Architects Michael Ostwald and Kim Williams ("Kim Williams (architect)"), considering the relationships between architecture and mathematics, point out that both fields, as commonly understood, may appear to be loosely connected, since architecture is a profession that deals with the practical question of designing and constructing buildings, while mathematics is the pure study of numbers and other abstract objects. However, they argue, both disciplines are strongly connected, and have been since classical antiquity.

In ancient Rome, Vitruvius described an architect as a man who knew enough of a variety of other disciplines, primarily geometry, to enable him to supervise craftsmen skilled in all other necessary areas, such as bricklayers and carpenters. The same applied in the Middle Ages, when university-trained architects learned arithmetic, geometry, and aesthetics along with the basic curriculum of grammar, logic, and rhetoric (the liberal arts) in elegant classrooms built by master builders who had mentored many craftsmen. A master builder at the top of his profession received the title of architect or engineer. In the Renaissance, the liberal arts (arithmetic, geometry, music and astronomy) became an extra program of polymaths like Leon Battista Alberti. Similarly, in England, Sir Christopher Wren, known today as an architect, was first a noted astronomer.[3]​.

Williams and Ostwald, reviewing the interaction between 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 completely new ideas; be "reactionary", not introducing changes; or be revivalist "Revivalism (architecture)"), drawing on styles from the past. They argue that architects have avoided seeking inspiration from mathematics in retrospect, which would explain why in revivalist periods, such as neo-Gothic architecture in 19th-century England, architecture had little connection with mathematics. Likewise, they point out that in reactionary periods such as Italian Mannerism from around 1520 to 1580, or the Baroque and Palladian movements of the 19th century, mathematics was hardly taken into consideration. In contrast, revolutionary movements at the turn of the century such as futurism and constructivism actively rejected old ideas, adopting mathematics as a conceptual basis and leading to modern architecture "Modernism (philosophical and cultural movement)"). Also towards the end of the century, architects quickly took advantage of fractal geometry, as well as aperiodic tessellations, to provide buildings with attractive coatings.[4]​.

Architects use mathematics for several reasons, leaving aside the necessary use of mathematics in building engineering.[5]​ Firstly, they use geometry because it defines the spatial form of a building.[6]​ Secondly, they use mathematics to design shapes that can be considered beautiful or harmonious.[7]​ Since the time of the Pythagoreans with their religious philosophy of number, architects[8]​ of classical Greece, ancient Rome, the Islamic world and the Italian Renaissance They were concerned with providing their creations with a sense of proportion "Proportion (architecture)") in accordance with mathematical, aesthetic and, sometimes, religious principles. the structure of tall buildings.[1]​.

Ancient Rome

Vitruvius, the influential architect of ancient Rome, argued that the design of a building such as a temple depends on two qualities, proportion and symmetry. Proportion ensures that each part of a building relates harmoniously to all the others. symmetry, according to Vitruvius, means something closer to modularity than mirror symmetry, since the concept relates to the assembly of the parts (modules) in relation to the entire building. In his Basilica of Fano, he uses proportions of small integers, especially triangular numbers (1, 3, 6, 10, ...) to provide the structure with the so-called Vitruvian modulation.[16]​ Therefore, the ratio between the width and length of the Basilica is 1:2; the corridor around it is as tall as it is wide, 1:1; the columns are five feet thick and fifty feet high, 1:10.[9]​.

Vitruvius mentioned three fundamental qualities of architecture in his work De architectura (c. 15 BC): firmness, usefulness and beauty, a sentence that would become a famous phrase. These three principles can be used as categories to classify the ways in which mathematics is used in architecture. Firmness encompasses the use of mathematics to ensure that a building remains standing, hence the mathematical tools used in design and to foundation the construction, for example, to ensure the stability and behavior of the structure. Utility is derived in part from the effective application of mathematics, reasoning and analyzing spatial and other relationships in a design. Beauty is an attribute of the resulting building, a product of the incorporation of mathematical relationships in the proportions of the building; includes aesthetic, sensitive and intellectual qualities.[17]​.

The Pantheon of Rome has survived intact, illustrating classical Roman structure, proportion and decoration. The main structure is a dome, the crown of which was left open like a circular oculus to let in light. It has a short colonnade, topped with a triangular pediment. The height of the oculus and the diameter of the inner circle are the same, 43.3 meters (142.1 ft), so the entire interior would fit exactly inside a cube, and the interior could hold a sphere of the same diameter.[18] These dimensions make more sense when expressed in ancient Roman measurements: the dome reaches 150 Roman feet in diameter;[19] the oculus measures 30 Roman feet in diameter; and the entrance is 40 Roman feet high.[20]​ The Pantheon remains the largest mass concrete dome in the world.[21]​.

