Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

architecture |ˈärkiˌtek ch ər| noun

  • The art or practice of designing and constructing buildings. The style in which a building is designed or constructed, esp. with regard to a specific period, place, or culture.

mathematics |maθ(ə)ˈmatiks| plural noun

  • The abstract science of number, quantity, and space. Mathematics may be studied in its own right (pure mathematics), or as it is applied to other disciplines such as physics and engineering (applied mathematics).

Introduction

What is the nature of the relationship between architects and mathematicians or between architecture and mathematics? As they are commonly understood these two groups seem to have few obvious connections. The word ‘architecture’ is used to describe either the practice of creating buildings or a particular class of constructed—architectural—objects. In contrast, the word ‘mathematics’ denotes a domain of pure or applied knowledge that is associated with the study or use of abstract objects such as numbers and shapes or forms. Professions, like architecture, tend to be isolated and controlled, restricting membership to experts who have been awarded particular qualifications and have fulfilled certain criteria (Fournier 2000). Instead, disciplines, like mathematics, are ways of grouping and identifying bodies of knowledge and expertise that are both pertinent within the discipline and which might also be applied in other fields (Klein 1996). However, despite these apparent differences, the distance between these two—between a profession and a discipline and between an object (of design) and a subject (of study)—is far less than many would assume. In order to make sense of the contemporary view which considers architecture and mathematics as dissimilar pursuits, it is useful to trace a brief history of the growth of disciplines and professions over time; a history which, most importantly, reminds us that in the past these two were very closely connected.

In the world of ancient Greece, learned professionals were trained in one of three fields: religion, ethics, or the human condition. While some distinction was made between the ways students were educated in each of these fields, a sound foundation in both philosophy and mathematics was considered a necessity (Marrou 1956). At the same time, the acts of creating a chair, designing a house or constructing a ballista were all tasks that could be undertaken by artisans. Such skilled workers were trained in the practical processes of making (stone-cutting, woodwork) along with a range of knowledge domains including both geometry and metrology (Kostof 1977a).

Later, in Roman times, Vitruvius tells us that the architect was neither a scientist nor a craftsman, but one who had sufficient knowledge of a range of scientific fields—first and foremost geometry, but also history, philosophy, music, medicine, law and astronomy—to be able to oversee the work of all other disciplines (Vitruvius 2009: 5).

A similar pattern of relationships is found in medieval Europe where the first universities recognised graduates in law, divinity and medicine. In the medieval university students were expected to complete preparatory studies on the trivium—grammar, logic and rhetoric—along with studies in arithmetic, geometry and aesthetics (Janin 2008). Seated in rows of carved pews, in high-roofed tapestry-lined halls, these students were able to directly observe the geometric mysteries of space and form. But the very halls they inhabited during these studies had been created by master builders and teams of craftsmen and labourers with a fund of geometric knowledge. Evidence of this knowledge is found in Villard de Honnecourt ’s Li vre de Portraiture, where he wrote, “…in this book … you are able to find the technique of representation as the discipline of geometry requires and instructs” (Barnes 2009: 35). The most senior of the master builders, often the direct descendants of artisan families, were given the title ‘architect’ or ‘engineer’ (Kostof 1977b). These master builders were trained in both the arts and sciences of design and construction. Furthermore, in both the Classical and Medieval eras, the mathematical disciplines provided a critical foundation for construction (Fitchen 1961; James 1981). This is why, for much of this era, there was little distinction between professions and disciplines, or between architects and mathematicians. This close and productive relationship between architecture and mathematics was to continue for several centuries before reaching its most visible apogee during the Renaissance.

During the Renaissance the Medieval educational foundation found in the trivium was expanded to encompass a second tier of four arts (the quadrivium) that typically included studies of arithmetic, geometry, music and astronomy. The archetypical ‘Renaissance man’ was expected to have both a broad and comprehensive knowledge across these seven subjects, along with their potential application in at least one, and possibly more, of the following pursuits; science, art, medicine and architecture. Leon Battista Alberti was one such polymath; an author, artist, poet and linguist who had also mastered optics, perspective and cryptography (Williams et al. 2010). Today, Alberti is best known as an architect, because it was in the application of these ideas to design, in both written and built work, that he achieved his most enduring success. For a similar reason Christopher Wren is also regarded as one of the world’s great architects, even though he originally distinguished himself in astronomy, physics and mathematics and only began to design buildings when he was already a respected scientist (Bennett 1982). Robert Hooke is another example of the natural philosopher-cum-architect. Even the great Isaac Newton demonstrated an amateur, but informed, interest in architectural theory. Such cases suggest that for centuries architecture and mathematics were closely related, and equally respected, areas of inquiry.

However, despite the examples of Wren and others, the seventeenth century was more generally marked by the rise in power of guilds and colleges who sought to define and preserve their members’ interests (Melton 2001). For example, it was around this time that the Freemasons formulated a series of rules of membership and practice that sought to protect the knowledge and skills of the stonemasons. Amongst the earliest articles of Freemasonry is a set of practical and symbolic rules showing the essential relationship between architecture and geometry (Berman 2012). Thus, while the tradition of the ‘Renaissance man’ was still being valorised in the eighteenth century, the guilds and technical colleges remained the driving force which gradually separated professions from disciplines and, inadvertently, increased specialisation began to distance architecture from mathematics (Clarke 1994).

