Math

Ah, not this one again. The folks at Iglu Cruises have put together a helpful infographic to explain various features of the Gulf of Mexico oil spill (via Deep Sea News). Here’s the bit where they compare the recent spill (which, by the way, is still ongoing at a fantastic rate) to previous oil spills. Click for full resolution.

Doesn’t make the current fiasco seem so bad, does it? That little blob on the left looks a lot smaller than the blob right next to it, representing Saddam Hussein’s dump of oil into the Persian Gulf during the first Gulf War. In fact, when you think about it, it looks a lot smaller. Which is weird, when you look at the numbers and see that the current spill is 38 million gallons (as of May 27), while the Iraqi spill was 520 million gallons, a factor of about 14 times bigger. The blob representing Iraq’s spill seems a lot more than 14 times the size of the blob for the current spill. You don’t think — no, they couldn’t have done that. Could they?

Yes, they did. When measure the diameter of the circle representing the Iraqi spill, I get about 360 pixels (in the high-res version), while the smaller spill is about 26 pixels — a factor of about 14 larger. But that’s the diameter, not the area. The area of a circle, as many of us learned when we were little, is proportional to the square of its radius: A = π r². The radius is just half the diameter, so the area is proportional to the diameter squared, not to the diameter. In other words, that big blob is about (14)² = 196 times the area of the little one, when it should be only 14 times bigger.

I remember reading on some other blog about this same mistake being made in a completely different context, but I have no recollection of where. (Update: it was at Good Math, Bad Math, sensibly enough.) Probably won’t be the last time.

Here’s a fun logic puzzle (see also here; originally found here). There’s a family resemblance to the Monty Hall problem, but the basic ideas are pretty distinct.

An eccentric benefactor holds two envelopes, and explains to you that they each contain money; one has two times as much cash as the other one. You are encouraged to open one, and you find $4,000 inside. Now your benefactor — who is a bit eccentric, remember — offers you a deal: you can either keep the $4,000, or you can trade for the other envelope. Which do you choose?

If you’re a tiny bit mathematically inclined, but don’t think too hard about it, it’s easy to jump to the conclusion that you should definitely switch. After all, there seems to be a 50% chance that the other envelope contains $2,000, and a 50% chance that it contains $8,000. So your expected value from switching is the average of what you will gain — ($2,000 + $8,000)/2 = $5,000 — minus the $4,000 you lose, for a net gain of $1,000. Pretty easy choice, right?

A moment’s reflection reveals a puzzle. The logic that convinces you to switch would have worked perfectly well no matter what had been in the first envelope you opened. But that original choice was complete arbitrary — you had an equal chance to choose either of the envelopes. So how could it always be right to switch after the choice was made, even though there is no Monty Hall figure who has given you new inside information?

…

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Physicists and mathematicians who think about higher-dimensional spaces are, if they allow their interest to somehow become public knowledge, inevitably asked: “How can you visualize more than three dimensions of space?” There are at least three correct answers: (1) You can’t. (2) You don’t have to; manipulating abstract symbols is enough to help you figure things out. (3) There are tricks to help you pseudo-visualize higher-dimensional objects by cleverly projecting them into three dimensions; see here and here.

But really, why can’t we visualize things in more than three dimensions of space? Could a Flatlander, living in a world with only two spatial dimensions, learn to visualize our three-dimensional world? Could we somehow, through practice or direct intervention in the brain, train ourselves to truly visualize more dimensions?

I can think of a couple of explanations why it’s so hard, with different ramifications. One would be simply that our imaginations aren’t good enough to project our consciousness into a constructed world so very different from our own. Could you, for example, really imagine what it’s like to live in two dimensions? Sure, you can visualize Flatland from the outside, but what about asking what it’s like to really be a Flatlander? The best I can do is to imagine a line, flickering with colors, surrounded by darkness on either side. But the darkness is still there, in my imagination.

The other possible explanation is that the process of visualization takes up a three-dimensional space in our actual brain, preventing us from “tuning a dimensionality knob” on our imaginations. The truth is certainly more complicated than that (and I’m not experts, so anyone who is should chime in); the visual cortex itself is effectively two-dimensional, but somehow our brain reconstructs a three-dimensional image of the space around us.

Maybe this could be a new tantric discipline: visualization in higher dimensions. Or maybe the Maharishi already offers a course?

Following Scott Aaronson’s advice, I instructed the good folks at Amazon to send me a copy of The Princeton Companion to Mathematics. (In exchange for money, of course.) It’s sprinkled with gems like this, in the article on “Differential Topology” by my former professor Clifford Taubes:

If you are with me so far, suppose now that an advanced alien en route from Arcturus to the galactic center kidnaps you and drops you into some unknown, 2n-dimensional manifold. You suspect that it is Sⁿ x Sⁿ, but you are not sure.

Come on, the screenplay practically writes itself! I’m seeing Ewan McGregor, maybe Natalie Portman. Russell Crowe as the alien. SEEx could help with some of the mathy stuff. If any studio executives are reading this, call me, I’d be happy to bang out a treatment.

Seriously, the book is great fun, and as Scott says it’s surprisingly readable. Not really a popularization; neither equations nor high-level abstractions are shied away from. (After months of jousting with the “grammar checker” in Microsoft Word, I now deploy sentence fragments and the passive voice out of sheer spite.) But put into the hands of the right ambitious high-school student, it could be life-changing.

p.s. You haven’t really lived until you’ve seen Cliff Taubes do his little dance to illustrate the concept of “quantum fluctuations.”