This appears to be a season for books about architecture and mathematics. Earlier this year a book by Jane and Mark Burry, *The New Mathematics of Architecture *(Thames & Hudson, 2011), tracked down some significant recent buildings that display in their construction rather sophisticated mathematics such as the use of Möbius bands, Klein bottles, Sierpiński sieves, and the Costa-Hoffman-Meeks surface. The Hahn volume does not try to catalogue the most sophisticated uses of mathematics or anticipate the use of more and more ideas from topology, differential geometry, the study of minimal surfaces, and such. It goes back and starts with the Greeks and advances eventually to the Sydney Opera House, the bridges and train stations of Santiago Calatrava, and the Guggenheim Museum in Bilbao. It includes explanations of the mathematics involved, generally more classical and more accessible to students than that used in the Burry and Burry book. This new work makes a nice complement to the earlier volume, however, and it could also serve as a text (or at least a supplementary text) for a course for liberal arts students on the appearance of mathematics in everyday life. Both books are handsomely produced and lavishly illustrated — and, not least, printed on high quality paper. The Hahn book goes further in being useful to recommend to students, however. It includes in each chapter a set of problems ranging from the very elementary to those that would be appropriate as stimulating challenges for mathematics majors. The author also uses these problems to introduce the readers to buildings that are not touched on in the text itself, but add considerably to the range of mathematical ideas used in building design.

Starting with antiquity the author introduces us to some of the standard questions, largely from mechanics, about weight-bearing and stress and the remarkable uses of science and mathematics in designing buildings with great aesthetic appeal: the Parthenon, the Pantheon in Rome, Romanesque cloisters, the Hagia Sophia, the Palladian villas, Gothic cathedrals, and beyond. Much of this will be familiar to many readers. But detailed and illustrated arguments on the role of the voussoirs versus that of the keystone in a Roman arch, for example, will clarify some basic facts that govern building design. They’re a bit closer to physics than to mathematics, but that’s fine. The author wisely avoids the trap of going too far in the direction of ascribing too much significance to the alleged appearance of the golden section in classical buildings, or in the applicability of Fibonacci numbers in these areas, claims that border on numerology or mysticism at times.

As the author argues in his introduction, he is intertwining two stories: “The mathematics provides clarifying insights into the architecture, and, in turn, the architecture is a stage that gives visibility to applications of abstract mathematics.” He tries in the first several chapters to stay close to high school mathematics, but eventually he moves into calculus and beyond. But what he covers suffices for most of the early architectural topics: “domes, arches, columns, and beams.”

In the Renaissance much was made of the problem of keeping domes from collapsing: the challenges Brunelleschi faced, for example, in designing and building the Duomo (Santa Maria del Fiore) in Florence. There was also much interest in using perspective and the study of vanishing points in pictures. Naturally, one of the illustrations has to be the picture, sometimes attributed to Piero della Francesca, now in the Palace of the Duke of Urbino (with a similar one in the Walters Art Gallery in Baltimore), illustrating clusters of buildings about a piazza where the challenge to the viewer is one of finding the vanishing point.

Whereas the Burry and Burry volume is largely looking ahead to the future — building on the models of mid-20^{th} century work of Eero Saarinen (the TWA building at Kennedy Airport, the Gateway Arch in St. Louis, the Dulles Airport Terminal outside Washington, DC) to the work of Frank Gehry with his Guggenheim Museum in Bilbao and the Disney Concert Hall in Los Angeles, and many others along the way, the Hahn volume tends to describe daring “mathematical buildings” up to the mid- to late-20^{th} century. The Sydney Opera House gets extensive treatment and it is fascinating to read the author’s explanations of the curved surfaces on that iconic building that has come to represent Sydney, if not Australia. It had a rocky start, however, experiencing cost overruns, since so much of the structure required custom-made materials. Little could be bought “off the shelf.” And, as beloved as the building is, it is not without detractors, even on grounds other than financial.

The Bauhaus maxim “Form follows function” followed by so many of the great architects of the 20^{th} century (Mies van der Rohe, Le Corbusier, Walter Gropius) may be evolving to “Function follows form.” One of the first to exhibit this was the Sydney Opera House. The concert spaces are shaped in such a way that there is little room on the sides of the stage for wings, making set storage between acts and performances very difficult; productions end up being designed to use such modest sets that some of the grandeur of opera is lost. And as glorious as the view across the water to the opera house is, suggesting great white sails, there are other interpretations possible. The sweeping curved surfaces are covered with small tiles, suggesting scales, so some have proposed that it is a reptilian creature crawling up out of the sea and up to no good! This tendency to design the exterior of a building to have dramatic impact is also evident in Gehry’s Disney Concert Hall, where some ceilings have to be so low that the spaces become oppressive or there are structural supports rising up in hallways impeding the flow of the audience from one part of the building to another.

Hahn tends to avoid discussing some of the aesthetic versus engineering limitations in favor of explaining how things work. And in that area he can be very helpful to the experienced architecture fan and the novice as well. And, as mentioned earlier, his problems often expand the range of the actual text to include such buildings as Gaudí’s La Sagrada Familia and aspects of the Mosque at Córdoba, though he seems more interested in the dome there over the Mihrab than that dazzling array of columns and arches throughout the building, surely one of the great architectural wonders of the world.

By restricting himself to the “great buildings” referred to in his title, the author has passed up some remarkable structures that use mathematics well within the time period he covers and require no mathematics beyond calculus: the McDonnell Planetarium in Forest Park, St. Louis (a hypocycloid of one sheet designed by HOK), St. Mary’s (Roman Catholic) Cathedral in San Francisco (seemingly hyperbolic paraboloids, the work of Pietro Belluschi and Pier Luigi Nervi), and catenaries (which he does cover with his material on the Gateway Arch) but where he could also cite the former Federal Reserve Bank in Minneapolis and the lowly Carmel Mission in California, dating from an entirely different era. There are good structural arguments to be made in each of these cases.

The mathematical sections are well illustrated and pictures of buildings abound. There is a special section of colored plates, including a beautiful shot of a few of the seemingly thousands of arches in the Córdoba mosque. Following the text is a helpful glossary of architectural terms, a list of references and an index. No effort has been spared to make this an informative and aesthetically pleasing book. And the problems are fun too.

Gerald L. Alexanderson is the Michael and Elizabeth Valeriote Professor in Science at Santa Clara University. He has served the MAA as President, Secretary, and editor of *Mathematics Magazine.*