Why I Love Math, Part 1: Because I Love Clever Movies

It’s no secret, I’m a mathy person. In fact, I just spent several hours of my free time this weekend trying to learn some new math, even though I didn’t need it for work.

Understandably, this seems ridiculous to a lot of folks. I probably get asked, “how can you enjoy this stuff?” more often than I get asked any other question about math or physics.

Well, one of the reasons I love math is that I love surprises, and math is full of them. Two mathematical concepts that initially look very different might turn out to be exactly the same. A problem that looks literally impossible might turn out to have a really easy solution, if you’re clever about it. Or a very simple idea might turn out to have much more important consequences than you’d ever have thought.

Whenever I find a situation like this in mathematics, I get a real thrill. But it’s not some esoteric, high-level pleasure reserved for academics. On the contrary, it’s the same sense of “Oh wow! Oh awesome!” that anyone can get from watching a good, clever movie.
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(Geographic) Overcompensation and Projection

Buzzfeed posted something a couple weeks ago that’s been in my head ever since: maps of the US with the outline of other countries superimposed. Sometimes, the result is uninstructive:

Image

Japan is distinctly smaller than the US. I bet you knew that.

Sometimes, however, It really took me by surprise. Read more of this post

It’s All the Same Mushroom… Sort Of

I'm starting to make a concerted effort to make sure the pictures I use on this blog don't violate copyright laws. Consequently, they're getting a lot more dull so far...

The other day  I tried to look up Portobello mushrooms on wikipedia, and I was redirected to the page for the mushroom species  Agaricus bisporus. There, I was presented with a list of all the different names they can go by: White, button, crimini, baby bella, portobello, swiss mushrooms, champignons…everything I could think of  except Oyster, Shiitake, and Porcini. They were, apparently, all the same.

I was a bit taken a back, vaguely annoyed about all the times I’d paid extra for a fancier sounding mushroom, and extremely curious. I did a cursory investigation and found plenty of pages superficially confirming what I’d just read. I filed it away in the brain-folder I use for “did-you-know factoids to impress people with at dinner parties.”

But that’s also where things got a little more complicated. After I typed the title of the post, I realized I had basically given the way the twist ending and would need some more information to serve as filler. As I dug further into the topic, however, I discovered that my initial investigation had led me astray. Read more of this post

The High Probability of Coincidence

Coincidences are quite likely.

The odds of any particular coincidence will be quite small. But there are so many possible coincidences that the odds of any coincidence are very high.

Take any deck of cards. Shuffle it, ideally 7 times*. What are the odds that you find a pair of queens on the top of the deck? About half a percent, it turns out. Very small.

CardShuffleBut what are the odds of finding a pair of any kind, anywhere in the deck? I don’t know actually. That’s hard to calculate. But it’s much, much larger.  Go ahead, check for yourself. I just tried it five different times myself, and never failed to find at least one pair. Often more than one.

This is probably fairly obvious. But you’d be surprised how easy it is to lose sight of it.

Before a presidential election, for example, news magazines tend to fill up with stories about weird things that can predict the results. They’re played for laughs, of course. But when you see these articles discussed in comments, you’ll find lots of readers determined to explain “why” these coincidences work.

Some of them may have explanations, it’s true.** But the point is, these coincidences don’t need explaining because their existence is not unlikely.

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Walking on Water (Now in Slow Motion)

Alright, I lied, this post isn’t strictly speaking about walking on water. It’s about walking on water and cornstarch–a combination you’re probably familiar with if you’ve ever been a kid or raised a kid. When I was in elementary school we called it “Oobleck,” a reference to this book. I’m sure it goes by a lot of different names, but that’s the one I’ll use here.

Oobleck, famously, flows like a liquid unless it is exposed to sudden force, at which point it “seizes up” and acts much more like a solid. As a result, you can walk on it, if you do it quickly enough, because the sudden forces of your footfalls make it go momentarily stiff. But you can’t stand on it, because while you’re still exerting a force on the Oobleck with your weight, that force is no longer sudden.

It’s all summarized nicely by the antics of these nice gentlemen on some sort of Barcelona-Based, Bill-Nye style science show. You don’t have to speak Spanish to appreciate it, but if you do, please tell me, is Google correctly translating the name of the show (“Hormiguero Cientifico”) as “Scientific Ant Hill”?

