**Mathematics and architecture** are related, since,
as with other arts,
architects use
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 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.

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.^{
[3]}

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.^{
[4]}

Architects use mathematics for several reasons, leaving aside the necessary use of mathematics in the
engineering of buildings.^{
[5]} Firstly, they use geometry because it defines the spatial form of a building.^{
[6]} Secondly, they use mathematics to design forms that are
considered beautiful or harmonious.^{
[7]} From the time of the
Pythagoreans with their religious philosophy of number,^{
[8]} 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.^{
[9]}^{
[10]}^{
[11]}^{
[12]} Thirdly, they may use mathematical objects such as
tessellations to decorate buildings.^{
[13]}^{
[14]} 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.^{
[1]}

The influential ancient Roman architect Vitruvius argued that the design of a building such as a temple depends on two qualities, proportion and *symmetria*. Proportion ensures that each part of a building relates harmoniously to every other part. *Symmetria* in Vitruvius's usage means something closer to the English term modularity than
mirror symmetry, as again it relates to the assembling of (modular) parts into the whole building. In his Basilica at
Fano, he uses ratios of small integers, especially the
triangular numbers (1, 3, 6, 10, ...) to proportion the structure into
(Vitruvian) modules.^{
[a]} 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.^{
[9]}

Vitruvius named three qualities required of architecture in his *
De architectura*, c. 15 B.C.: firmness, usefulness (or "Commodity" in
Henry Wotton's 16th 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.^{
[16]}

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, 43.3 metres (142 ft), so the whole interior would fit exactly within a cube, and the interior could house a sphere of the same diameter.^{
[17]} These dimensions make more sense when expressed in
ancient Roman units of measurement: The dome spans 150
Roman feet^{
[b]}); the oculus is 30 Roman feet in diameter; the doorway is 40 Roman feet high.^{
[18]} The Pantheon remains the world's largest unreinforced concrete dome.^{
[19]}

The first Renaissance treatise on architecture was Leon Battista Alberti's 1450 *
De re aedificatoria* (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:√2, while the diameter of the sphere which circumscribes the cube gives 1:√3.^{
[20]}^{
[21]} 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.^{
[12]}

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 (books 1 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.^{
[23]}

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*.^{
[24]} 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.^{
[25]} Palladio permitted a range of ratios in the *Quattro libri*, stating:^{
[26]}^{
[27]}

There are seven types of room that are the most beautiful and well proportioned and turn out better: they can be made circular, though these are rare; or square; or their length will equal the diagonal of the square of the breadth; or a square and a third; or a square and a half; or a square and two-thirds; or two squares.

^{ [c]}

In 1615,
Vincenzo Scamozzi published the late Renaissance treatise *L'idea dell'architettura universale* (The Idea of a Universal Architecture).^{
[28]} 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.^{
[29]}

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.^{
[30]}^{
[31]}^{
[32]}

The early twentieth century movement
Modern architecture, pioneered^{
[d]} by Russian
Constructivism,^{
[33]} used rectilinear
Euclidean (also called
Cartesian) geometry. In the
De Stijl movement, the horizontal and the vertical were seen as constituting the universal. The architectural form consists of putting these two directional tendencies together, using roof planes, wall planes and balconies, which either slide past or intersect each other, as in the 1924
Rietveld Schröder House by
Gerrit Rietveld.^{
[34]}

Modernist architects were free to make use of curves as well as planes.
Charles Holden's 1933
Arnos station has a circular ticket hall in brick with a flat concrete roof.^{
[35]} In 1938, the
Bauhaus painter
László Moholy-Nagy adopted
Raoul Heinrich Francé's seven
biotechnical elements, namely the crystal, the sphere, the cone, the plane, the (cuboidal) strip, the (cylindrical) rod, and the spiral, as the supposed basic building blocks of architecture inspired by nature.^{
[36]}^{
[37]}