Over the ensuing 200 years, in parallel with the emergence of new technology and the need for more focussed trades and skills, disciplines and professions became increasingly specialised and their roles began to change (Duffy and Hutton 1998). For example, established in 1794 the French École Polytechnique was at the forefront of approaches to training a new, elite class of technocrats. Embracing a scientific disposition, the curriculum included issues of aesthetic perception, positivism and rationalism. In part because of this educational focus, this era marked a growing separation between architects and engineers (Picon 1992). The French Enlightenment was also one of the last periods wherein architects, still trained in descriptive geometry, directly contributed to mathematical knowledge. In particular, the discipline of stereotomy was developed largely by architects to allow stone blocks to be cut and assembled in complex forms (Warren 1875).

By the early years of the twentieth century architectural education was split between an atelier-based model, which traced its origins to the École des Beaux-Arts, and the technical college model, which acknowledged a lineage to the apprenticeship system (Cuff 1991; Crinson and Lubbock 1994). The atelier system, modelled on fine art practice, included geometry in its core curriculum but the role of arithmetic was less apparent (Draper 1977). The technical college system included both science and mathematics in its syllabus along with more extensive applications of geometry. Significantly though, despite the apparent differences between these systems, both effectively positioned mathematics as a secondary discipline which merely served to buttress the education of architects (Boyer and Mitgang 1996). In a comparable way, the discipline of mathematics divided itself into a ‘pure’ and an ‘applied’ strand, with the former being regarded as the path for specialists, and the latter for those who sought to engage more directly with other fields (Davis et al. 1995). This pattern was repeated around the world with the combination of increasing specialisation and the desire for professional recognition gradually separating and isolating different knowledge domains and disciplines from each other (Fournier 2000).

Such was the compartmentalisation of knowledge that occurred in the early part of the twentieth century that British scientist and novelist Charles Percy Snow famously criticised the rise of two distinct and separate cultures—science and humanities—each seemingly unaware of the basic values and lessons of the other (Snow 1998). Snow’s observations, derived from his identification of the growing separation between disciplinary groups, were both widely reported and criticised (Carafoli et al. 2009). Certainly it was becoming harder for a person to be qualified in two or more fields and the era of the peripatetic scholar was effectively over. This was true even within the discipline of mathematics itself; Henri Poincaré was considered the “last universal mathematician”. Yet, the growing accessibility of knowledge, no longer protected by professional guilds or enshrined in the lore of esoteric societies, meant that rather than fostering the divide between two distinct cultures, there were potentially a multitude of secondary connections to be made between different groups, each creating new sub-cultures (Nicolescu 2002). However, such new transdisciplinary groups face a twofold problem: visibility and recognition (Doucet and Janssens 2011). In the first instance, while important connections exist between fields, like architecture and mathematics, they are often rendered invisible by contemporary educational practices and the legal implications of professional ethics (Sokolowski 1991). In the second, sustained research must take place before such hidden associations can be recognised, investigated and celebrated. This is especially the case in contemporary society where these same two problems of visibility and recognition continue to hinder our capacity to engage with transdisciplinary knowledge.

Today, the extent to which architects are formally trained in mathematics is probably lower than in any previous period in history (Ostwald and Williams 2008). The degree to which mathematicians directly engage with building design and construction is at an equally low ebb. This is unfortunately true of the general public as well, since no formal architectural education at all is offered in public schools, while at least of minimum of mathematics is taught. Yet, this situation is not entirely as it seems. Advances in computing have placed mathematical techniques and processes at the fingertips of every young architect in the world, providing a means of using complex geometry which was previously unavailable (Szalapaj 2005; Littlefield 2008; Ostwald 2012). In the last decade buildings have been produced which model non-linear dynamical systems, are covered in fractals or aperiodic tiles, are roofed in complex membranes and are optimised for energy performance and wind-load using Boussinesq or Bernoulli equations (Ostwald 2006; Burry and Burry 2012). All of these developments rely on advances in mathematics yet, paradoxically, the distance between the architectural profession and the mathematical disciplines has seemingly never been greater. The problem is, as stated previously, that a myriad of connections continue to exist between architecture and mathematics, just as they have done for several millennium, but in order to understand and appreciate these connections—to perceive and recognise both their historical and theoretical significance—academics and professionals must be willing to engage in transdisciplinary scholarship. This is where the present work has an important role to play.

The relationship between architecture and mathematics is most visible and recognisable when members of these two groups cross the divide and productively work with concepts, themes and topics that have been developed in the other field. Thus, for architects to talk with authority about geometry or arithmetic requires both a willingness and a capacity to traverse disciplinary boundaries. Similarly, a mathematician venturing into the realm of architectural history, theory and design must engage with a field that has its own language and traditions. The present, two-volume, edited collection represents the work of some 100 authors who have made such important transdisciplinary incursions. They have investigated the complex interplay of connections between architecture and mathematics and have engaged with bodies of disciplinary and professional knowledge that are often separated in contemporary society. These authors have worked to reconnect two fields that were once closely reliant on each other and, in doing so, blaze a path so that other scientists, scholars, professionals and gifted amateurs may follow.

At the start of this chapter two questions were raised. The first asked: “What is the nature of the relationship that exists between architects and mathematicians?” The short answer is that the two share a common intellectual heritage and similar values and concerns. Both work with a highly structured system of symbols that support each other hierarchically to achieve an edifice. Mathematicians, in fact, speak quite often of “the mathematical edifice”. Here is Bertrand Russell on the study of mathematics:

The discovery that all mathematics follows inevitably from a small collection of fundamental laws is one which immeasurably enhances the intellectual beauty of the whole; to those who have been oppressed by the fragmentary and incomplete nature of most existing chains of deduction this discovery comes with all the overwhelming force of a revelation; like a palace emerging from the autumn mist as the traveller ascends an Italian hillside, the stately storeys of the mathematical edifice appear in their due order and proportion, with a new perfection in every part (Russell 2009: 67).

