Mathematics and architecture

thumb|upright=1.3|"The Gherkin",

Mathematics and architecture are related, since architecture, like some other arts, uses mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a building; from the Pythagoreans of the sixth century BC onwards, to create architectural forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings.

In ancient Egypt, ancient Greece, India, and the Islamic world, buildings including pyramids, temples, mosques, palaces and mausoleums were laid out with specific proportions for religious reasons. In Islamic architecture, geometric shapes and geometric tiling patterns are used to decorate buildings, both inside and outside. Some Hindu temples have a fractal-like structure where parts resemble the whole, conveying a message about the infinite in Hindu cosmology. In Chinese architecture, the tulou of Fujian province are circular, communal defensive structures. In the twenty-first century, mathematical ornamentation is again being used to cover public buildings.

In Renaissance architecture, symmetry and proportion were deliberately emphasized by architects such as Leon Battista Alberti, Sebastiano Serlio and Andrea Palladio, influenced by Vitruvius's De architectura from ancient Rome and the arithmetic of the Pythagoreans from ancient Greece.

At the end of the nineteenth century, Vladimir Shukhov in Russia and Antoni Gaudí in Barcelona pioneered the use of hyperboloid structures; in the Sagrada Família, Gaudí also incorporated hyperbolic paraboloids, tessellations, catenary arches, catenoids, helicoids, and ruled surfaces. In the twentieth century, styles such as modern architecture and Deconstructivism explored different geometries to achieve desired effects. Minimal surfaces have been exploited in tent-like roof coverings as at Denver International Airport, while Richard Buckminster Fuller pioneered the use of the strong thin-shell structures known as geodesic domes.

Connected fields

thumb|upright|In the [[Renaissance, an architect like Leon Battista Alberti was expected to be knowledgeable in many disciplines, including arithmetic and geometry.]]

The architects Michael Ostwald and Kim Williams, considering the relationships between architecture and mathematics, note that the fields as commonly understood might seem to be only weakly connected, since architecture is a profession concerned with the practical matter of making buildings, while mathematics is the pure study of number and other abstract objects. But, they argue, the two are strongly connected, and have been since antiquity. In ancient Rome, Vitruvius described an architect as a man who knew enough of a range of other disciplines, primarily geometry, to enable him to oversee skilled artisans in all the other necessary areas, such as masons and carpenters. The same applied in the Middle Ages, where graduates learnt arithmetic, geometry and aesthetics alongside the basic syllabus of grammar, logic, and rhetoric (the trivium) in elegant halls made by master builders who had guided many craftsmen. A master builder at the top of his profession was given the title of architect or engineer. In the Renaissance, the quadrivium of arithmetic, geometry, music and astronomy became an extra syllabus expected of the Renaissance man such as Leon Battista Alberti. Similarly in England, Sir Christopher Wren, known today as an architect, was firstly a noted astronomer.

Williams and Ostwald, further overviewing the interaction of mathematics and architecture since 1500 according to the approach of the German sociologist Theodor Adorno, identify three tendencies among architects, namely: to be revolutionary, introducing wholly new ideas; reactionary, failing to introduce change; or revivalist, actually going backwards. They argue that architects have avoided looking to mathematics for inspiration in revivalist times. This would explain why in revivalist periods, such as the Gothic Revival in 19th century England, architecture had little connection to mathematics. Equally, they note that in reactionary times such as the Italian Mannerism of about 1520 to 1580, or the 17th century Baroque and Palladian movements, mathematics was barely consulted. In contrast, the revolutionary early 20th-century movements such as Futurism and Constructivism actively rejected old ideas, embracing mathematics and leading to Modernist architecture. Towards the end of the 20th century, too, fractal geometry was quickly seized upon by architects, as was aperiodic tiling, to provide interesting and attractive coverings for buildings.

Architects use mathematics for several reasons, leaving aside the necessary use of mathematics in the engineering of buildings. Firstly, they use geometry because it defines the spatial form of a building.

