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How to make a rectangle out of a circle

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The key to thinking mathematically about curved shapes is to pretend they’re made up of lots of little straight pieces. That’s not really true, but it works … as long as you take it to the limit and imagine infinitely many pieces, each infinitesimally small. That’s the crucial idea behind all of calculus.

That's from a surprisingly terrific blog post about, well, math. Almost makes me want to dig out my high school textbooks.

Almost.

By Ezra Klein  |  April 8, 2010; 8:56 AM ET
 
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Comments

I don't think of myself as a math nerd but thats not a rectangle...

Posted by: spotatl | April 8, 2010 9:13 AM | Report abuse

But you can't circle a square. It's been proven impossible!

Posted by: DDAWD | April 8, 2010 9:19 AM | Report abuse

CircRect?

Posted by: pj_camp | April 8, 2010 9:38 AM | Report abuse

Umm, yeah, that's pretty much the idea behind integral calculus. Differential calculus is about slopes rather than areas--kind of the photo negative.

You humanities types really are adorable. I remember a nascent playwright that hung around our student theatre at my Unnamed Engineering School and liked to lecture me about how we all lacked a "well-rounded education." And somehow it never occurred to her that her lack of exposure to chemistry, physics, higer mathematics, and so forth, might have left her own education with some rough edges as well.

Go read "Godel, Escher, Bach" next time you want a wild trip but don't want to actually drop acid. If you know philosophy, it will teach you math, and if you know math, it will teach you philosphy.

Posted by: ckbryant | April 8, 2010 9:38 AM | Report abuse

That's not a rectangle. In the limit, it's a paralleleped.

Posted by: xtalguy | April 8, 2010 9:46 AM | Report abuse

I read every one of the NYT math articles and sadly it always reminds me of how much I've forgotten from my calculus classes. I do remember though, while a math instructor was helping me with a problem, asking her how long did take you to figure this stuff out? And her reply was, "after about the third semester of teaching it".

Posted by: philphil40 | April 8, 2010 9:48 AM | Report abuse

spotatl, as the wedges get thinner and thinner, the shape formed by lining them up becomes closer and closer to a rectangle.

Posted by: maralenenok | April 8, 2010 9:48 AM | Report abuse

"And somehow it never occurred to her that her lack of exposure to chemistry, physics, higer mathematics, and so forth, might have left her own education with some rough edges as well."

I remember a couple of business types asking me what I thought the proper requirements were to be a programmer. I replied, English Poetry.

Don't waste your time with math.

Posted by: leoklein | April 8, 2010 10:01 AM | Report abuse

This is a bit deceptive. Pi is a transcendental number, which means it cannot be written in any finite number of terms involving those commonly used mathematical operations you learned in calc (power, roots, adding, etc.). So a better statement would be that this shows you can get as close to squaring a circle as you would like, reduce error, by finding more and more decimal places of Pi.

Posted by: chrisgaun | April 8, 2010 10:52 AM | Report abuse

@leoklein: lol! Too true. My best subjects were always reading and spelling, I failed Calculus twice in college, and I'm a programmer. People often look incredulous when I tell them that; even moreso when I tell them that math really isn't important for most corporate programming gigs.

I think this post just convinced me to buy the set of Calc DVDs from the Teaching Company. Screw textbooks.

Posted by: BigTunaTim | April 8, 2010 10:54 AM | Report abuse

Another "Godel, Escher, Bach" reference. This is one of the most overrated books of the previous century (and now just about the only thing Borders carries in it's philosophy section). When Escher, to cite one example, draws an "endlessly rising staircase", I rather suspect the image is playing on the interpretive anomalies created in rendering a three-dimensional pattern in 2-dimensional space. Hofstadter treats it as a profound paradox. And the book never rises above that sophomoric level, in music, logic, or "art". (Plane tesselated with frogs, anyone?)

Posted by: sadtosay | April 8, 2010 10:57 AM | Report abuse

*My best subjects were always reading and spelling, I failed Calculus twice in college, and I'm a programmer. People often look incredulous when I tell them that; even moreso when I tell them that math really isn't important for most corporate programming gigs.*

This is precisely the reason that I left my corporate programming job.