Le Corbusier proposed an
anthropometric
scale of proportions in architecture, the
Modulor, based on the supposed height of a man.^{
[38]} Le Corbusier's 1955
Chapelle Notre-Dame du Haut uses free-form curves not describable in mathematical formulae.^{
[e]} The shapes are said to be evocative of natural forms such as the
prow of a ship or praying hands.^{
[41]} The design is only at the largest scale: there is no hierarchy of detail at smaller scales, and thus no fractal dimension; the same applies to other famous twentieth-century buildings such as the
Sydney Opera House,
Denver International Airport, and the
Guggenheim Museum, Bilbao.^{
[39]}

Contemporary architecture, in the opinion of the 90 leading architects who responded to a 2010
World Architecture Survey, is extremely diverse; the best was judged to be
Frank Gehry's Guggenheim Museum, Bilbao.^{
[42]}

Denver International Airport's terminal building, completed in 1995, has a
fabric roof supported as a
minimal surface (i.e., its
mean curvature is zero) by steel cables. It evokes
Colorado's snow-capped mountains and the
teepee tents of
Native Americans.^{
[43]}^{
[44]}

The architect
Richard Buckminster Fuller is famous for designing strong
thin-shell structures known as
geodesic domes. The
Montréal Biosphère dome is 61 metres (200 ft) high; its diameter is 76 metres (249 ft).^{
[45]}

Sydney Opera House has a dramatic roof consisting of soaring white vaults, reminiscent of ship's sails; to make them possible to construct using standardized components, the vaults are all composed of triangular sections of spherical shells with the same radius. These have the required uniform
curvature in every direction.^{
[46]}

The late twentieth century movement
Deconstructivism creates deliberate disorder with what
Nikos Salingaros in *
A Theory of Architecture* calls random forms^{
[47]} of high complexity^{
[48]} by using non-parallel walls, superimposed grids and complex 2-D surfaces, as in Frank Gehry's
Disney Concert Hall and Guggenheim Museum, Bilbao.^{
[49]}^{
[50]} Until the twentieth century, architecture students were obliged to have a grounding in mathematics. Salingaros argues that first "overly simplistic, politically-driven"
Modernism and then "anti-scientific" Deconstructivism have effectively separated architecture from mathematics. He believes that this "reversal of mathematical values" is harmful, as the "pervasive aesthetic" of non-mathematical architecture trains people "to reject mathematical information in the built environment"; he argues that this has negative effects on society.^{
[39]}

New Objectivity: Walter Gropius's Bauhaus, Dessau, 1925

Geodesic dome: the Montréal Biosphère by R. Buckminster Fuller, 1967

Uniform curvature: Sydney Opera House, 1973

Deconstructivism: Disney Concert Hall, Los Angeles, 2003

The
pyramids of
ancient Egypt are
tombs constructed with mathematical proportions, but which these were, and whether the
Pythagorean theorem was used, are debated. The ratio of the slant height to half the base length of the
Great Pyramid of Giza is less than 1% from the
golden ratio.^{
[51]} If this was the design method, it would imply the use of
Kepler's triangle (face angle 51°49'),^{
[51]}^{
[52]} but according to many
historians of science, the golden ratio was not known until the time of the
Pythagoreans.^{
[53]} The Great Pyramid may also have been based on a triangle with base to hypotenuse ratio 1:4/π (face angle 51°50').^{
[54]}

The proportions of some pyramids may have also been based on the
3:4:5 triangle (face angle 53°8'), known from the
Rhind Mathematical Papyrus (c. 1650–1550 BC); this was first conjectured by historian
Moritz Cantor in 1882.^{
[55]} It is known that right angles were laid out accurately in ancient Egypt using
knotted cords for measurement,^{
[55]} that
Plutarch recorded in *
Isis and Osiris* (c. 100 AD) that the Egyptians admired the 3:4:5 triangle,^{
[55]} and that a scroll from before 1700 BC demonstrated basic
square formulas.^{
[56]}^{
[f]} Historian Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem," but also notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle's sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain; he guesses that the ancient Egyptians probably knew the Pythagorean theorem, but "there is no evidence that they used it to construct right angles."^{
[55]}