The second question was more general and inclusive: “How are the subjects of architecture and mathematics connected?” The answer to this involves a consideration of several thousand years of history, along with an investigation of philosophy, number and shape, construction and material science. Issues of representation, meaning, religion, culture and ethics are also pertinent to this question. Thus, to begin to approach this larger topic, the following section of this chapter describes the structure of the work, which alternates between historical and theoretical themes.

The Structure of the Work

The chapters that comprise these present volumes have been arranged following a predominantly chronological approach. We chose this convention because it best represents the way architecture and mathematics—but especially architecture—has developed. The architecture of any given historical period is usefully seen in relation to the one that preceded it, perhaps as a natural development or outgrowth, but also potentially as a rebellion against it. Often the underlying reason for a change in style is a shift in thinking, or in knowledge or technique. Thus we have interspersed the historically grouped chapters with groups relating to ideas and theory. We believe in this way it is possible for ideas and concepts, along with built works and processes, to juxtapose and illuminate each other.

The relationship between architecture and mathematics is both longstanding and complex, with the two being bound together in a multitude of practical, representational and contingent ways. On a practical level, the design and construction process for a building relies on mathematics for measurements, timelines, weights and structural calculations (Salvadori 1968; Swallow et al. 2004). In a different way, architects have used numbers and shapes to represent—through symbolic, metaphoric or semiotic means—a broad range of themes that are socially and culturally significant (Rowe 1947; Preziosi 1979; Evans 1995). A parallel tradition in architecture uses geometry and other branches of mathematics to analyse the way designers approach form (Stiny 1975), to investigate spatial hierarchies (Hillier 1995) or measure visual and phenomenal properties (Benedikt 1979). All of these examples involve different types of connections between architecture and mathematics. Furthermore, throughout history there have been various periods characterised by particularly dominant types of relationships between architecture and mathematics.

In the ancient Greek and Roman worlds architects repeatedly used combinations of shapes and numbers to evoke spiritual or cosmic themes. For example, the Roman architect Vitruvius suggested that because the human body possesses distinct geometric proportions, an architecture that is produced in accordance with those proportions represents a microcosm of the divine universe (Rykwert 1996). In parallel with this symbolic application of geometry, the challenges of supporting construction and trade throughout a rapidly expanding empire led engineers and architects to develop a standard system of measurement (Kostof 1977a). In contrast, during the Renaissance systems of proportion based on the musical ratios were the focus of much aesthetic debate. With the rise of Baroque architecture, a broader geometrical vocabulary including ovals and ellipses was used to reinforce the combined spiritual and experiential power of space (Norberg-Schulz 1971). In the eighteenth century French Enlightenment architects designed buildings in the shape of monumental Phileban solids (spheres, cones, and cubes) in an attempt to encourage contemplation of higher order scientific or philosophical principles (Vidler 2006). However, by the nineteenth century widespread concerns with health standards and overpopulation encouraged architects to embrace industrial techniques, along with the new ways of managing production scheduling and understanding material tolerances and limits. In our own day, algorithmic and computational approaches to architecture have embedded mathematics in every line of every CAD model or BIM file, and in every form or shape evolved or generated in the computer (Szalapaj 2005). While measurement remains of practical importance in contemporary architecture, it is no longer as critical as it was in the ancient world. Similarly, while several Renaissance treatises suggest that design could be parametrically determined, the power of computing has allowed this once peripheral notion to become increasingly important. Thus, throughout history the relationship between architecture and mathematics has shifted and changed, with some key moments in this relationship being embodied in a single building of a period or the work of a particular architect, but at other times being only understood retrospectively through the efforts of historians and theorists.

The structure of the present book reflects this interweaving of the theory and history of architecture and mathematics. The first volume commences, in Part I, with a set of chapters that are concerned with overarching theories connecting architecture and mathematics, before progressing, in Part II, to the examination of a particular period in architectural history: the ancient world prior to 1000 ad. This transition from theory to history and back again is repeated for the remainder of the two volumes. For example, in Part III, a series of chapters focus on the historic importance of mathematics in creating systems of measurement and structural stability. Thereafter, returning to the historic time line, in Part IV, architectural examples drawn from 1100 ad to 1400 ad (from Medieval to Romanesque) are analysed. This alternating structure, which knits a historical chronology of buildings and architects together with theories that were of relevance to the era, is repeated throughout the two volumes. The present work can therefore be read either historically or thematically, but it is only in the combination of the two—through viewing the complete fabric and not just its warp or weft—that the wealth and profundity of connections between architecture and mathematics can be grasped.

The theory strand that connects the two volumes—Parts I, III, V, VII, X and XI—examines the significance of essential concepts including measurement, proportion, symmetry and representation. Theories and applications of tiling—both periodic and aperiodic—fractals and scanning technology are examined in the penultimate theory section. In the final of these parts, computational and parametric theories are described along with the philosophical implications of these developments in the context of architectural history. While this is the shorter of the two strands that connect the two volumes of the present work, it frames the historical work, providing a foundation for ideas developed throughout the chapters and a context for the changing role of mathematics in architecture.

The history strand comprises the majority of the two volumes—Parts II, IV, VI, VIII and IX. It commences with Neolithic and Copper age construction, before progressing through examples of ancient Egyptian, Greek, Roman and Mayan architecture. The Renaissance and Baroque eras are covered in the intermediate sections along with Islamic, Christian and Ottoman design. Architects and theorists whose works are considered in this section include such giants as Michelangelo Buonarroti, Andrea Palladio , Francesco Borromini and Christopher Wren , along with lesser-known figures such as Juan Bautista Villalpando and Antonio Rodrigues . The final section in this strand is focussed on the twentieth century and the first decade of the twenty-first century. It considers examples of Modern, Organic, Postmodern and Computer-Generated architecture. Works and texts by Frank Lloyd Wright , Marcel Breuer , Le Corbusier , Louis Kahn and Oscar Niemeyer , amongst many others, are considered in this part.

