Arch Remarks

'' A bridge [...] is a very special thing. Haven't you seen how delicate they are in relation to their size? They soar like birds; they extend and embody our finest efforts; and they utilize the curve of heaven. When a catenary of steel a mile long is hung in the clear over a river, believe me, God knows. [...] the catenary, this marvelous graceful thing, this joy of physics, this perfect balance between rebellion and obedience, is God's own signature on earth. I think it pleases Him to see them raised.''

Mark Helprin - Winter's Tale
1983 Harcourt Brace Jovanovich, Inc.

Through the magic of the WWW, you can see the Brooklyn Bridge live! from which the above photo was taken. (I guess so you'll know if it collapses...) For more about the bridge, click on it. The history of its construction is a great book. On the other side of the Atlantic, the history of the Clifton Bridge is a good online story.

It's a fact that the curving main cables of a suspension bridge closely follow a parabola, that mathematical curve that shows up all the time in introductory physics. Proving this, however, is more difficult than the standard examples of parabolas like the path of a thrown ball. You're used to the parabola opening down, like the water jets under the force of gravity, but in the bridge the parabola opens upwards because there are more forces involved.

But not everything's a parabola: a cable by itself, not holding a bridge, just hanging from its ends, like a power line, follows a different curve called a catenary from the Latin word for "chain". It looks pretty similar. Galileo thought it was a parabola, and it took a few generations until it was proved otherwise: here's some history and interactive Java catenaries. Why should anyone care about the difference between the two curves? Because the curves are a direct result of the forces acting on matter, and understanding which curve is present in a specific situation means understanding the forces that work there.

Of course, a catenary can open down, too, like the parabola. But here, the physical examples are a little different. The bridge and power line and water jet were sorta "automatic" curves, if ya know what I mean: you set up the situation and the matter just follows the curve. But the opening-down catenary has to be deliberately constructed. For instance, an opening-down catenary is one of the shapes an arch can take, though arches are made in many different shapes. The catenary arch is physically special, though, for reasons to be revealed.

An architect is someone who builds arches. (OK, actually it's Greek for "master builder") A great one, Eero Saarinen, designed the 630 ft St. Louis Gateway Arch as a catenary. (he also designed the Dulles Airport terminal with catenaries) (Extra credit for those who think they already know all this: the photo here is a view at an angle to the St. Louis Arch: is the shape still a catenary? How about an oblique view of a suspension bridge, like at the upper left ... is it still a parabola?) If it's daylight, you can see the St. Louis Arch live! (again, I guess so you'll know if it collapses...must be a lot of suspicious people out there) Incidentally, you can ride up inside the arch: the history of the tram design is a good online story.

There are natural arches; can you guess how they form? (Check it out at the National Park link ... you'll have to dig) You might think this is an exception to what I said above, but no: this is an "automatic" arch, but it's not a catenary.

We mentioned that these pretty curves really follow from the physical forces that make them. Would you like to see how this works? Then
follow the force arrow!

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