Vaastu Shastra, the ancient
Indian canons of architecture and town planning, employs symmetrical drawings called
mandalas. Complex calculations are used to arrive at the dimensions of a building and its components. The designs are intended to integrate architecture with nature, the relative functions of various parts of the structure, and ancient beliefs utilizing geometric patterns (
yantra), symmetry and
directional alignments.^{
[57]}^{
[58]} However, early builders may have come upon mathematical proportions by accident. The mathematician Georges Ifrah notes that simple "tricks" with string and stakes can be used to lay out geometric shapes, such as ellipses and right angles.^{
[12]}^{
[59]}

The mathematics of
fractals has been used to show that the reason why existing buildings have universal appeal and are visually satisfying is because they provide the viewer with a sense of scale at different viewing distances. For example, in the tall
gopuram gatehouses of
Hindu temples such as the
Virupaksha Temple at
Hampi built in the seventh century, and others such as the
Kandariya Mahadev Temple at
Khajuraho, the parts and the whole have the same character, with
fractal dimension in the range 1.7 to 1.8. The cluster of smaller towers (*shikhara*, lit. 'mountain') about the tallest, central, tower which represents the holy
Mount Kailash, abode of Lord
Shiva, depicts the endless repetition of universes in
Hindu cosmology.^{
[2]}^{
[60]} The religious studies scholar William J. Jackson observed of the pattern of towers grouped among smaller towers, themselves grouped among still smaller towers, that:

The ideal form gracefully artificed suggests the infinite rising levels of existence and consciousness, expanding sizes rising toward transcendence above, and at the same time housing the sacred deep within.

^{ [60]}^{ [61]}

The
Meenakshi Amman Temple is a large complex with multiple shrines, with the streets of
Madurai laid out concentrically around it according to the shastras. The four gateways are tall towers (
gopurams) with fractal-like repetitive structure as at Hampi. The enclosures around each shrine are rectangular and surrounded by high stone walls.^{
[62]}

Pythagoras (c. 569 – c. 475 B.C.) and his followers, the Pythagoreans, held that "all things are numbers". They observed the harmonies produced by notes with specific small-integer ratios of frequency, and argued that buildings too should be designed with such ratios. The Greek word *symmetria* originally denoted the harmony of architectural shapes in precise ratios from a building's smallest details right up to its entire design.^{
[12]}

The
Parthenon is 69.5 metres (228 ft) long, 30.9 metres (101 ft) wide and 13.7 metres (45 ft) high to the cornice. This gives a ratio of width to length of 4:9, and the same for height to width. Putting these together gives height:width:length of 16:36:81, or to the delight^{
[63]} of the Pythagoreans 4^{2}:6^{2}:9^{2}. This sets the module as 0.858 m. A 4:9 rectangle can be constructed as three contiguous rectangles with sides in the ratio 3:4. Each half-rectangle is then a convenient 3:4:5 right triangle, enabling the angles and sides to be checked with a suitably knotted rope. The inner area (naos) similarly has 4:9 proportions (21.44 metres (70.3 ft) wide by 48.3 m long); the ratio between the diameter of the outer columns, 1.905 metres (6.25 ft), and the spacing of their centres, 4.293 metres (14.08 ft), is also 4:9.^{
[12]}

The Parthenon is considered by authors such as
John Julius Norwich "the most perfect Doric temple ever built".^{
[64]} Its elaborate architectural refinements include "a subtle correspondence between the curvature of the stylobate, the taper of the
naos walls and the *entasis* of the columns".^{
[64]} *
Entasis* refers to the subtle diminution in diameter of the columns as they rise. The stylobate is the platform on which the columns stand. As in other classical Greek temples,^{
[65]} the platform has a slight parabolic upward curvature to shed rainwater and reinforce the building against earthquakes. The columns might therefore be supposed to lean outwards, but they actually lean slightly inwards so that if they carried on, they would meet about a kilometre and a half above the centre of the building; since they are all the same height, the curvature of the outer stylobate edge is transmitted to the
architrave and roof above: "all follow the rule of being built to delicate curves".^{
[66]}