This is in fact a rather “old” clip by internet standards. What caught my attention this week is this newer video showing the same trick, but in super slow motion:

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Quantum Superposition: Where the Magic Begins

Alright, here’s my first attempt to talk about quantum physics while still staying within my 500-word limit.* Wish me luck and brevity!**

Recall: a quantum object is any extremely small object–like a single atom. They’re interesting because, unlike a “normal” sized object (physicists call them “classical”) a quantum object can have two or more different, seemingly contradictory properties at the same time– at least as long as you’re not looking at it.

Compare, for example, a classical penny and a “quantum penny.” Suppose you can’t decide between having sushi or  tempura for dinner. If you flip a regular penny and cover it with your hand, you’re done. Underneath your hand is either a head or a tail. Now, you won’t know which one it is until you look, but it is definitely one or the other, just waiting there to be seen.

A quantum penny would not have to behave this way. You could flip a quantum penny, cover it, and insist (correctly) that you hadn’t made a decision yet: underneath your hand, the quantum penny is no more committed than you are to raw versus battered fish. It is essentially both “heads” and “tails,” since both are possible. This is what we call a “superposition” of the two. But never fear; you won’t be left staring at a menu all night. As soon as you lift your hand and look, it will randomly settle (physicists say “collapse”) into one or the other.

You may feel like there’s no functional difference between saying the coin has been heads the whole time and saying that it only decided to be heads when you lifted your hand, since both appear identical whenever you’re looking. But there’s a real, testable way to see the difference.

Suppose I gave you ten coins, and said that they were either all fair coins, or a mixture of some coins with two heads and some with two tails. If I tossed all ten, could you tell me whether the coins were fair? I don’t think so. Even if you got 9 heads, you might suspect they were mostly two-headed, but you couldn’t be sure it wasn’t just the result of fair flips.

But here’s the thing about quantum coins: Given the same setup, you can build an apparatus that takes the coins and uses quantum mechanics to tell the difference.*** But even more amazingly, the machine only works if you don’t look at the coins before feeding them in. If you look, the coins “collapse” to being just heads and tails, like the classical case. But if you don’t do that — if you leave them in their superposition–you can apparently learn something about them that you couldn’t otherwise. Hence, there was something special about their state before you peeked.

All clear? If you’re quantum, and you’re considering complementary states of being (heads or tails, up or down, Beatles fan or terrible human being) then whenever no one’s looking, you can be all of them at once. That’s the essence of quantum superposition, and in many ways the essence of quantum mechanics itself.

Boom! 500 words exactly :-).

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* This little introduction doesn’t count.

** Neither do footnotes.

*** The above is a very broad analogy for a famous kind of quantum experiment called a “double slit” experiment, in case you want to read more about the details. I’m doing my best to recast it in the more comfortable “coin” picture, but the actual set up is physically quite different. For experts (and if you are one, please feel free to critique), the parallel I’m trying to draw is this: each side of the coin represents one of two paths taken by a particle in a 2-slit experiment, so each fair coin collectively represents a particle randomly allowed to travel through either slit. An “unfair” coin corresponds to a case where one slit is blocked, guaranteeing one path over the other. Run these two with billiard balls (targeting random slits vs. randomly closing off one slit at a time) and you won’t see a difference in their detected locations. Run it with electrons and you will, since only the first case will allow an interference pattern to form. What’s more, if you “look at the coins before they’re detected” (i.e., if you measure which slit the electron enters along the way) you will destroy your ability to distinguish the cases because no pattern will form. Hence, there is something different between the unobserved quantum state and the classical analogue.

Less is More?

Last week, I wrote way too much about quantum teleportation. In response, several different people (not just you John!) suggested that instead of trying to explain such a complex topic, I should start with something smaller. Like quantum entanglement, which you really have to understand before you can appreciate teleportation. So I set out to write a small primer on quantum entanglement.

Then, after several paragraphs of explaining the background information you need to appreciate entanglement, I realized I’d need to take yet a further step back.

All this is to say, meet my new goal for WLOG blog: For the rest of the year, my new goal is to keep all my posts under 500 words. Stop laughing! I think I can do it. But wish me luck– this goes against the very nature of my being as a Stony Brook Babbler….

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