In what follows, we provide an overview of the first six parts of the present work which collectively make up Volume I. A separate introduction and overview is provided at the start of Volume II (Chap. 48) covering material in that work and offering a framework for understanding why certain eras offer a multitude of examples of connections between architecture and mathematics, while others have a relative paucity.

Part I: Connections Between Architecture and Mathematics

The first part of the present volume comprises a series of chapters which consider a range of themes pertaining to the overarching relationship between architecture and mathematics. In ‘Can there be any relationships between Mathematics and Architecture?’ (Chap. 2), the eminent engineer and mathematician Mario Salvadori offers a provocative reading of the relationship between pure mathematics and what he calls ‘concrete real architecture’. Salvadori, in a personal address to the reader, illustrates pure mathematics using examples of Euclidean and non-Euclidean geometries, and concludes that the beauty of mathematics is that ‘it is totally free, it is abstract’. In contrast, architecture must be constructed and it is because of this property that, in a mathematical sense, the two are incomparable. But then, Salvadori reverses his own argument to demonstrate that the connections between mathematics and architecture are so many that ‘if mathematics had not been invented, architects would have had to invent it themselves.’

In Chap. 3, “Mathematics in, of and for Architecture” we editors of this publication, Michael Ostwald and Kim Williams , begin to formulate an answer to Salvadori ’s question; ‘can there be any relationships between mathematics and architecture?’ Starting with a review of the two famous origin myths of architecture, then though a consideration of more contemporary modes of architecture practice, the chapter identifies three types of relationships. The first type, covers geometric or numeric properties that are intentionally designed into or demonstrated in architecture. The second type encompasses analytical methods for quantifying or determining various properties of architecture. The final includes practical tools for the support of architectural design, construction, generation and conservation.

In ‘Relationships between History of Mathematics and History of Art’ (Chap. 4), which is similarly broad in its scope, Clara Silvia Roero examines the relationship between the history of art in general and the history of mathematics. Her chapter proposes three levels of connection between art and mathematics—a surface or substrate level, the conscious or unconscious application of principles, and finally a higher-level demonstration of knowledge.

While the close relationship between architecture and mathematics was especially prevalent during the Renaissance, in ‘Art and Mathematics Before the Quattrocento: A Context for Understanding Renaissance Architecture’ (Chap. 5), Stephen Wassell demonstrates that this achievement should be understood by considering the development of mathematics in previous eras. Starting with Neolithic speculative geometry, and then progressing through mathematical principles developed in Ancient Greece and Rome, Wassell proposes that the practical and metaphysical foundation for Renaissance architecture can be attributed to much earlier developments in both architectural and mathematical knowledge.

In ‘The Influence of Mathematics on the Development of Structural Form’ (Chap. 6), Holger Falter examines the role played by mathematics in the development of structural form. By tracing examples from different eras, Falter argues that not only is mathematics (in the form of structural mechanics) essential to every design, but that each era developed different architectural structures in response to changing cultural conceptions and mathematical developments. While Falter acknowledges that geometry often served important symbolic functions in architecture, he argues that the structural and material properties cannot be forgotten.

Part II: Architecture from 2000 bc to 1000 ad

Histories of architecture typically suggest that it was around 9000 bc, in the early Mesolithic era, that the first consciously designed structures were built. Prior to that time natural features in the landscape, including caves, hollows and rock overhangs, provided rudimentary shelter from both the environment and predators. However, in the Mesolithic era small communities began to create simple, but often extensive, structures from naturally weathered stone that could be moved and arranged to create earthen mounds or barrows. By selectively covering these mounds in branches and soil, and inhabiting the voids between the larger rocks, these structures became the equivalent of artificial caves. The remains of a large number of these barrows have been excavated throughout the last century, with some evidently serving community or ceremonial purposes, while others acted as burial mounds (Spikins 2008).

In the early Neolithic era, dating from around 5000 bc, a number of communities in different parts of the world developed a capacity to create simple masonry structures (Hofmann and Smyth 2013). Some of the most extensive of these featured large, walled enclosures made of crude mud bricks which were sometimes finished in a type of plaster. For example, in Southern Anatolia in Turkey the Chalcolithic (Copper Age) people of Çatalhöyük constructed a dense cellular architecture, wherein a network of broadly orthogonal masonry walls was roofed with timber shingles. At around the same time, wattle and daub structures with thatched roofs over timber beams began appearing across both Eastern and Western Europe. In Germany, England and Ireland more elaborate timber structures, including long houses, tombs and cisterns were also being constructed (Cruickshank 1996).

In parallel with the latter part of the European Neolithic period, in North Africa, the higher temperatures allowed for the development of large-scale techniques for casting and drying mud-bricks. It was this advance which lead to the rise of Sumerian architecture (Kramer 1963). Using clay-based masonry the Mesopotamian culture developed methods that allowed them to stack layers of bricks, typically without mortar, to produce buttressed walls and stepped pyramids, or ziggurats. Supplemented with a limited amount of timber and occasionally clad in coloured stone, these buildings were large in scale, but the friable nature of the masonry core of these structures rendered them impermanent.