Secondly, they use mathematics to design forms that are considered beautiful or harmonious. From the time of the Pythagoreans with their religious philosophy of number, architects in ancient Greece, ancient Rome, the Islamic world and the Italian Renaissance have chosen the proportions of the built environment – buildings and their designed surroundings – according to mathematical as well as aesthetic and sometimes religious principles. Fourthly, they may use mathematics in the form of computer modelling to meet environmental goals, such as to minimise whirling air currents at the base of tall buildings. Thus the Basilica's width to length is 1:2; the aisle around it is as high as it is wide, 1:1; the columns are five feet thick and fifty feet high, 1:10.

thumb|upright|Floor plan of the Pantheon

Vitruvius named three qualities required of architecture in his De architectura, : firmness, usefulness (or "Commodity" in Henry Wotton's 17th century English), and delight. These can be used as categories for classifying the ways in which mathematics is used in architecture. Firmness encompasses the use of mathematics to ensure a building stands up, hence the mathematical tools used in design and to support construction, for instance to ensure stability and to model performance. Usefulness derives in part from the effective application of mathematics, reasoning about and analysing the spatial and other relationships in a design. Delight is an attribute of the resulting building, resulting from the embodying of mathematical relationships in the building; it includes aesthetic, sensual and intellectual qualities.

The Pantheon

The Pantheon in Rome has survived intact, illustrating classical Roman structure, proportion, and decoration. The main structure is a dome, the apex left open as a circular oculus to let in light; it is fronted by a short colonnade with a triangular pediment. The height to the oculus and the diameter of the interior circle are the same, , so the whole interior would fit exactly within a cube, and the interior could house a sphere of the same diameter. These dimensions make more sense when expressed in ancient Roman units of measurement: The dome spans 150 Roman feet); the oculus is 30 Roman feet in diameter; the doorway is 40 Roman feet high. The Pantheon remains the world's largest unreinforced concrete dome.

Renaissance

thumb|Facade of [[Santa Maria Novella, Florence, 1470. The frieze (with squares) and above is by Leon Battista Alberti.]]

The first Renaissance treatise on architecture was Leon Battista Alberti's 1450 (On the Art of Building); it became the first printed book on architecture in 1485. It was partly based on Vitruvius's De architectura and, via Nicomachus, Pythagorean arithmetic. Alberti starts with a cube, and derives ratios from it. Thus the diagonal of a face gives the ratio 1:, while the diameter of the sphere which circumscribes the cube gives 1:. Alberti also documented Filippo Brunelleschi's discovery of linear perspective, developed to enable the design of buildings which would look beautifully proportioned when viewed from a convenient distance.]]

The next major text was Sebastiano Serlio's Regole generali d'architettura (General Rules of Architecture); the first volume appeared in Venice in 1537; the 1545 volume (books1 and 2) covered geometry and perspective. Two of Serlio's methods for constructing perspectives were wrong, but this did not stop his work being widely used.

thumb|right|[[Andrea Palladio's plan and elevation of the Villa Pisani ]]

In 1570, Andrea Palladio published the influential I quattro libri dell'architettura (The Four Books of Architecture) in Venice. This widely printed book was largely responsible for spreading the ideas of the Italian Renaissance throughout Europe, assisted by proponents like the English diplomat Henry Wotton with his 1624 The Elements of Architecture. The proportions of each room within the villa were calculated on simple mathematical ratios like 3:4 and 4:5, and the different rooms within the house were interrelated by these ratios. Earlier architects had used these formulas for balancing a single symmetrical facade; however, Palladio's designs related to the whole, usually square, villa. Palladio permitted a range of ratios in the Quattro libri, stating:

In 1615, Vincenzo Scamozzi published the late Renaissance treatise L'idea dell'architettura universale (The Idea of a Universal Architecture). He attempted to relate the design of cities and buildings to the ideas of Vitruvius and the Pythagoreans, and to the more recent ideas of Palladio.

Nineteenth century

thumb|upright=0.5|[[Hyperboloid structure|Hyperboloid lattice lighthouse by Vladimir Shukhov, Ukraine, 1911]]

Hyperboloid structures were used starting towards the end of the nineteenth century by Vladimir Shukhov for masts, lighthouses and cooling towers. Their striking shape is both aesthetically interesting and strong, using structural materials economically. Shukhov's first hyperboloidal tower was exhibited in Nizhny Novgorod in 1896.

Twentieth century

thumb|left|[[De Stijl's sliding, intersecting planes: the Rietveld Schröder House, 1924]]

The early twentieth century movement Modern architecture, pioneered