The golden ratio was known in 300 B.C., when
Euclid described the method of geometric construction.^{
[67]} It has been argued that the golden ratio was used in the design of the Parthenon and other ancient Greek buildings, as well as sculptures, paintings, and vases.^{
[68]} More recent authors such as Nikos Salingaros, however, doubt all these claims.^{
[69]} Experiments by the computer scientist George Markowsky failed to find any preference for the
golden rectangle.^{
[70]}

The historian of Islamic art Antonio Fernandez-Puertas suggests that the
Alhambra, like the
Great Mosque of Cordoba,^{
[71]} was designed using the
Hispano-Muslim foot or *codo* of about 0.62 metres (2.0 ft). In the palace's
Court of the Lions, the proportions follow a series of
surds. A rectangle with sides 1 and √2 has (by
Pythagoras's theorem) a diagonal of √3, which describes the right triangle made by the sides of the court; the series continues with √4 (giving a 1:2 ratio), √5 and so on. The decorative patterns are similarly proportioned, √2 generating squares inside circles and eight-pointed stars, √3 generating six-pointed stars. There is no evidence to support earlier claims that the golden ratio was used in the Alhambra.^{
[10]}^{
[72]} The
Court of the Lions is bracketed by the Hall of Two Sisters and the Hall of the Abencerrajes; a regular
hexagon can be drawn from the centres of these two halls and the four inside corners of the Court of the Lions.^{
[73]}

The
Selimiye Mosque in
Edirne, Turkey, was built by
Mimar Sinan to provide a space where the
mihrab could be see from anywhere inside the building. The very large central space is accordingly arranged as an octagon, formed by eight enormous pillars, and capped by a circular dome of 31.25 metres (102.5 ft) diameter and 43 metres (141 ft) high. The octagon is formed into a square with four semidomes, and externally by four exceptionally tall minarets, 83 metres (272 ft) tall. The building's plan is thus a circle, inside an octagon, inside a square.^{
[74]}

Mughal architecture, as seen in the abandoned imperial city of
Fatehpur Sikri and the
Taj Mahal complex, has a distinctive mathematical order and a strong aesthetic based on symmetry and harmony.^{
[11]}^{
[75]}

The Taj Mahal exemplifies Mughal architecture, both representing
paradise^{
[76]} and displaying the
Mughal Emperor
Shah Jahan's power through its scale, symmetry and costly decoration. The white marble
mausoleum, decorated with
pietra dura, the great gate (*Darwaza-i rauza*), other buildings, the gardens and paths together form a unified hierarchical design. The buildings include a
mosque in red sandstone on the west, and an almost identical building, the Jawab or 'answer' on the east to maintain the bilateral symmetry of the complex. The formal
charbagh ('fourfold garden') is in four parts, symbolising the four
rivers of Paradise, and offering views and reflections of the mausoleum. These are divided in turn into 16 parterres.^{
[77]}

The Taj Mahal complex was laid out on a grid, subdivided into smaller grids. The historians of architecture Koch and Barraud agree with the traditional accounts that give the width of the complex as 374 Mughal yards or
gaz,^{
[g]} the main area being three 374-gaz squares. These were divided in areas like the bazaar and caravanserai into 17-gaz modules; the garden and terraces are in modules of 23 gaz, and are 368 gaz wide (16 x 23). The mausoleum, mosque and guest house are laid out on a grid of 7 gaz. Koch and Barraud observe that if an octagon, used repeatedly in the complex, is given sides of 7 units, then it has a width of 17 units,^{
[h]} which may help to explain the choice of ratios in the complex.^{
[78]}

