Letters: The Gateway Arch is NOT a Parabola

E-mail from listeners prompts two corrections: One about Latin confusion and the other about the Gateway Arch in St. Louis. MANY listeners wrote in to point out that the Arch is not a parabola, as we said on air. It's a catenary curve. Stanford math professor Keith Devlin explains the difference.

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LYNN NEARY, host:

It's time for your letters.

A couple of corrections have come our way. Last week Scott Simon talked with Professor Paul Collins about the mysterious murder of Maria Martin in 1828. The crime was a sensation in Great Britain, every lurid detail reported by a journalist, James Curtis. One copy of Mr. Curtis's book on the crime was bound in the murderer's own skin. It's called anthropodermic binding.

Scott suggested that the term was Latin, but Demosanese Cazanes(ph) in Severna Park, Maryland wrote, anthropodermic is a Greek and not a Latin word, as suggested by Scott. It's meaning: made of human skin. Anthrop is human, and derma, skin. Mr. Cazanes should know. He's all Greek too.

And we're still trying to dig out of the piles of mail that came in when Scott Simon said this last week...

SCOTT SIMON: I'm Scott Simon. On this date in 1965, workers topped out the final section of the Gateway Arch in St. Louis. The world's best know parabola stands...

NEARY: Well, it turns out that the Gateway Arch is not a parabola. Robert Osserman(ph) wrote from Berkeley, California: I'm sure the Math Guy will be able to straighten you out on why the gateway is not a parabola, as stated by Scott Simon, but a catenary.

Well, we took him up on that challenge and invited the Math Guy to join us this morning. And Keith Devlin is with us now. Thanks so much for being with us, Keith.

Professor KEITH DEVLIN (Stanford University): Hi, Lynn. Thanks for the invitation.

NEARY: So the Gateway Arch is not a parabola but a catenary curve. Now, can you explain the difference for us?

Prof. DEVLIN: Right. Well, first I should say that Scott is not alone in making that mistake. No less a person than the great Galileo, the founder of modern science, made essentially the same mistake back in the 17th century. The shape is indeed a catenary. I mean Galileo wasn't talking about the St. Louis arch, that's for sure. It's a catenary. It's the shape you get if you take a chain, a long, thin chain, and you hold it out between your two hands or between to posts and let it sag in the middle. It's the shape the chain will take when it sags down due to gravity. That means if you turn it upside down, it's the most efficient shape if you want to make a large stone arch like the St. Louis arch.

Back in the 17th century, Galileo was thinking about what shape you'd get if you hung a chain between two points. He looked at it, he thought, yeah - like Scott did - he thought, that looks like a parabola, and Galileo thought it was a parabola. However, about 20 or 30 years later, it was shown that it's not a parabola. It's indeed one of these curves called a catenary.

NEARY: Well, I'm sure that I would not have made the same mistake that Scott Simon made or that Galileo made. But I assume it's easy to make it because they look alike.

Prof. DEVLIN: They look alike to the naked eye. In fact, it took quite a long time to find out what the equation was of a catenary. The parabola, as any high school student should know, and I hope does know, you get a parabola if you take the equation y2=ax2a, is a constant. A catenary is a much more complicated equation. In fact, you need to use calculus in order to figure out the equation of a catenary. Galileo was too soon; calculus hadn't been invented then. And Scott has maybe forgotten his calculus already, so he wasn't able to do it either. But it's actually quite a complicated curve.

And in fact, it's not just that people confuse the two. There is a connection, a rather interesting connection between a parabola and a catenary. If you take a parabola and you roll it along a straight line, then the focus of the parabola, which is like the equivalent of the center of a circle, the focus of a parabola will trace out a catenary, or a catenary - depends whether you're British or American as to how you pronounce it, I think.

NEARY: You know, I was about to make that point before we get any more mail on this question.

(Soundbite of laughter)

Prof. DEVLIN: Well, I understand that catenary or catenary is actually from a Latin word, not a Greek word - although, I may stand to be corrected. It is the Latin word, comes from the Latin word for chain. And it was an interesting question of mathematical physics in the 17th century: what exactly is the curve you get when you hang a chain between two points. Is it a parabola? Or is it, as we subsequently found out, a catenary, or a catenary?

NEARY: This is all way too highbrow for me.

Prof. DEVLIN: We don't even know how to pronounce the word, let alone write down the equation.

(Soundbite of laughter)

NEARY: We've been told that the Gateway Arch in St. Louis is not even a true catenary or catenary curve.

Prof. DEVLIN: No, it can't be, because it's got thickness, for one thing. And the thickness varies. It starts out pretty darn thick and it gets thinner as you - I've actually been there and stood at the bottom, and looked up, but I've never taken the tram ride inside. But it's not a mathematical catenary in the true sense. However, it is designed around the idea of catenary, simply because that is the most efficient shape that equals out of the pressure throughout the whole arch. That's why a chain hangs down that way. It's the most efficient way for hanging things down. It's the most efficient way for building arches that go upwards.

NEARY: You've even managed to change your pronunciation during the course of this interview.

Prof. DEVLIN: I'm a quick learner, Lynn. I'm a very quick learner. I hope Scott is too.

NEARY: Well, thank you so much for your expertise, Keith. We appreciate it.

Prof. DEVLIN: Okay. My pleasure. Thanks for having me on again.

NEARY: Keith Devlin is the executive director of the Center for the Study of Language and Information at Stanford University. His most recent book is called The Math Instinct: Why You're a Mathematical Genius Along with Lobsters, Birds, Cats and Dogs.

And you can send us your letters. Go to our Web site, npr.org, and click on Contact Us.

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