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Why Mathematicians Can’t Find the Hay in a Haystack

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The first time I heard a mathematician use the phrase, I was sure he’d misspoken. We were on the phone, talking about the search for shapes with certain properties, and he said, “It’s like looking for hay in a haystack.”

“Don’t you mean a needle?” I almost interjected. Then he said it again.

In mathematics, it turns out, conventional modes of thought sometimes get turned on their head. The mathematician I was speaking with, Dave Jensen of the University of Kentucky, really did mean “hay in a haystack.” By it, he was expressing a strange fact about mathematical research: Sometimes the most common things are the hardest to find.

“In many areas of mathematics you’re looking for examples of something, and examples are really abundant, but somehow any time you try to write down an example, you get it wrong,” said Jensen.

The hay-in-a-haystack phenomenon is at work in one of the first objects that kids encounter in mathematics: the number line. Points on the number line include the positive and negative integers (such as 2 and –29), rational numbers (ratios of integers like $latex \frac{3}{2}$ and $latex \frac{1}{137}$) and all irrational numbers — those numbers, like pi or $latex \sqrt{2}$, that can’t be expressed as a ratio.

Irrational numbers occupy the vast, vast majority of space on a number line — so vast, in fact, that if you were to pick a number on the number line at random, there is literally a 100 percent chance that it will be irrational.*

Yet despite their overwhelming presence, we almost never encounter irrational numbers in our daily lives. Instead we count with whole numbers and follow recipes with fractions. The numbers we know best are the extremely rare numbers, the special numbers — the needles in the haystack.

The hay is hard to find precisely because it’s so unexceptional. Rational numbers have the distinctive property that it’s possible to write them down. This calls them to our attention. Irrational numbers have an infinite decimal expansion. You couldn’t write one down even with an endless amount of time. That these numbers lack the exceptional property of “write-down-able-ness” is what makes them nearly invisible to our way of seeing.

“We’re looking with a magnet, and you’re not going to find hay with magnet; you’re only going to find needles,” said Dhruv Ranganathan, a mathematician who is in the midst of a move from the Massachusetts Institute of Technology to the University of Cambridge.

The search for hay in a haystack characterizes many different areas of math, including the subject of my most recent Quanta article, “Tinkertoy Models Produce New Geometric Insights.” There I wrote about mathematicians who are investigating the relationship between geometric shapes and the equations used to describe them. In rare cases, objects can be expressed by simple equations. These are the needles, the shapes we know best: lines, parabolas, circles, spheres.

The overwhelming preponderance of shapes resist such elegant formulation. They may be everywhere, but because you can’t write down the equations that describe them, it’s hard to establish that even a single one of them exists.

In my article, I explained how techniques from a field called “tropical geometry” serve as an especially sly way of deducing the existence of these ubiquitous geometric objects — the ones that, like the irrational numbers, are everywhere, even if you can’t write them down.

In mathematics it often happens that either something doesn’t exist, or it exists in abundance. The nature of those abundant objects might make them hard to detect, but if you’re a mathematician and you believe they’re there, and you believe they constitute almost all of everything, your task is straightforward: Find just one.

It’s as if you were convinced the oceans were filled with water, said Ranganathan, but every time you took a sample, you came up with something else — a shell, a rock, a plant. Yet to start to believe your hypothesis was correct, you’d hardly need to empty the sea.

“All you have to do is find any water,” he said. “One droplet of water will do.”



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RyanAdams
2 days ago
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Central Indiana
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jlvanderzwan
5 days ago
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I kind of wonder if this isn't a bit of a perspective problem, like saying "there is infinitely more space than there is stuff occupying that space, yet we can't touch it"

What's your favorite piece of trivia?

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There's good trivia and bad trivia. Bad trivia is mere fact (e.g. the population of Toronto is .....). Good trivia is entertaining, surprising, insightful. What's your favorite bit of good trivia?
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RyanAdams
13 days ago
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Central Indiana
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I played every game about monkeys on Itch.io

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I could detail the circumstances that led me here, to this place, at this moment in time, but it is a tangled web we weave, and the machinations of fate are as unknowable to us as the span of a gorilla’s back to the tiny flea, or something. Did you know there are 49 games about monkeys on Itch.io? I have decided to play them all. (more…)

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RyanAdams
13 days ago
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Monkey.
Central Indiana
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How constraints lead to creativity: making music for Super Nintendo games

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In this short video, Evan Puschak talks about how music is made for Super Nintendo games. That system was first released in 1990 and the audio chips could only hold 64 KB of information, only enough room for beeps, boops, and very short samples. But composers like David Wise, whose soundtrack for the Donkey Kong Country series of games is on many lists of the best video game music, were able to make the SNES sing despite its limited capabilities.

Tags: David Wise   Evan Puschak   music   Nintendo   video   video games
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RyanAdams
30 days ago
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Central Indiana
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Windows 95 Is Now Available As an App. Literally.

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Windows 95 is now an app you can download on Windows, Mac, or Linux. Yep.

The post Windows 95 Is Now Available As an App. Literally. appeared first on Thurrott.com.

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peelman
31 days ago
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i guess my comment would be that this is a good example of why Slack has so many problems. because this is the kind of shit their developers undertake as a side project.
Seymour, Indiana
RyanAdams
32 days ago
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Central Indiana
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Six things I just learned about Inspector Gadget

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Inspector Gadget costume available from HalloweenCostumes.co.uk.

Last week at Gen Con I had the pleasure of learning a bunch of dumb stuff I didn’t know about Inspector Gadget! And now I will share it all with you.

