Architecture and mathematics from antiquity to the future, Volume I: Antiquity to the 1500s by edited Kim Williams and Michael J Ostwald/Architecture and mathematics from antiquity to the future, Volume II: The 1500s to the future by edited Kim Williams and Michael J Ostwald

In De architectura, the only architectural treatise to have survived from Classical Antiquity, Vitruvius encouraged architects to equip themselves with a knowledge of multiple branches of learning, including philosophy, history, medicine, law, music, geometry, and astronomy: not to perfection, so as to become a master practitioner in any of those fields, but enough to know the theory, the principles or the ratio, common to all knowledge. Particularly invaluable was mathematics, especially geometry, which taught the architect the use of the rule and compass, the means by which designs could be laid out on the site. These geometer's tools allowed for the mediation between the realms of theory and practice, between design and building.

This traditional relationship between architecture and mathematics, taken for granted for centuries, has suffered from increasing specialization in the disciplines. With professionalization and the expansion of knowledge in each field, it has become untenable to sustain the image of the architect as a hybrid of Archilochus's proverbial fox and hedgehog. With specialization, it has become similarly difficult to imagine a multi-faceted figure such as the early modern mathematician, simultaneously engaging in theoretical and practical mathematics, finding numerous outlets (navigation, optics, surveying, etc.) for the application of his wide field of knowledge. Fortunately, scholarship of the last decades has begun to address the degree of impoverishment of our understanding of the past as a consequence of this forgetting of the multidisciplinary nature of knowledge and creation, of the relationship between architecture and mathematics. It follows an earlier revival of interest that began with the publication of Rudolf Wittkower's highly influential Architectural principles in the age of humanism (Warburg Institute, 1949), one of the first studies to focus on the preoccupations with geometry and harmonic proportions in Renaissance architecture. Historian of science Jim Bennett's pioneering work on the diversity of early modern mathematical practice and its relationship to the development of modern science has further elucidated the epistemological and philosophical dimensions of mathematics. Architecture and geometry in the age of the Baroque (University of Chicago Press, 2000) by George Hersey, the exhibition and catalogue Compass & rule: architecture as mathematical practice in England, 1500–1750 (Yale University Press, 2009) by Anthony Gerbino and Stephen Johnston, and The geometry of creation: architectural drawing and the dynamics of Gothic design (Ashgate, 2011) by Robert Bork are just some of the more recent monographs focusing on the relationship between architecture and mathematics.

The work under review here, Architecture and mathematics from antiquity to the future (Birkhäuser, 2015) edited by Kim Williams and Michael J Ostwald, is one of the latest additions to the scholarship: compiled in two volumes spanning more than 1400 pages in ninety-two chapters, it is also one of the most ambitious. Williams has been exceptionally active in promoting the study of the relationship between the two fields, organizing the biennial Nexus conference since 1996, publishing its proceedings under the ‘Nexus: Architecture and Mathematics’ series, and founding the Nexus Network Journal in 1999. She has also contributed two collaborative translations; with Stephen R Wassell, of the sixteenth-century mathematician Silvio Belli's On ratio and proportion (Kim Williams Books, 2002), and with Wassell and Lionel March, of the Mathematical works of Leon Battista Alberti (Birkhäuser, 2010).

In their Preface, Williams and Ostwald caution that Architecture and mathematics from antiquity to the future is not ‘a comprehensive encyclopaedia of the history and theory of every facet of the relationship between architecture and mathematics’. While such disclaimers are standard in collected works, in this case it is more than valid: for despite its size, chronological and regional span, and indeed encyclopaedic title, this two-volume set is in fact a compilation of ‘the most highly cited works’ selected out of the seven volumes of the now out-of-print Nexus conference proceedings published up to 2010, when the format was abandoned in favour of making the papers more widely available via the journal instead. The editors have used the opportunity to re-arrange the papers ‘both thematically and chronologically to trace key moments in the history and theory of architecture and mathematics, from antiquity to the present day, along with predictions for the future’ (p. vi), but it appears they have not included any other articles from the Nexus Network Journal nor sought work done by scholars outside of their network. While it is understandable that the editors would not want to publish articles readily available in their journal and may have felt they already had plenty of material to work with, a more carefully-worded title could have made it clearer that the book is limited to selections from conference proceedings rather than a survey of the most recent scholarship on the subject.

Despite this limitation, there is plenty on offer among the ninety-two chapters by authors from diverse backgrounds that include, in addition to architecture and mathematics, engineering, physics, chemistry, philosophy, art, and music. The first volume, which covers the period between 2000 bc and 1500s ad, has an equally vast geographical span, thus in addition to chapters devoted to discussions of various theoretical approaches to the relationships between mathematics, art, architecture, and their histories, it includes studies of specific practices in Ancient Egypt, Greece and Rome, Medieval and Renaissance Europe, the Islamic world, Africa, India, China, Japan, and Mesoamerica. The mathematics of construction techniques, and theories of proportion and symmetry are also addressed in this first volume. The second volume covers the period after the 1500s, and except for a chapter on Ottoman architecture, mostly in the European and North American context—although the last section on the applications of computer sciences could be considered global given the current ubiquity of computer-aided design in architecture. Some of the chapters address the work of individual architects, such as Andrea Palladio, Francesco Borromini, Christopher Wren, Robert Hooke, Frank Lloyd Wright, and Le Corbusier, while the last two sections on contemporary issues include chapters on the more instrumental applications of mathematics, for example, the geometry of soap bubbles, Penrose tilings, or the use of linear algebra for generating architectural forms. The volume concludes with two views from philosophical and historical perspectives, addressing some of the ethical implications of the use of computers in architecture.

With interceding chapters, the editors do their best to contextualize and order the ninety-two chapters into a cohesive work. However, the different backgrounds of the contributors, and their use of methodologies specific to their fields, leave the nature and quality of the scholarship inconsistent, making it possible to find, for instance, historical studies based on primary sources and ahistorical morphological analyses of buildings in the same volume. Such juxtapositions may be partly intentional as one of the stated objectives of the editors is to reconnect architecture and mathematics to encourage ‘transdisciplinary scholarship’ by ‘scientists, scholars, professionals and gifted amateurs’ (p. 5).

It is easy to see how this work would be useful to architects and architectural educators who are as concerned with understanding the past as they are with building the future; for them it presents several alternative historical and theoretical contexts of the relationship between architecture and mathematics which has been pushed to the foreground during the past decades with the increasing use of computer-aided design in their profession (cf. the Archaeology of the digital exhibition curated by Greg Lynn at the Canadian Centre for Architecture). Historians of mathematics, too, will no doubt find useful material here for their research, especially if they are interested in the more practical concerns that have shaped the development of their field, and how the overlaps between architecture and mathematics manifested themselves in actual buildings and artworks. They may also discover more theoretical approaches to practice; that is, how mathematics has played a mediatory role between knowledge and creation for architects, and why, since Vitruvius, they have been so keen on their compasses and rules.