The Mathematics of Architecture

Rob Hardy

In his autobiography, Bertrand Russell wrote of his passion for mathematics, "I have tried to apprehend the Pythagorean power by which number holds sway about the flux." The Pythagoreans held all things to be numbers, and it wouldn't be surprising if Alexander J. Hahn, a professor of mathematics, held them in the same reverence. If all things are numbers, so all buildings are numbers, and Hahn's Mathematical Excursions into the World's Great Buildings (Princeton University Press) is an attempt to show mathematical and architectural principles together through history. Both subjects are vast, but Hahn hits highlights of both as he ties the two together. This is a good-looking volume, large format, with plenty of illustrations consisting of old and new pictures of famous buildings and force diagrams and curves in the Cartesian plane. It also could be a text for a course in mathematics; every chapter ends in a list of "Problems and Discussions," exercises to be solved by the student, with answers not given. Since these range from the Pythagorean theorem to full scale calculus, they may be more than the general reader (such as this one) might want to attempt. Some of the math within the text is daunting, too, but Hahn's explanations are clear and his enthusiasm is obvious.

There are themes that run through the chapters here to make the book a consistent whole. One is the idea of the arch. Builders didn't use arches in the beginning; at Stonehenge, for instance, a horizontal lintel rested on two uprights. The Parthenon and plenty of other classic structures use columns and lintels, too, and Hahn explains the forces involved. Stone does very well under compression; the uprights are hardly deformed at all by the weight they have to carry. The stone of the lintel, however, is not only compressed, but is pulled apart by tension at the bottom; if the tension is high enough, the stone sags, and if it is higher still, it pulls itself apart. To better hold up weight between two uprights, the Romans used the arch, typically the half-circle version that you can see in many of their buildings and aqueducts. You will here find analysis of the forces on the "voussoirs" (stones making the arch, with the keystone the top one), the friction between them, and the outward thrust.

The Romans did little calculation for making their arches, but of course used the mathematics of geometry. The Gothic arch was made of circles, too, but of segments meeting each other at a point rather than a complete half circle, and such arches had less outward thrust. (Hahn describes how there were few calculations made for the magnificent Gothic cathedrals but that geometry told all: "The Gothic concept of structure relied on geometric and numerical relationships and not on considerations of loads, thrusts, and stresses.") The idealized arch is one that supports its own weight and gets no support except at its very base. You can see this sort of ideal in the Gateway Arch in St. Louis. (It is often reported to have the curve of a parabola, which it resembles, but the two curves are mathematically distinct.) Hahn does the calculations to show that the ideal arch is what is called a catenary, from the Latin word for chain; the way a chain hangs in a U shape when its two ends are supported is a catenary curve. When Christopher Wren, with the help of Robert Hooke, came to design St. Paul's Cathedral in London, he and Hooke could not have estimated mathematically the loads and thrusts, but they did know what structural shapes best resisted those forces. It was Hooke's idea to make the middle dome in the shape of an inverted hanging chain. It is under the famous dome you see on the outside and over the painted dome of the inside, so it is not noticeable, but it is essential for carrying their weights. Antonio Gaudi used catenary arches in many corridors, but he also paid close attention to hanging chains; his impossibly complicated model of his masterpiece La Sagrada Familia is shown here; it is upside down and consists of an astonishing number of connected loops of string with attached weights, and it looks like his cathedral upside down.

Related to the arch is the dome; a dome may be regarded as an arch spun on its axis. Like the arch, it has a tendency to bow outward, known as hoop stress, and to contain the stress, chains are often integral within the bottom wall of the dome. If the chains do their job perfectly, the dome simply sits, as Hahn says, like the lid on a pot. The calculations of such stress can be done now, but when St. Peter's in Rome was being built, all they knew was to put big chains around the bottom. There weren't enough; cracks were found in the dome shortly after it was finished, and they got worse. The mathematician and engineer who repaired the dome used extra chains, after analyzing the forces of the dome by representing them with a hanging chain. There are pages here as well on Brunelleschi's famous dome in Florence, the domes of San Marco in Venice, the Pantheon in Paris, and the US Capitol. The latter dome was a work of cast iron and calculation, and though it has had cracks and breaks, it is still sound almost a century and a half after it was completed.

More pages are spent here on the Sydney Opera House than any other modern building. Its designer, Jørn Utzon, submitted a sketch of its sail-like halls and the idea was approved, but the shape of the billowing roofs had not been considered in detail. People loved the shape, but how to achieve it was not clear. Utzon considered the parabola and then the ellipse, but ran into problems. There were constraints of time and money, so the huge shells were going to have to be constructed in pieces, and the pieces would have to be mass produced. A shell based on a paraboloid or an ellipsoid would not do because these shapes curve differently in every region. The only shape that curves the same way everywhere on its surface is a sphere, and Utzon had a flash of inspiration that saved the project. There are large "sails" and small ones in the building, but they are all essentially triangles drawn on one big sphere of fixed radius. The details of the construction and its geometry are described here; this seems to have been an ideal teamwork between the architect and the engineers who were to build the complex.

This is a handsome book that ought to be enjoyed by anyone who is interested in architecture or in the practical application of (sometimes advanced) mathematics. The union of the two disciplines has been put into fast forward by the computer, whose application to architecture is considered in the book's final pages. Only rudimentary computer design aids were available to the architect and builders of the Sydney Opera House, but those shells would never have been made without even this moderate computer assistance. By the time Frank Gehry was designing the fantastic Guggenheim Museum in Bilbao, computers were integral in visualizing and making models of the prospective building, and then computers helped cut and mill the parts to go directly into and onto the building. When this book is revised in twenty years, there is no telling how the math will have changed the appearance and engineering of the buildings that will have sprung up by then.