrong and admit it, but at other times, it reflects a deep yearning for something solid to cling to, in a world full of clashing perspectives and contested facts.

Numbers from Games

Something very much like nothing anyone had ever seen before came trotting down the stairs and crossed the room.

“What is that?” the Duke asked, palely.

“I don’t know what it is,” said Hark, “but it’s the only one there ever was.”

— James Thurber, “The 13 Clocks”

Why was a Cambridge University Fellow and Lecturer named John Conway, on an unremarkable day in the late 1960s, lying on his back, waving his feet in the air, and giggling?

To be fair, it was a decade in which many people did crazy things. In the U.S., Conway’s fellow-academic Timothy Leary was giving LSD to Harvard undergradutes, while some of Conway’s fellow-Liverpudlians, talented lads who called themselves The Beatles, were causing musical mania on both sides of the Atlantic. But Conway’s performance was for an audience of none (not counting himself), and the thing that had caused him such hilarity was neither a drug nor a catchy melodic hook but a psychedelic mathematical insight — specifically, the realization that, in an arcane but rigorous sense, four times four equals not sixteen but six.

Why Names Matter

I just went through my lesson plan for an upcoming lecture on number-sequences and replaced the name “Fibonacci” by the name “Hemachandra”. By the time you finish reading this essay, you’ll know why I did it, and if you’re a teacher, I hope you’ll do it too. [Note added on November 19: I might now go back again and change “Hemachandra” to “Virahanka”; see the Endnotes.]

To the extent that we can reconstruct the story of the famous sequence

1,2,3,5,8,13,21,…

from historical sources, the tale starts with the ancient Indian poet and mathematician Pingala (a contemporary of Euclid’s, give or take a century). For Pingala, these numbers arose from exhaustive consideration of the rhythmic possibilities of Sanskrit poetry. If you want a six-beat poetic phrase built out of short (1-beat) syllables and long (2-beat) syllables, how many possibilities are there? The answer turns out to be 13, so that’s the sixth term of Pingala’s sequence. Likewise, if one is playing the tabla, there are 13 different six-beat drumming patterns one can build from 1-beat and 2-beat components. (The 1-beat and 2-beat components are often rendered vocally as “dhin” and “dha” respectively, so that the two most dissimilar six-beat patterns would be the leisurely “dha, dha, dha” and the rapid-fire “dhin-dhin-dhin-dhin-dhin-dhin”.)

Here There Be Dragons

Last week my daughter asked me about weird bases. “Do bases have to be integers?” “Do they have to be real?” (She’s heard about complex numbers such as i, the infamous square root of minus one.) Then she asked “How would you write 256 in base i+1?”

She swears that she made up the question on the spot, but the answer is suspiciously nice, as we can see by starting with i+1 and repeatedly squaring. If we multiply i+1 by itself we get (i+1)·(i+1) = i·i + i·1 + 1·i + 1·1, or –1 + i + i + 1; the –1 and the 1 cancel, so we get (i+1)² = 2i. Square both sides: (i+1)⁴ = (2i)(2i) = (2)(2)(i)(i) = (4)(–1) = –4. Square again: (i+1)⁸ = (–4)(–4) = 16. Square one last time: (i+1)¹⁶ = (16)(16) = 256. So 256 equals 1 times (i+1)¹⁶, plus 0 times (i+1)¹⁵, plus 0 times (i+1)¹⁴, plus …, plus 0 times (i+1)¹, plus 0 times (i+1)⁰, and we conclude that the base i+1 representation of 256 is 1 followed by sixteen 0’s: 10000000000000000. (What are the chances?)

I posted this on Twitter, and someone wrote “Impressive; how old is she?”.  My daughter just turned 13, and 13 is (16)+(–4)+(1) which equals (i+1)⁸ + (i+1)⁴ + (i+1)⁰, so I wrote back “She just turned 100010001.”

But then I got to thinking: do I really know how to write every positive integer in base i+1? Or did I just get lucky?

Miracle People: A “Genius Box” Postscript

This past summer the Journal of Humanistic Mathematics published a revised version of my essay “The Genius Box“. In the original 2018 version I had asked “What are we doing when we call someone a genius?” and I had tried to show the ways in which having a special category of people called geniuses is harmful. At the journal’s request I added some new material to the published version, and put in a short new section called “Myth and Countermyth” that showed how one version of The Genius (the lightning-fast thinker) can give way to an antithetical version (the slow, deep thinker) without really fixing the problem with certain people being called geniuses in the first place.

While putting the finishing touches on the published version, I came across a relevant quote from a major twentieth-century physicist who was on a first-name basis with most of the people hailed as geniuses in the twentieth-century physics community. Here’s what he said about the pioneers of quantum physics and about himself:

Reckoning and Reasoning

I don’t like arithmetic, maybe in part because I’m not especially good at it. But what I dislike more than the feeling of doing arithmetic is the fact that so many people think math is nothing but arithmetic.

So let’s start with arithmetic — because if I’m trying to undermine the view of math as a mule train, there’s no better place to start than the place where the tethers feel tightest.

Suppose someone asked you to compute 997 + 998 + 3 + 2. How might you do it? You could use the standard one-size-fits-all procedure for adding up a list of nonnegative integers illustrated below (I’m omitting the cross-outs and carries).

But if the niceness of the final answer leads you to suspect that there’s a slicker way, you’re right: 997 + 998 + 3 + 2 equals (997 + 3) + (998 + 2), which equals 1000 + 1000, which equals 2000. This alternative path will lead you to the right answer because addition satisfies laws: the commutative law, which guarantees that changing the order of the terms doesn’t change the sum (so that 997 + 998 + 3 + 2 equals 997 + 3 + 998 + 2) and the associative law¹, which guarantees that how you group the terms doesn’t change the sum (so that 997 + 3 + 998 + 2 equals (997 + 3) + (998 + 2)).

Unlike human laws, which constrain the behavior of people, number laws constrain the behavior of numbers and thereby free people to solve problems in flexible ways.

In Praise of Pedantry

Abbott: Funny thing, Lou: there’s an infinite number of numbers.

Costello: How many?

Abbott: An infinite number.

Costello: What infinite number?

Abbott: Oh no, there is no infinite number.

Costello: But you just said there was!

Abbott: No I didn’t. I said there’s an infinite number of —

Costello: There! You said it again!

— Abbott and Costello, in a number of their movies (specifically, the number zero)

Earlier this year I dug a mathematico-linguistic rabbit hole on Twitter when I wrote:

Dozens of people posted on that thread with very different takes on my question. One of them, Fred Klingener, reported to me the following actual example of infinity in real life:

Another respondent, Akiva Weinberger, dug a secondary rabbit-hole of his own for his linguistics pals on Facebook, who have their own brand of nerdiness distinct from, but parallel to, the nerdiness of math folks.

After jumping down both holes and crawling through all the tunnels, I emerged blinking into the light of day with renewed appreciation of the way different people can use different words to talk about the same thing.