Hyperbolic Cosine & The Arch

St. Louis, MO
7/6/2010

Sometimes, people try to tell me the arch in St. Louis is a parabola. I am pretty sure it is not. A better model for the curve of the arch is the catenary – the path a hanging chain marks out. Granted, to get the Gateway to the West, you would have to turn the catenary over. This can easily be done with a quick multiplication by -1.

The catenary is the graph of hyperbolic cosine, or cosh. Cosh, in turn, can be defined in terms of the exponential function, e^x. In sloppy math-ese, one might say arch(x) = -cosh(x). (I forgot the negative on that park bench. Really though, orientation is arbitrary. Stand on your head, and arch is cosh.) Really though, the arch is not a graph of hyperbolic cosine. The graph of cosh is two-dimensional at best. The arch standing there above the Mississippi is at least three-dimensional.

Experts, cited on Wikipedia, agree that the Arch is not a true catenary; it is not simply cosh.

Saarinen originally conceived the Arch as being thinner at the top than at its bases so that the form would seem to soar toward the heavens. He experimented with two catenaries one inside the other for the intrados (inside) and extrados (outside) of the Arch, but he was dissatisfied with the sculptural appearance feeling that it was too severe. Saarinen and his associates ended up choosing a weighted catenary (with bases heavier than apex) that could be seen only as an imaginary line going through the center of the Arch. Thus, neither the extrados nor the intrados of the arch was a catenary. That choice was rooted in the type of visual logic the Greeks used in their architecture where appearance took precedence over precise mathematical logic of architectural forms.

From Laura Soullière Harrison (1985) (PDF), National Register of Historic Places Inventory-Nomination: Jefferson National Expansion Memorial Gateway Arch / Gateway Arch; or “The Arch”.

If you want to read more about cosh, check out the definition of hyperbolic cosine here.