Just as architecture developed throughout the Mesolithic, Neolithic and Sumerian eras, so too did mathematics. Simple attempts to record the passing of time were common in early agrarian cultures. The need to plant and harvest crops at appropriate times led to the need to measure and predict the seasons and thus the first calendars were developed. The desire to trade with neighbouring lands was responsible for the Sumerians developing simple weights and measures. However, the first great developments in both architecture and mathematics can be traced to the Ancient Egyptians. Responsible for a variation of the decimal system, for surveying and the astronomical calendar, the Egyptian civilisation developed both architecture and mathematics to a new level of refinement (Rossi 2004).

Spanning broadly from around 3000 bc to 500 ad, Egypt provided one of the most enduring civilisations of the ancient world. In this era, however, Egypt was not alone in making both mathematical and architectural advances; both the Ancient Indian Vedic texts and Ancient Mayan (Mesoamerican) hieroglyphs reveal similar levels of advancement, with the latter inventing a base-20 (vigesimal rather than decimal, or base-10) system of counting. It is this age, when the Egyptian, Mayan and Indian cultures developed systems of counting, measuring and geometry, which is the focus of the opening four chapters in Part II of the present work. The majority of the latter chapters in Part II address the architecture and mathematics of the Ancient Greek and Roman civilisations.

In ‘Old Shoes, New Feet and the Puzzle of the First Square in Ancient Egyptian Architecture’ (Chap. 7), Peter Schneider examines the origins of the first square, one of the most primitive and recurring geometric figures in architecture. Schneider finds evidence of a simple, yet sophisticated technique for constructing the square, in the ancient Egyptian measures of the 20 digit remen and the 28 digit cubit, suggesting that this allowed architects to construct a primal square which could be readily replicated. In ‘Geometric and Complex Analyses of Maya Architecture: Some Examples’ (Chap. 8), Gerardo Burkle-Elizondo , Nicoletta Sala and Ricardo David Valdez-Cepeda undertake a detailed geometric analysis of 26 pyramids constructed by the Mayan civilization. Using fractal analysis they examine the relationship between the formal properties of these structures and their cosmological purpose. Again with regard to Mesoamerica and the stepped pyramids of the first century ad, in ‘A New Geometric Analysis of the Plan of the Teotihuacan Complex in Mexico’ (Chap. 9), Mark Reynolds offers a geometric analysis of the plan of the Teotihuacan complex in Mexico. Rather than considering the significance of these structures in an astronomical or archaeological sense, Reynolds uses plan overlays to examine the organization of these culturally significant buildings.

George Gherveghese Joseph proposes a different cultural relationship between architecture and mathematics in ‘The Geometry of Vedic Altars’ (Chap. 10). Ancient Indian systems of weights and measures can be traced in the construction methods and forms of a series of sacrificial altars (vedi). Through an analysis of these altars, Joseph demonstrates that while the origin of Indian mathematics has often been linked to largely theological or symbolic purposes, the degree to which this culture was able to solve practical problems in geometry has been underestimated.

Site planning is the focus of Graham Pont ’s chapter, ‘Inauguration: Ritual Planning in Ancient Greece and Italy’ (Chap. 11). Drawing on a theory of planning by polar coordinates, Pont considers 29 ancient sites, noting the evidence for a system of site planning which involves the division of sites using visual arcs. Such a ritual system, based on visual relations as defined from distinct points in space, might explain why many Greek towns and sites have complex angled plans, which have resisted standard geometric analysis techniques.

In ‘The Geometry of the Master Plan of Roman Florence and its Surroundings’ (Chap. 12), Carol Martin Watts describes the Roman town planning practice of laying out major streets in accordance with cardinal points. Thereafter she offers a possible explanation for the planning of both the city and countryside of Florence, which is in accordance with Roman practices, relies on a range of clear geometric processes and responds to symbolic concerns about genius loci. Continuing the focus on Roman architecture, in ‘Architecture and Mathematics in Roman Amphitheaters’ (Chap. 13), Sylvie Duvernoy examines a particular elliptical shape used almost exclusively for the design of amphitheatres. The geometric basis for Roman amphitheatres has been the subject of a body of specialised research, leading to debates about the difference between oval and elliptical forms. Starting with amphitheatres from the late Republic and early Roman Empires, Duvernoy traces a series of simple geometric relations. Then, using measured surveys of the amphitheatres of Pompeii, Roselle, and Veleia, she analyses the nature of the geometric curves which provide the basis for each. Geometry in Roman architecture is also the topic of Carol Martin Watts’s second chapter in Part II, ‘The Square and the Roman House: Architecture and Decoration at Pompeii and Herculaneum’ (Chap. 14). Through an analysis of the Roman house and its decoration, Watts uncovers several geometric systems and operations (including a variation of the ‘sacred cut’) in the proportional relationships found in the overall site planning, spatial organisation, and decorations and tiling of the house. The penultimate chapter in this part continues the close analysis of geometry in Roman architecture. In ‘The Quadrivium in the Pantheon of Rome’ (Chap. 15), Gert Sperling examines the Pantheon in Rome, reviewing several famous interpretations of its form, and partially rejecting the cosmological connections that have been made between the building’s dome and the heavens. Taking into account a survey of the metrical dimensions of the complex, Sperling proposes that the geometry of the Pantheon not only reflects the heavens, but is also derived from the mathematical knowledge of the heavens recorded in the ancient quadrivium.

In ‘“Systems of Monads” in the Hagia Sophia : Neo-Platonic Mathematics in the Architecture of Late Antiquity’ (Chap. 16), Helge Svenshon and Rudolf Stichel examine the relationship between mathematics and design in one of the most iconic buildings of the first millennium. Hagia Sophia or Ayasofya in Istanbul (formerly Constantinople), was built in the fifth century for the Byzantine emperor Justinian. Famous for its enormous dome and the majesty of its interior, significantly, the original designers commissioned for Hagia Sophia were not architects, but were rather a scientist and a mathematician; respectively, Isidore of Miletus and Anthemius of Tralles.