Like everyone in the world, I watched the television program as a kid, and chuckled along to the antics of the terrible lawman and his orphan ward and cryptid dog. I knew about the two live-action movies as well, though I’ve never actually watched them all the way through. And I vaguely knew that there were some various other Gadget adaptations out there.

NOW, however, I know all that AND these six other things, which I will share with you now:

THING ONE: Inspector Gadget had a moustache in the original pilot, but DiC (the production company) was sued by MGM because he too closely resembled their own Inspector Clouseau.

Rather than re-animating the pilot, a line of dialogue was added to later showings, in which Penny comments on the moustache and Gadget says: “It’s so that nobody will recognize me. I’m on vacation, absolutely, totally and completely off duty.”

Here’s the original version of the opening theme with the moustache present: [On YouTube]

THING TWO: In the 1999 live-action movie, there is a scene in which Inspector Gadget covers a lady with 100 gallons of toothpaste. It’s just as terrible as it sounds:

I agree with one of the YouTube commenters: “Why would they even put that much toothpaste in him anyway? How often is he gonna come across a situation that requires a crap load of toothpaste like that?”

THING THREE: There have been THREE spinoff television shows since the original:

Gadget & the Gadgetinis (2002–2003), in which the Inspector has been promoted to Lieutenant and Penny has built him two miniature robot sidekicks. [On YouTube]

Gadget Boy & Heather (1995–1998), in which the brain of an adult policeman has been placed into the body of a child for some reason, and the child given all the typical crimefighting gadgets. The villain is an 8-armed spider-lady named Spydra. [On YouTube]

A current CGI version of Inspector Gadget (2015–present), a “sequel” to the original, presently available on Netflix in the US. In this version, Dr. Claw has a teenaged nephew, whom Penny has a crush on. [On YouTube]

There have also been two animated movies: 2002’s Inspector Gadget’s Last Case [wiki/YouTube], which features Jaleel White (Urkel) as Gadget’s car, and 2005’s Inspector Gadget’s Biggest Caper Ever [wiki/YouTube], which features Bernie Mac as Gadget’s car.

THING FOUR: Inspector Gadget once met the Mario Brothers:

For this live-action segment, Gadget was played by Maurice LaMarche, who would go on to voice Gadget in several of his other animated incarnations once Don Adams had retired from the character (and from this world).

If anyone can make it through the entire thing, let me know what happens.

THING FIVE: There are, surprisingly, not that many interesting covers of the Inspector Gadget theme song on YouTube, but this one done in Mario Paint is rather charming:

I particularly like its judicious use of the Yoshi sound effect.

THING SIX: Speaking of the Inspector Gadget theme song, it turns out that the one we in the English-speaking world all know so well is merely a VASTLY INFERIOR ITERATION when you learn that in other languages, IT HAS MORE LYRICS.

Here’s the French version:

It doesn’t have the “Go, Gadget, go” breakdown in the middle, but oh là là, DOES IT EVER HAVE LYRICS.

Our friendly local Francophone at Gen Con broke them down for us. There is a lot of repetition, but in summary and with some idiomatic translation, it goes something like this:

Eh la qui va là // Hey there who is that
Inspecteur Gadget // Inspector Gadget
Eh la ça va pas // It’s not going well
Ouh Ouh! // Ooh ooh!
Oh la je suis là // Hey there I am here
Inspecteur Gadget // Inspector Gadget

C’est moi que voilà // It’s me, I am here
Inspecteur Gadget // Inspector Gadget
Ca va être la joie // It’s going to be fun
Ouh Ouh! // Ooh ooh!
Au nom de la loi // In the name of the law
Moi je vous arrête // I will arrest you
Je vous arrête là! // I’m gonna stop you there!

[CHORUS]
Go Go, Gadget à main // Go Go Gadget hand
Flash, Gadget au chapeau // Flash! Gadget in hat
Hé ho, Gadget au poing // Hey ho, Gadget fist
Oh la, Elastico-Gadget // Oh that’s Elastico-Gadget

Les bandits sont là // There the villains are
Inspecteur Gadget // Inspector Gadget
Ils n’échapperont pas // They will not escape
Ouh Ouh! // Ooh ooh!
Si l’inspecteur fait gaffe // If the Inspector watches out
Fait gaffe aux gadgets // Watches out for his gadgets
Qui marchent ou marchent pas // Which may or may not work

[CHORUS]
Go Go, Gadget à main // Go Go Gadget hand
Là-haut, Gadget au chapeau // Up there, Gadget in hat
Go Go, Gadget au poing // Go Go Gadget fist
Qui c’est? // Who is that?
C’est Elastico-Gadget // That’s Elastico-Gadget

Et puis patatrac // And then, bang!
Inspecteur Gadget // Inspector Gadget
Voilà le chef qu’est là // Look, here is the Chief
Ouh Ouh! // Ooh ooh!
Salut chef c’est moi // Hello, Chief, it’s me
Inspecteur Gadget // Inspector Gadget
Ça n’en finit pas // It just never ends!

I’m super charmed by the mention of both Gadget “hand” and “fist” (they’re different?), and then the mysterious “Elastico-Gadget.” It sounds like it could be his springy legs, but the video shows his helicopter hat at that point! WHAT IS EVEN GOING ON.

If anyone can advise if either the Spanish version or German version communicates any new insight, please leave a comment on this post…

Until then, I’ll be humming “C’est moi que voilà, Inspecteur Gadget” for the foreseeable future.

YOU’RE WELCOME

Edited to add: Lots of good comments on the Twitter thread for this post, too!

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RyanAdams
39 days ago
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Central Indiana
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