The
Christian
patriarchal
basilica of
Haghia Sophia in
Byzantium (now
Istanbul), first constructed in 537 (and twice rebuilt), was for a thousand years^{
[i]} the largest cathedral ever built. It inspired many later buildings including
Sultan Ahmed and other mosques in the city. The
Byzantine architecture includes a nave crowned by a circular dome and two half-domes, all of the same diameter (31 metres (102 ft)), with a further five smaller half-domes forming an
apse and four rounded corners of a vast rectangular interior.^{
[79]} This was interpreted by mediaeval architects as representing the mundane below (the square base) and the divine heavens above (the soaring spherical dome).^{
[80]} The emperor
Justinian used two geometers,
Isidore of Miletus and
Anthemius of Tralles as architects; Isidore compiled the works of
Archimedes on
solid geometry, and was influenced by him.^{
[12]}^{
[81]}

The importance of water
baptism in Christianity was reflected in the scale of
baptistry architecture. The oldest, the
Lateran Baptistry in Rome, built in 440,^{
[82]} set a trend for octagonal baptistries; the
baptismal font inside these buildings was often octagonal, though Italy's largest
baptistry, at Pisa, built between 1152 and 1363, is circular, with an octagonal font. It is 54.86 metres (180.0 ft) high, with a diameter of 34.13 metres (112.0 ft) (a ratio of 8:5).^{
[83]}
Saint Ambrose wrote that fonts and baptistries were octagonal "because on the eighth day,^{
[j]} by rising, Christ loosens the bondage of death and receives the dead from their graves."^{
[84]}^{
[85]}
Saint Augustine similarly described the eighth day as "everlasting ... hallowed by the
resurrection of Christ".^{
[85]}^{
[86]} The octagonal
Baptistry of Saint John, Florence, built between 1059 and 1128, is one of the oldest buildings in that city, and one of the last in the direct tradition of classical antiquity; it was extremely influential in the subsequent Florentine Renaissance, as major architects including
Francesco Talenti, Alberti and Brunelleschi used it as the model of classical architecture.^{
[87]}

The number five is used "exuberantly"^{
[88]} in the 1721
Pilgrimage Church of St John of Nepomuk at Zelená hora, near
Žďár nad Sázavou in the Czech republic, designed by
Jan Blažej Santini Aichel. The nave is circular, surrounded by five pairs of columns and five oval domes alternating with ogival apses. The church further has five gates, five chapels, five altars and five stars; a legend claims that when
Saint John of Nepomuk was martyred, five stars appeared over his head.^{
[88]}^{
[89]} The fivefold architecture may also symbolise the
five wounds of Christ and the five letters of "Tacui" (Latin: "I kept silence" [about secrets of the
confessional]).^{
[90]}

Antoni Gaudí used a wide variety of geometric structures, some being minimal surfaces, in the
Sagrada Família,
Barcelona, started in 1882 (and not completed as of 2015). These include hyperbolic
paraboloids and
hyperboloids of revolution,^{
[91]} tessellations,
catenary arches,
catenoids,
helicoids, and
ruled surfaces. This varied mix of geometries is creatively combined in different ways around the church. For example, in the Passion Façade of Sagrada Família, Gaudí assembled stone "branches" in the form of hyperbolic paraboloids, which overlap at their tops (directrices) without, therefore, meeting at a point. In contrast, in the colonnade there are hyperbolic paraboloidal surfaces that smoothly join other structures to form unbounded surfaces. Further, Gaudí exploits
natural patterns, themselves mathematical, with
columns derived from the shapes of
trees, and
lintels made from unmodified
basalt naturally cracked (by cooling from molten rock) into
hexagonal columns.^{
[92]}^{
[93]}^{
[94]}