Part III: Theories of Measurement and Structure

The eight chapters in Part III are centred on theories of measurement and the role of mathematics in the construction and structuring of architecture. In ‘Measure, Metre, Irony: Reuniting Pure Mathematics with Architecture’ (Chap. 17), Robert Tavernor takes as his starting point the argument that systems of measurement are representative of the values of the culture that defined them. Thus, in the ancient world the measures and proportions of the human, male, body were often used for theological, rather than practical reasons. Tavernor illuminates the fundamental irony that the systems of measurement developed and used by architects and mathematicians are rarely so universal as they seem. In ‘Facade Measurement by Trigonometry’ (Chap. 18), Paul Calter examines a different way of measuring the built environment, derived from surveying and optimised for the consideration of historic buildings and ruins.

Mark Wilson Jones describes his motivation for writing ‘Ancient Architecture and Mathematics: Methodology and the Doric Temple’ (Chap. 19) as a desire to bring greater rigour to the measurement of ancient Greek and Roman architecture. In particular Jones is critical of proportional studies of façades and plans that are neither precise nor objective enough. Jones offers seven criteria for evaluation of the efficacy of a particular set of measures and their interpretation, each of which is illustrated with specific examples from the ancient world.

The act of measuring buildings is also the focus of ‘Calculation of Arches and Domes in Fifteenth-Century Samarkand’ (Chap. 20) by Yvonne Dold Samplonius. This chapter features a reading of the great mathematician al-Kashi’s work, Key of Arithmetic, an important text about the measurement of buildings. The techniques developed by al-Kashi were used to determine the surface area of vaults and domes in order to support the practical construction (and estimates of costs and materials) of architectural work. The mathematics of domes and vaults is also the subject of ‘Curves of Clay: Mexican Brick Vaults and Domes’ (Chap. 21) by Alfonso Ramírez Ponce and Rafael Ramírez Melendez . The authors, father and son, analyse the properties of traditional Mexican brick vaults and domes that were constructed using brick stacking techniques that required no scaffolding or centering.

The structural properties of architecture are the primary topic of ‘Mathematics and Structural Repair of Gothic Structures’ (Chap. 22) by Javier Barrallo and Santiago Sanchez-Beitia . Founded in the proposition that architectural education should include an introduction to mathematics, Barrallo describes processes for the restoration and maintenance of Gothic buildings. While Barrallo’s chapter is concerned with the European tradition, in ‘Mathematics of Carpentry in Historic Japanese Architecture’ (Chap. 23) Izumi Kuroishi describes how Heinouchi Masaomi , a nineteenth-century master carpenter and mathematician, developed a theory of Kikujutu (stereotomy) and an application of the Japanese mathematical system, Wasan, to carpentry. Kuroishi’s research into Masaomi’s theory demonstrates both the technical and cultural significance of mathematics in the construction process.

The final two chapters in this section continue the examination of different cultural traditions of geometric construction and measurement. In ‘On Some Geometrical and Architectural Ideas from African Art and Craft’ (Chap. 24), Mozambique mathematician Paulus Gerdes suggests that the practice of weaving large baskets or surfaces represents the early stages of architectural development in Africa. Gerdes’s work undertakes a close analysis of geometric patterns and construction methods found in woven mats, which were often used to line or decorate traditional structures. Continuing the cross-cultural survey, in ‘Design, Construction and Measurement in the Inka Empire’ (Chap. 25) William Sapp commences by noting that the lack of consistency in Inkan architectural features (doors, niches, windows) has lead to the suggestion that there were no clear mathematical rules for generating architectural dimensions and proportions. Sapp partially refutes this position demonstrating how the use of plumb bobs and a proportional system, both known to have been in use by the Inka civilisations, can explain the way their architecture was laid out.

Part IV: Architecture from 1100 ad to 1400 ad

The twelfth century ad saw both the end of the late Romanesque architecture and the rise of the first Gothic buildings that characterised so much European design at that time. For the next 300 years developments in Early English and Muscovite architecture occurred in parallel with more advanced Gothic brick and stone structures until, around the beginning of the fifteenth century, the first Renaissance buildings began to be completed. Part IV of the present volume covers this 300-year period, which broadly coincides with the rise and fall of the Gothic. However, of the 11 chapters in Part IV only four are about Gothic architecture and only five are set in central Europe. The remainder of Part IV features two chapters about developments in the northern parts of the continent (Ireland and the Baltic Sea), two chapters about developments in Asia (India and China) and two, North Africa (Egypt and Persia).

In ‘Vastu Purusha Mandala’ (Chap. 26) Vini Nathan describes the geometric diagram, part of the Vastu Shastra doctrine, which was developed in medieval India to provide a set of rules for translating cosmic or theological ideas and values into architectural form. Nathan initially describes the way in which the Vastu purusha mandala was used in India for the planning of buildings and towns. Thereafter she argues that the geometric logic of the mandala had a much greater influence on the cultural milieu of India than previously thought. While ostensibly serving a function similar to the Indian Vastu Shastra, the Yingzao fashi, was a twelfth-century Chinese manual of building and construction standards and techniques. In ‘Algorithmic Architecture in Twelfth-Century China: The Yingzao Fashi’ (Chap. 27), this historic rule-based system is used by Andrew I-kang Li as the basis for a shape grammar; a set of rules which, collectively, provide an algorithmic basis for understanding, and potentially replicating, a particular approach to design.

