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# Mathematics: Handmaiden to Architecture

BOOK REVIEW: Mathematical Excursions to the World’s Great Buildings. By Alexander J. Hahn. Princeton University Press, Princeton, New Jersey, 2012, 317 pages, \$49.50.

Some years ago, while my son and his family were living in St. Louis, I visited the city and its Gateway Arch for the first time. The Arch was then already a great tourist attraction. (Its construction was begun in 1963 and completed in 1965.) On a later trip, which was professional, I was shown around the city by Professor ABC, a local mathematician. He naturally took me to see the Gateway Arch, and we went inside. Later, standing outside and looking up at the Arch, I asked my companion about the mathematics of the Arch:

What mathematics do you see in the Arch, or what mathematics does it make you think of?
I see a parabola, he replied. No, I think it must be a catenary.
Nothing else?
Nothing. Should there be something else?

Gaudí’s designs for La Sagrada Familia, Hahn says, “relied on careful and elaborate studies of complex systems of loaded strings.” One of his models is shown here. Photo by Cleftref.
I answered in a non-committal fashion, but in my mind I did not give Professor ABC a passing grade.

I return to the Gateway Arch and its mathematics below.

Alexander Hahn, a professor of mathematics at the University of Notre Dame and a prolific writer, has drawn on a vast and eclectic store of knowledge to produce an eyebrow-raising book filled with pictures and diagrams, including two dozen images in full color. In fact, Hahn has produced a half dozen books rolled into one. What we have here is a potpourri, a gallimaufry, a salad bar, or perhaps a celebration of mathematics in its infinite variety. Taking a cue from Stephen Leacock’s Lord Ronald, Hahn flung himself upon his mathematical horse and rode madly off in all directions.

Allow me to detail some of the directions.

Excursions is of “coffee table” size and quality. You can dip into its pages and view with enjoyment the many fine reproductions of famous buildings and then read how the author has linked them to mathematics. You may very well have visited some of these buildings: St. Peter’s in Rome, the U.S. Capitol Building, the Guggenheim Museum in Bilbao, La Sagrada Familia in Barcelona. . . . I’m surprised that he skipped Frank Lloyd Wright’s helical Guggenheim on Fifth Avenue in New York City.

Alternatively, you may consider giving Excursions to a young mathematical aspirant of junior high school age. He or she will find in it a Gradus ad Parnassum, so to speak, to the rudiments of Euclidean geometry, to elementary calculus, to statics, and even to a bit of numerical analysis. This textbook aspect is rounded out by the inclusion of many problems that the aspirant could work through.

If you are familiar with three-dimensional analytic geometry or with computer graphics, you will find considerable elucidation and delight within these theories. Thus, the “sails” or the “waves” of the famous Sydney Opera House are composed of a sequence of spherical triangles of varying sizes.

The soaring vaults of La Sagrada Familia in Barcel-ona, Gaudí’s masterpiece, “are at once playful, surreal, and inspiring,” Alexander Hahn writes in the book under review. Begun in the 1880s, construction of the church continues today. Photo by Sarika Bedi.Illustrations from Mathematical Excursions to the World’s Great Buildings.
If you are interested in Russian ecclesiastical history, you will be intrigued by the story of how Peter the Great and Charles-Augustin de Coulomb of electric charge fame (1736–1806), years apart, were involved in the design of St. Isaac’s Cathedral in St. Petersburg. Hahn devotes five pages to a discussion of Coulomb’s Essay on Problems of Statics (1773).

You may have read conjectures as to how the huge and mysterious heads on Easter Island in the middle of the Pacific Ocean were lifted into position. If so, you will be interested to know that the obelisk in front of St. Peter’s Basilica in Rome, standing 80 feet high and weighing more than 700,000 pounds, was moved in 1586 to its present location—a distance of 260 yards. “Five hundred mathematicians, engineers and others came to present proposals about how best to move the obelisk,” Hahn writes. The accompanying image from the book shows the scaffolding and suggests how the work was done.

In another illustration, you will see how the square roots of the successive integers can be arranged in a spiral that this reviewer dubbed the “spiral of Theodorus.”* How this is relevant to architecture is not explained. Nor is the inclusion of Gauss’s list of the regular polygons that can be constructed with ruler and compass.

“Use a word three times and it is yours” was a daily feature years ago in my city’s newspaper. In the glossary to Excursions you will come across the following words: squinch, groin, pendentive, and voussoir; propose to colleagues that they use these words and watch them blanch.

As promised, I return to the Gateway Arch. The author gives it short  shrift, paying much more attention to other famous buildings, e.g., the Hagia Sophia in Istanbul or the Colosseum in Rome. He describes the Gateway Arch as a “compressed catenary” and lets it go with the formula $$–A~\mathrm{cosh}(Bx/b) + (h + A)$$. Now that’s a formula for a plane curve. But the Arch is a solid, or rather a hollow shell large enough to house a tram that carries visitors. The cross sections of the shell are equilateral triangles tapering from 54 feet at the two bases to 17 feet at the top. Should Professor ABC have known all this? Should the author of Excursions have coined a special name for a 3-D squashed catenary with equilateral cross sections?

To complete the story of the mathematics employed to describe the Arch in its full scope, we should really add the hundreds, perhaps thousands, of informal computations, now hidden from view, that were performed during the design and structural stages in the offices of Eero Saarinen and Hannskarl Bandel. All this was done in the late 1970s, I tell my students, assuring them that there was mathematical life before the advent of MATLAB or computer graphics.

*Spirals: From Theodorus to Chaos, Philip J. Davis, A.K. Peters, 1993.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at philip_davis@brown.edu.