Math is fundamental to understanding algorithmic thinking. However, math is incidental to programming in the same way that it's incidental to auto repair: sure, the design engineers can probably do a lot of calculus, but repair and assembly, essentially what a lot of programming jobs are, don't require it.

And if you're in IT, some of the best people who do that haven't even finished college.

Posted by: constans | April 8, 2010 11:53 AM | Report abuse

It's all an Ezra Kleinian metaphor. Think of the rectangle as the perfectly designed and executed policy undertaking. The circle is the set of things that are politically possible. Can the political process, the deal making, the log rolling, the nut cutting etc slice up the circle with sufficient fineness and accuracy as to approximate the rectangle of perfect policy. Not often, but in the limit it can be done.

Posted by: bdballard | April 8, 2010 12:06 PM | Report abuse

Well thanks a lot Ezra. I just wasted a good portion of my morning reading all the postings from that blog. Great stuff! I really liked the Feb. 14 post on multiplying negatives. I think I can justify that as work-related because of it's comparison to political relationships.

Posted by: bettyahrens | April 8, 2010 12:16 PM | Report abuse

I happened upon the NYT article series a couple of days ago and delightedly devoured the series. The one on "rock groups" took me right back to 7th grade math ('63)and the point where I began to feel math was separate from my interests. One diagram in the rock groups article brought back to me that point and illuminated what I should have seen then, but didn't. I'm apparently a visual learner and I should have been incorporating this approach all along. The series motivated me to buy three books (two math books and a novel) mentioned in the articles and I'm anxiously awaiting their arrival. It's never too late to appreciate the beauty of mathematics.

Posted by: jslamen | April 8, 2010 12:48 PM | Report abuse

Ah well, you programmers scoffing at calculus don't live in the real world do you. It's just 1s and 0s in your universe, no? What's lovely about calculus is that you can take an abstract law like the conservation of energy and get Bernoulli's equation, which explains how an airplane wing provides lift, using quantities we can physically measure like velocity, pressure, volume, mass. Now that's poetry becoming prose, my friends.

Posted by: Lonepine | April 8, 2010 1:01 PM | Report abuse

Ah well, you programmers scoffing at calculus don't live in the real world do you. It's just 1s and 0s in your universe, no? What's lovely about calculus is that you can take an abstract law like the conservation of energy and get the very applied Bernoulli's equation, which explains how an airplane wing provides lift, using quantities we can physically measure like velocity, pressure, volume, mass. Now that's poetry become prose, my friends.

Posted by: Lonepine | April 8, 2010 1:56 PM | Report abuse

Poetry, forsooth. I'm the last person on Earth who's going to under-rate the value of poetry, but a serious programmer needs a little exposure to combinatorics and a glancing familiarity with logarithms, calculus, formal languages, and high-school geometry (mostly because geometry is taught through proving theorems--if you'd rather study logic, then good on you). Such a person may not be a very capable graphical interface designer, but she will understand what she is doing and why in a way that no up-jumped English major with a copy of "Teach Yourself Visual Basic in 17 and 1/2 Minutes" can. Sooner or later, a programmer is going to need to understand what a pointer is: no integrated visual development environment (Platinum Pro Edition) is going to protect you from that for long.

Of course, a great programmer can also just be a natural genius at this stuff, which is the orgin of the dropout-master programmer. But I wouldn't advise anyone to drop out of school in order to further their career as a computer programmer. (Gates counts, probably, but Jobs doesn't: Jobs is a deal-maker, designer and pitch-man, not a programmer.)

I have seen that flummoxed look on far too many soi-disant programmers when their pretty GUI tools desert them. Good code is beautiful, and so is a poem, but so is an elegant proof. And the elegant proof is far closer to what we're really doing when we slam bits around.

I refer to the far more eloquent (besides which he's a better programmer than I) Joel Spolsky: "The Perils of JavaSchools." http://www.joelonsoftware.com/articles/ThePerilsofJavaSchools.html

Posted by: ckbryant | April 8, 2010 3:30 PM | Report abuse

David Foster Wallace wrote a wonderful book, "Everything and More : a Compact History of Infinity", that richly rewards the work necessary to get through it. Highest recommendation:

http://www.amazon.com/Everything-More-Compact-Infinity-Discoveries/dp/0393326292

Posted by: SamPenrose | April 9, 2010 12:04 AM | Report abuse

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