The way the Vastu purusha mandala operated in Indian culture to mediate between the heavens and the earth has clear parallels to the use of stereographic projections of the Heavenly Sphere in parts of Europe. In ‘The Celestial Key: Heaven Projected on Earth’ (Chap. 28) by Niels Bandholm , the placement and proportions of 15 medieval churches (constructed between 1150 and 1250) on the island of Bornholm in the Baltic Sea are examined, and potentially explained, using stereographic projection. Late Medieval architecture remains the focus of the following two chapters. In ‘Friedrich II and the Love of Geometry’ (Chap. 29), Heinz Götze revisits the Castel del Monte , describing its geometry and tracing several precedents for its form in navigational charts, wind stars and mosaics. The Holy Roman Emperor Friedrich II built the Castel del Monte in the thirteenth century in southern Italy. This small castle features a particular nested octagonal plan structure, which has fascinated architects and mathematicians for many hundreds of years. In ‘Metrology and Proportion in the Ecclesiastical Architecture of Medieval Ireland’ (Chap. 30), Avril Behan and Rachel Moss examine the extent to which a close, empirical analysis of medieval Irish window tracery can illuminate the backgrounds, training and work practices of the masons who constructed them. This study offers a different way of thinking about both architecture and measurement, not for symbolic purposes, but for supporting historical and archival research into construction.

Window tracery is also the topic of ‘The Cloisters of Hauterive ’ (Chap. 31) by Benno Artmann , although his focus is on Gothic church windows in the Cistercian monastery of Hauterive near Fribourg, Switzerland. Gothic tracery is constructed geometrically from an elaborate combination of circular arcs and straight lines. Artmann uncovers a series of complex geometric patterns and rules which represent a departure from the early Gothic tracery.

The next pair of chapters is concerned with the architecture and geometry of the Islamic world. The first of these commences with a reading of the twelfth-century Persian text On interlocks of similar or complementary figures, an important treatise of the era which describes 68 geometric constructions for use by artisans. In ‘The Use of Cubic Equations in Islamic Art and Architecture’ (Chap. 32), Alpay Özdural uncovers a series of techniques that were historically used by artisans to solve complex geometric relations. Such rich geometric relationships, which are found in North African and Islamic decoration, are also present in the geometric order of the Sultan Hassan Floor in Cairo. In ‘Explicit and Implicit Geometric Orders in Mamluk Floors: Secrets of the Sultan Hassan Floor in Cairo’ (Chap. 33), Gulzar Haider and Muhammad Moussa use a detailed measured survey of the largest of these Mamluk floors to describe a computer-assisted analysis of the geometric orders in the design.

The final three chapters in this part are all focussed on the architecture of Italy. The first of these is about the Palazzo della Signoria in Florence, the second Milan Cathedral and the last the Baptistery and the Campanile of Pisa . In ‘The Sequence of Fibonacci and the Palazzo della Signoria in Florence’ (Chap. 34), Maria Teresa Bartoli describes the significance of the Fibonacci sequence (a sequence of numbers wherein each is the sum of the two preceding values) and which converges to ϕ, the golden section, and traces its geometric presence (in the form of the Fibonacci rectangle, which is also an approximation of the golden section) in the Palazzo della Signoria. Through this analysis Bartoli demonstrates the extent to which the Fibonacci sequence, when expressed as a set of geometric proportions, was significant in society at that time. In ‘What Geometries in Milan Cathedral?’ (Chap. 35) Elena Marchetti and Luisa Rossi Costa show a similar point of convergence on the use of geometric proportional systems in architecture. Milan cathedral, with its late Gothic structure and early Renaissance façade, was completed in the fourteenth century. Marchetti and Rossi Costa examine the cathedral, seeking evidence for any of the common proportional systems including the golden mean and other ‘metallic’ numbers. In ‘The Symmetries of the Baptistery and the Leaning Tower of Pisa ’ (Chap. 36), physicist David Speiser describes and demonstrates the 15-fold and 30-fold symmetry of the penta-decagonal campanile (the ‘Leaning Tower’) at Pisa, before examining the even more striking 12-fold symmetry in the plan of the Baptistery at Pisa.

Part V: Theories of Proportion, Symmetry, Periodicity

The Renaissance saw the rise of a particular fascination with relationships between parts, involving theories of proportionality, symmetry and periodicity, many of which were either explained or rationalised using musical or harmonic notions. Part V contains five chapters, the first two of which are concerned with the use of music as a means of connecting architecture and mathematics. In ‘Musical Proportions at the Basis of Systems of Architectural Proportions both Ancient and Modern’ (Chap. 37), Jay Kappraff provides an overview of the way in which musical proportions have been used to shape architectural form. A recurring theme throughout this time was the conviction that the application of certain ratios or proportions would endow a design with both an overarching sense of unity as well as a distinct harmony between its component parts. Through this process Kappraff identifies three significant proportional systems—Alberti ’s musical ratios, the Roman application of the ‘sacred cut’ and Le Corbusier ’s Modulor—all of which have a similar purpose, but use ratios in different ways. In ‘From Renaissance Musical Proportions to Polytonality in Twentieth Century Architecture’ (Chap. 38), Radoslav Zuk describes a related proportional system derived from consonant musical intervals and traces its evolution. Zuk argues that for such proportional relationships between architecture and music to be meaningful, they must incorporate the three-dimensional properties of architecture, as well as the more common two-dimensional relationships found in plans and elevations.