The 1971
Cathedral of Saint Mary of the Assumption, San Francisco has a
saddle roof composed of eight segments of hyperbolic paraboloids, arranged so that the bottom horizontal cross section of the roof is a square and the top cross section is a
Christian cross. The building is a square 77.7 metres (255 ft) on a side, and 57.9 metres (190 ft) high.^{
[95]} The 1970
Cathedral of Brasília by
Oscar Niemeyer makes a different use of a hyperboloid structure; it is constructed from 16 identical concrete beams, each weighing 90 tonnes,^{
[k]} arranged in a circle to form a hyperboloid of revolution, the white beams creating a shape like hands praying to heaven. Only the dome is visible from outside: most of the building is below ground.^{
[96]}^{
[97]}^{
[98]}^{
[99]}

Several medieval
churches in Scandinavia are circular, including four on the Danish island of
Bornholm. One of the oldest of these,
Østerlars Church from c. 1160, has a circular nave around a massive circular stone column, pierced with arches and decorated with a fresco. The circular structure has three storeys and was apparently fortified, the top storey having served for defence.^{
[100]}
^{
[101]}

The vaulting of the nave of Haghia Sophia, Istanbul

*( annotations*), 562The octagonal Baptistry of Saint John, Florence, completed in 1128

Fivefold symmetries: Jan Santini Aichel's Pilgrimage Church of St John of Nepomuk at Zelená hora, 1721

Passion façade of Antoni Gaudí's Sagrada Família, Barcelona, started 1882

Oscar Niemeyer's Cathedral of Brasília, 1970

Central column of Østerlars Nordic round church in Bornholm, Denmark

Islamic buildings are often decorated with
geometric patterns which typically make use of several mathematical
tessellations, formed of ceramic tiles (
girih,
zellige) that may themselves be plain or decorated with stripes.^{
[12]} Symmetries such as stars with six, eight, or multiples of eight points are used in Islamic patterns. Some of these are based on the 'Khatem Sulemani' or Solomon's seal motif, which is an eight-pointed star made of two squares, one rotated 45 degrees from the other on the same centre.^{
[102]} Islamic patterns exploit many of the 17 possible
wallpaper groups; as early as 1944, Edith Müller showed that the Alhambra made use of 11 wallpaper groups in its decorations, while in 1986
Branko Grünbaum claimed to have found 13 wallpaper groups in the Alhambra, asserting controversially that the remaining four groups are not found anywhere in Islamic ornament.^{
[102]}

The complex geometry and tilings of the muqarnas vaulting in the Sheikh Lotfollah Mosque, Isfahan, 1603–1619

Louvre Abu Dhabi under construction in 2015, its dome built up of layers of stars made of octagons, triangles, and squares

Towards the end of the 20th century, novel mathematical constructs such as fractal geometry and aperiodic tiling were seized upon by architects to provide interesting and attractive coverings for buildings.^{
[4]} In 1913, the Modernist architect
Adolf Loos had declared that "Ornament is a crime",^{
[103]} influencing architectural thinking for the rest of the 20th century. In the 21st century, architects are again starting to explore the use of
ornament. 21st century ornamentation is extremely diverse. Henning Larsen's 2011
Harpa Concert and Conference Centre, Reykjavik has what looks like a crystal wall of rock made of large blocks of glass.^{
[103]} Foreign Office Architects' 2010
Ravensbourne College, London is tessellated decoratively with 28,000 anodised aluminium tiles in red, white and brown, interlinking circular windows of differing sizes. The tessellation uses three types of tile, an equilateral triangle and two irregular pentagons.^{
[104]}^{
[105]}^{
[l]} Kazumi Kudo's
Kanazawa Umimirai Library creates a decorative grid made of small circular blocks of glass set into plain concrete walls.^{
[103]}

