Andrew Witt’s book *Formulations: Architecture, Mathematics, Culture* is a fascinating exploration of the influence of mathematics and the sciences in general on architecture and design. It is densely written, with extensive footnotes and quite a few illustrations. Still, I found it handy to have access to a search engine to explore many of the references further.

The author has an academic background in both mathematics and architecture. As a scholar of both, he demonstrates connections between mathematics and architecture (and, more generally, design) concentrating on the mid-twentieth century up to the time of digitization, especially the decades between 1940 and 1970. The mathematics involved includes topology, information theory, polyhedral geometry, matrix analysis, complex analysis, etc. Thus, we are taken far beyond the traditional connections of mathematics to architecture, namely proportion, golden rectangles, and numerology. According to the author, the influences of biology, chemistry, and physics on design occur through the lens of mathematics. Further, each chapter of the book should be regarded as a cabinet of ideas, designs, and formulations that are related to each other from mathematics, the sciences, and design, so that the reader can consider the connections before moving on to the next cabinet.

The text makes use of frequent illustrations to convey these connections. Chapter 2 explores how architects are able to use their instrumental knowledge of mechanical devices to draw complicated curves which they may not understand geometrically. The instruments discussed include Suardi’s geometrical pen, the helicograph, and the clampylograph. Some of these instruments appear in illustrations, and some of the curves drawn were for decorative patterns. Conic sections also provided the designs of vaults, windows, and embrasures. The text does a nice job of incorporating historical context. I was delighted to see a quotation here from Alfred North Whitehead: “It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking about what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations we can perform without thinking about them.” [page 38, from Whitehead’s *An Introduction to Mathematics*, 1911, page 61.]

Moreover, Chapter 3 illustrates the use of three-dimensional models in education of artists, including photographers, sculptors, visual artists, and designers, engineers, and architects from 1900 up to 1920 across the United States and Europe. Specific examples of models and designs used by architects and artists such as Antoni Gaudí, Frank Lloyd Wright, Max Ernst, and others are depicted and discussed along with the theory provided by mathematicians like Théodore Olivier, Felix Klein, Richard P. Baker and Albert Wheeler (including a mention of the MAA), and the collective known as Nicolas Bourbaki. The illustrations in both these chapters are effective in conveying the mathematical structure underlying these tools and works of art.

More practical applications of mathematics for architecture, engineering, and economics can be found in later chapters. Surveying, which at one time was in fact included in the curriculum of the mathematics department at my institution, is viewed here as a computational representation of reality, including height, length, and width, but also time, cost, and energy in Chapter 4. Both Carl Friedrich Gauss and Henri Poincaré were involved in national triangulation projects. Also, Poincaré’s three popular books were excerpted in architectural journals. Four-dimensional considerations, including those of American mathematician Washington Irving Stringham, appear here. Other mathematicians included in this chapter are Leon Battista Alberti and August Möbius, for example, while the mathematical topics of this chapter include mesh geometries and hyperdimensonal parameterizations. Chapter 8 introduces the discretization of space, following that of labor (by Henry Ford) and process (by Frederick Taylor), was in part a response to the great depression, with the thought of creating modular forms with a cubic standard. The author notes the cubic lattices in chemistry and standardized shipping containers as similar themes. The modularity led to the gamification of planning and the use of linear Diophantine equations to determine possibilities.

While some of the book applies mathematics to these practical considerations, the mathematics can also be an inspiration to the artists and architects. Topological ideas appear in architecture since at least the 1930s, but there was a greatly increased interest in the 1990s. As an example, there was a proposed rail station for Arnhem, the Netherlands, based on a Seifert diagram. The author includes a discussion of knots in mathematics, chemistry, and physics, as well as minimal surface films. The topologist Max Dehn taught designers and artists, Josef Albers shared ideas of gestalt coherence and topology with sculptural and architectural designers, and Sergio Buscemi used minimal surfaces in his design of the Ponte sul Basento in Potenza, Italy. Providing further evidence, some design schools included mathematical topics such as fractal geometry, curve theory, probability, and linear programming in their curricula. Network maps and Voronoi diagrams are also included the context of design decisions.

Less currently-popular design topics with their associated mathematics are also included. Hyperbolic and ruled-surface forms were used for their visual impact in marketing media for popular, architectural, engineering, and production press. These designs had largely disappeared in architecture by 1970 but continued in popular outlets such as the *Whole Earth Catalog*, the *Dome Cookbook*, and the* Inflatocookbook*. In another fad, the 1950s saw a new crystal mania, drawing on group theory, homeomorphisms, and lattice structuralism. Hermann Weyl’s Symmetry is mentioned as an influence with the 1951 Festival of Britain given as an example of the popularity. The author notes that approaching design via advanced algebra and geometric topics paralleled the rise of New Math in the American educational system. One popular artist to make an appearance in this chapter is M. C. Escher via his parquet deformations. Crystallographic architecture died out in the 1980s.

Readers may be interested to learn about the inspiration of mathematics to science and technology. Stereoscopy and stereophotograhy were of value to scientists, including in crystallography and for visualizing data sets. In Chapter 5, the Necker cube, drawings of James Clerk Maxwell, and anaglyphs, are additional examples. NASA’s Alan Schoen’s stereoviews of triply periodic minimal surfaces inspired architects and sculptors in the 1970s. Ultimately, the author considers these topics as the forerunners of virtual reality. Chapter 6 begins with cloud chambers that showed the paths of particles and begins a discussion of graphic images in physics spilling over into the visuality of design. The German art school, the Staatliches Bauhaus, argued for the unity of art and technology, while Walter Gropius sought to counter what he felt was overspecialization in the sciences. Perhaps spatial and optical perception might provide a basis for common ground between architecture and technoscience; this would include such things as perspective, optical illusions, and color consideration. Mathematicians and mathematical topics mentioned include Gottlob Frege, Bertrand Russell, and Lissajous curves. To conclude, the texts “speculates on the future of mathematics in the cultural mutations of architecture”[page 27]. It is clear that the author sees a future for the continued interaction of these two creative disciplines.

It would be good if every university library contained a copy of this book. Regularly, mathematics students seek to explore the connections between mathematics and art. This book will provide motivated students a plethora of suggestions for exploring connections of nontrivial mathematics to architecture and design.

Joel Haack is Emeritus Professor of Mathematics at the University of Northern Iowa. He can be reached at

joel.haack@uni.edu.