The following pair of chapters shift the focus away from proportions and ratios and towards issues of symmetry (the mirrored or translated version of a shape or form) and periodicity (the repetition of a shape or form at set intervals). In ‘Quasi-Periodicity in Islamic Geometric Design’ (Chap. 39), Peter Saltzman undertakes a detailed analysis of Islamic geometry and tiling, reviewing both published research and important examples. He concludes by noting the presence of complex quasi-periodicity within tiling fragments in fifteenth-century Iranian decoration and architecture. More abstract geometric notions also inform ‘The Universality of the Symmetry Concept’ (Chap. 40), István Hargittai and Magdolna Hargittai ’s chapter about the apparent universality of symmetrical form. The authors, both chemists, commence by describing the ubiquity of symmetry, before arguing that a better understanding and appreciation of different types of symmetry may assist the development of trans-disciplinary knowledge.

The final chapter in Part V contains one of the better-known counterarguments to the practice of seeking hidden geometric traces, constructions and proportions in historic buildings. In ‘Contra Divinam Proportione’ (Chap. 41), Marco Frascari and Livio Volpi Ghirardini argue that the presence and importance of the golden mean in historic architecture has been much exaggerated. They propose that such is the allure and simplicity of the golden mean that it has been uncritically adopted to explain a growing number of forms which neither closely conform to its geometry nor have any theoretical affinity with it. By considering the way architecture was historically constructed, Frascari and Volpi Ghirardini demonstrate the potential fallacy at the heart of many interpretations of historic buildings.

Part VI: Architecture from 1400 ad to 1500 ad

The fifteenth century saw the waning of the Gothic era, the growth in importance of Renaissance architecture, and the first stages of the Tudor style in England. The first four chapters in Part VI are about Leon Battista Alberti , his architecture and theory. The following two chapters also have a focus on the Italian Renaissance while the final chapter examines the geometry of Muqarnas vaulting.

Alberti ’s Sant’Andrea in Mantua is widely regarded as one of his most refined and complete works. In ‘Alberti’s Sant’Andrea and the Etruscan Proportion’ (Chap. 42), Michael Ytterberg uncovers an unusual proportional system in the completed building and demonstrates that this ratio can be accounted for by considering the Etruscan architectural tradition. Livio Volpi Ghirardini ’s chapter, ‘The Numberable Architecture of Leon Battista Alberti as a Universal Sign of Order and Harmony’ (Chap. 43), is also concerned with Alberti’s architecture, but with a particular emphasis on signs of harmony and number symbolism. Considering both Sant’ Andrea and San Sebastiano in Mantua, Volpi Ghirardini investigates two conflicting interpretations of Alberti’s proportional geometry. Volpi Ghirardini concludes by observing that Alberti relied on finite, but not musical, ratios and progressions, which produced ‘numerically proportionate triads, which are themselves proportionately interrelated.’ In ‘Leon Battista Alberti and the Art of Building’ (Chap. 44), Salvatore di Pasquale examines Alberti’s defence of the use of models, noting that while Alberti apparently viewed the design concept as a fixed or inviolate proposition, he was also aware of the importance of material and structural properties, using models to test basic principles. The next pair of chapters are centred on the city of Florence in Tuscany. The first of these analyses the geometric design of a single tombslab, while the second considers the geometric composition of an entire church. Kim Williams deciphers a geometric code, in ‘Verrocchio ’s Tombslab for Cosimo de’ Medici : Designing with a Mathematical Vocabulary’ (Chap. 45). Credited to Florentine sculptor Andrea del Verrocchio, the tombslab in the basilica of San Lorenzo features a set of complex geometrical forms and proportions. Williams (one of the editors of the present volume) analyses Cosimo de’ Medici’s tombslab and then compares its design with that of three pavements which were completed at a similar time: the Sistine Chapel , the Medici Chapel in the Palazzo Medici and the Chapel of the Cardinal of Portugal in S. Miniato al Monte. Williams’s conclusion stresses the way geometry was used to reinforce humanity’s symbolic centrality in the cosmos. In ‘A New Geometric Analysis of the Pazzi Chapel in Santa Croce’ (Chap. 46), Mark Reynolds undertakes a geometric analysis of a single work by Brunelleschi . Reynolds, whose analysis of the Teotihuacan complex was featured in an earlier chapter, uses tracing techniques, overlaid on plans, elevations and sections, to analyse architecture. While informed by the interpretations of past scholars, Reynolds’s approach applies a range of geometric constructions (like the ‘sacred cut’ or the vesica pisces) to measured drawings to seek an underlying order in the architecture. Here he develops evidence to support several common interpretations of the Pazzi Chapel’s construction using divided squares and circles, along with the suggestion that the altar space in the building may be constructed around the golden Section.

A muqarnas vault is one of the characteristic architectural forms found across North Africa and in Arabic and Islamic architecture. While examples of muqarnas vaulting have been traced to the tenth century, it was during the fifteenth century that some of the most complex were produced, and, not coincidentally, when the mathematician Ghiyath al-Din al Kashi developed a technique for measuring their surface area. In ‘Muqarnas, Construction and Reconstruction’ (Chap. 47), Yvonne Dold-Samplonius and Silvia Harmsen describe their development of a database of muqarnas dimensions and constructions to allow for the mapping of styles, regions and timeframes using subtle developments that occurred in the geometry and form of the vault.

Conclusion

Across 47 chapters, authored by 53 scholars of architecture, mathematics, engineering and philosophy, the present volume contains examinations of key theories, buildings and treatises which, between 2000 bc and 1500 ad, evidence the crucial relationships between architecture and mathematics. In the theory strand, overarching ideas have been introduced and analysed, while in the history strand, precise examples have been considered in great detail, often testing past ideas or proposing new ways of viewing famous buildings. Volume II, featuring a similar number of chapters and with a commensurately wide range of authors, spans the history and theory of architecture from 1500 ad to the present day.