Ravensbourne College, London, 2010

Harpa Concert and Conference Centre, Iceland, 2011

Kanazawa Umimirai Library, Japan, 2011

Museo Soumaya, México, 2011

The architecture of
fortifications evolved from
medieval fortresses, which had high masonry walls, to low, symmetrical
star forts able to resist
artillery bombardment between the mid-fifteenth and nineteenth centuries. The geometry of the star shapes was dictated by the need to avoid dead zones where attacking infantry could shelter from defensive fire; the sides of the projecting points were angled to permit such fire to sweep the ground, and to provide crossfire (from both sides) beyond each projecting point. Well-known architects who designed such defences include
Michelangelo,
Baldassare Peruzzi,
Vincenzo Scamozzi and
Sébastien Le Prestre de Vauban.^{
[106]}^{
[107]}

The architectural historian
Siegfried Giedion argued that the star-shaped fortification had a formative influence on the patterning of the Renaissance
ideal city: "The Renaissance was hypnotized by one city type which for a century and a half—from Filarete to Scamozzi—was impressed upon all utopian schemes: this is the star-shaped city."^{
[108]}

Coevorden fortification plan. 17th century

Neuf-Brisach, Alsace, one of the Fortifications of Vauban

In
Chinese architecture, the
tulou of
Fujian province are circular, communal defensive structures with mainly blank walls and a single iron-plated wooden door, some dating back to the sixteenth century. The walls are topped with roofs that slope gently both outwards and inwards, forming a ring. The centre of the circle is an open cobbled courtyard, often with a well, surrounded by timbered galleries up to five stories high.^{
[109]}

Architects may also select the form of a building to meet environmental goals.^{
[88]} For example,
Foster and Partners'
30 St Mary Axe, London, known as "
The Gherkin" for its
cucumber-like shape, is a
solid of revolution designed using
parametric modelling. Its geometry was chosen not purely for aesthetic reasons, but to minimise whirling air currents at its base. Despite the building's apparently curved surface, all the panels of glass forming its skin are flat, except for the lens at the top. Most of the panels are
quadrilaterals, as they can be cut from rectangular glass with less wastage than triangular panels.^{
[1]}

The traditional
yakhchal (ice pit) of
Persia functioned as an
evaporative cooler. Above ground, the structure had a domed shape, but had a subterranean storage space for ice and sometimes food as well. The subterranean space and the thick heat-resistant construction insulated the storage space year round. The internal space was often further cooled with
windcatchers.^{
[110]}

**^**In Book 4, chapter 3 of*De architectura*, he discusses modules directly.^{ [15]}**^**A Roman foot was about 0.296 metres (0.97 ft).**^**In modern algebraic notation, these ratios are respectively 1:1, √2:1, 4:3, 3:2, 5:3, 2:1.**^**Constructivism influenced Bauhaus and Le Corbusier, for example.^{ [33]}**^**Pace Nikos Salingaros, who suggests the contrary,^{ [39]}but it is not clear exactly what mathematics may be embodied in the curves of Le Corbusier's chapel.^{ [40]}**^**Berlin Papyrus 6619 from the Middle Kingdom stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other."**^**1 gaz is about 0.86 metres (2.8 ft).**^**A square drawn around the octagon by prolonging alternate sides adds four right angle triangles with hypotenuse of 7 and the other two sides of √49/2 or 4.9497..., nearly 5. The side of the square is thus 5+7+5, which is 17.**^**Until Seville Cathedral was completed in 1520.**^**The sixth day of Holy Week was Good Friday; the following Sunday (of the resurrection) was thus the eighth day.^{ [84]}**^**This is 90 tonnes (89 long tons; 99 short tons).**^**An aperiodic tiling was considered, to avoid the rhythm of a structural grid, but in practice a Penrose tiling was too complex, so a grid of 2.625m horizontally and 4.55m vertically was chosen.^{ [105]}

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- Nexus Network Journal: Architecture and Mathematics Online
- The International Society of the Arts, Mathematics, and Architecture
- University of St Andrews: Mathematics and Architecture
- National University of Singapore: Mathematics in Art and Architecture
- Dartmouth College: Geometry in Art & Architecture