*The Fractal Dimension of Architecture*, by Michael J. Ostwald and Josephine Vaughan, is the first in a series *Mathematics and the Built Environment*“ edited by Kim Williams. The series “highlight[s] the multifaceted connections between the disciplines of mathematics and architecture.” Both Ostwald and Williams have been leaders in exploring these connections.

In this particular book, Ostwald and Vaughan develop a protocol for using a box-counting approach to obtain the fractal dimension of a building’s design. The box-counting approach, first described by Benoit Mandelbrot in *The Fractal Geometry of Nature*, yields a measure of the complexity of building elevations and plans. This book is a case-study of the utility of this tool, displaying its strengths and weaknesses in testing hypotheses regarding the architecture of 85 homes from four continents designed by prominent architects between 1901 and the present day. The authors apply the tool sensitively to examine three broad hypotheses and five secondary hypotheses, appropriately concluding that “in the field of architecture, where cultural, social, and philosophical issues shape design, the use of a mathematical system of analysis will always require rigour (in its application) and sensitivity (to interpret the results) first, before it can be useful for scholars and practitioners.” [397] There can be many influences that affect the complexity of a building; it is to be expected that structures with similar fractal dimensions may be quite different, so a need for sensitivity is apparent.

Ostwald and Vaughan begin in Chapter 1 laying out the structure of the book, introducing fractal dimension, and distinguishing fractal dimension from fractal geometry. Chapter 2, which has a discussion of fractal geometry and architecture, is the last time in the book that fractal geometry and fractally-generated architecture are mentioned. Fractal geometry is *not* the point of this book; rather, the point is to consider the tool of fractal dimension in architectural analysis.

In Chapters 3–6, the authors develop the box-counting tool. In Chapter 3, they provide examples of the box-counting technique and provide a discussion of when it should be used and how the measurements can be made reliably. In practice, the computations are carried out by *ArchImage* software, so it is valuable to see here a few computations carried out by hand, offering the reader some insight into the process. In Chapter 4, the authors move on to discussions of what depictions of a building should be the starting point of an analysis and what features of a building should be included in the analysis. Chapter 5 continues a discussion of these practical considerations by analyzing the impact of a number of methodological variables.

Mathematicians will enjoy reading the results of the application of the box-counting method and the protocols developed here to compute the fractal dimension of actual fractals and comparing the computed values with the actual dimensions of these nine classical fractals — the Koch Snowflake, the Terdragon Curve, the Apollonian Gasket, the Minkowski Sausage, the Sierpinski Triangle, the Sierpinski Hexagon, the Fibonacci Word, the Pinwheel Fractal, and the Sierpinski Carpet. [116–127] Finally, in Chapter 6, Ostwald and Vaughan explain their selection of the houses to be analyzed, the preparation of the formal plans of the houses for analysis, the process of calculating their fractal dimensions, and the analysis of the data that that will be carried out in the next five chapters.

Chapters 7 through 11 present the analyses of the houses. The intention is to consider three major hypotheses: (1) As the complexity of social groupings and functions contained within the home has reduced over time, the fractal dimensions of plans and elevations should decrease to reflect this change; (2) Each stylistic genre or movement possesses a distinct visual character that is measurable using fractal dimensions; and (3) Individual architects will present distinctive patterns of three-dimensional formal and spatial measures across multiple designs. [370–371] To avoid providing spoilers, I’ll not share the conclusions of the considerations of the hypotheses here.

Within each of Chapters 7–11, five additional, more particular, hypotheses are also considered. There is a common format to these chapters: an introduction to the architectural period/style being considered, background on the particular architects whose houses were selected for analysis, the results of the analyses of a set of five houses for each of the architects (or, in the case of some architects whose output warrants, several sets of five houses exhibiting different styles). What follows is the consideration of a secondary hypothesis. For example, in Chapter 7, the authors provide an analysis of four sets of five houses, namely two sets by Le Corbusier (both pre-modern and modern), and one set each by Eileen Gray and by Mies van der Rohe. The secondary hypothesis is whether fractal dimension can provide insight into the slogan “Form follows function.” In Chapter 8, the authors consider three sets of houses by Frank Lloyd Wright, five each from the Prairie Style, the Textile-block period, and the Usonian period. (In complexity, the Textiles-block homes have a higher fractal dimension, followed by the Prairie Style homes, which were generally more complex than those of the Usonian period.) The chapter concludes with an analysis of walking into and through the Robie House in Chicago, to consider the hypothesis that the degree of complexity decreases as one is further into the house.

In Chapter 9, the authors consider houses designed by three members of the New York Five: Peter Eisenman, John Hejduk, and Richard Meier. To analyze frontality versus rotation in these houses, the fractal dimension of various presentations of the houses is discussed. The topic of Chapter 10 is post-modern houses by the team of Robert Venturi and Denise Scott Brown and by Frank Gehry. The secondary hypothesis in this chapter is to test whether the number of openings in a façade and the geometric modeling of the façade can shed light on the form versus function question. Finally, in Chapter 11, the authors consider houses by Japanese minimalists Kazuyo Sejima and the architectural firm Atelier Bow-Wow and by the Australian regionalists Peter Stutchbury and Glenn Murcutt, with both the early rural works and the later-career houses of Murcutt being considered. Whether fractal dimension can help test a claim of transparency in Murcutt’s houses is the subject of this chapter’s special hypothesis.

Ostwald and Vaughan provide their conclusions of the analyses of all 85 houses in Chapter 12.

There are several other noteworthy features of the book. First, the structure of the book itself is essentially fractal. Chapters 7–11 contain a discussion of an architectural period, followed by sections including discussions of individual architects and analyses of his/her/their individual houses, followed by a summary of the houses of a particular architect, followed by a summary of the houses from the particular period. The final chapter then provides a summary of all the houses. Second, the mathematics used here is not deep. Instructor-designed projects based on the material could reasonably appear in a liberal arts core course, and applications of box-counting could provide the basis of interdisciplinary undergraduate research projects. Finally, compared with other books I have reviewed recently, the text is both well-written and remarkably error free — I noticed only one misspelling, namely of “were” for “where.” Thank you, authors and editors!

Joel Haack is Professor of Mathematics at the University of Northern Iowa.