Renaissance

The first Renaissance treatise on architecture, "De re aedificatoria" (On the Art of Building) by Leon Battista Alberti, written in 1450, became the first printed book on architecture in 1485. It was based in part on Vitruvius's "De architectura" and, through Nicomanus, on Pythagorean arithmetic. Alberti begins with a cube, from which he deduces various proportions. Thus, the diagonal of a face gives the ratio 1:, while the diameter of the sphere circumscribing the cube gives 1:.[22]​[23]​ It also documented Filippo Brunelleschi's discovery of perspective, developed to allow the design of buildings that would look beautifully proportioned when viewed from a suitable distance.[12]​.

The next important text was "Regole generali d'architettura" (General Rules of Architecture) by Sebastiano Serlio; 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 incorrect, but this did not prevent his work from being widely used.[25]​.

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, with the help of some followers such as the English diplomat Henry Wotton, with his 1624 text entitled *"The Elements of Architecture".[26] The proportions of each room within the villa were calculated in simple mathematical proportions such as 3:4 and 4:5, and the different rooms within the house were interrelated by these proportions. Previous architects had used these formulas to balance the main façade; However, Palladio's designs related to the entire villa, generally square.[27] Palladio allowed a range of proportions in the Quattro libri, indicating that:[28]​[29]​.

In 1615, Vincenzo Scamozzi published the late Renaissance treatise titled *"L'idea dell'architettura universale" (The Idea of ​​a Universal Architecture).[31] Scamozzi 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.[32]

19th century

Vladimir Shukhov began using hyperboloids at the end of the century, designing masts, lighthouses and cooling towers with this shape. Its striking configuration is both aesthetically interesting and strong, using structural materials economically. The Shukhov Tower at Polybino, his first hyperbolic structure, was exhibited in Nizhny Novgorod in 1896.[33]​[34]​[35]​.

20th century

The early-century movement called modern architecture, presaged[37]​ by Russian constructivism "Constructivism (art)"),[36]​ used Euclidean (also called Cartesian) rectilinear geometry. In the movement called De Stijl, the horizontal and the vertical were considered universal elements. The architectural form consists of uniting these two directional tendencies, using roof planes, wall planes and balconies, which slide or intersect, as in Gerrit Rietveld's Schröder House (1924).[38]​.

Modernist architects were free to use curves and planes. The Arnos Grove station of the London Underground, designed by Charles Holden in 1933, has a circular brick access atrium, with a flat concrete roof. and the spiral), as the supposed basic building blocks of architecture inspired by nature.[40]​[41]​.

Le Corbusier proposed anthropometry as a scale relationship "Scale (relationship)") to establish proportions in architecture. He devised the system called Modulor, based on the estimated height of a man.[42]​ Le Corbusier's 1955 Notre Dame du Haut Chapel uses free-form curves, which cannot be described with mathematical formulas.[45]​ The curves are said to evoke natural shapes such as the bow of a ship or hands in prayer.[46]​ The configuration only applies to the largest scale: there is no hierarchy of details at smaller scales. and, therefore, a fractal dimension is not incorporated; The same applies to other famous buildings of the century, such as the Sydney Opera House, Denver International Airport or the Guggenheim Museum Bilbao.[43]​.

Contemporary architecture, in the opinion of the 90 leading architects who responded to the World Architecture Survey organized by the magazine Vanity Fair "Vanity Fair (magazine)") in 2010, is extremely diverse, with Frank Gehry's Guggenheim Museum in Bilbao being the highest-rated building.[47]​.

The Denver International Airport terminal building, completed in 1995, has a suspended sheet roof configured as a minimum surface (i.e., its mean curvature is zero) using steel cables. It evokes the snow-capped mountains of Colorado and the tents (teepees) of the Native Americans.[48]​[49]​.

Architect Richard Buckminster Fuller is famous for designing rigid shell structures, known as geodesic domes. The Biosphère de Montreal dome has a height of 61 meters (200.1 feet) and its diameter is 76 meters (249.3 feet).[50]​.

The Sydney Opera House has a spectacular roof, consisting of high white vaults, reminiscent of ship sails. In order to be built with standardized components, they were composed of triangular sections of spherical shells with the same radius, designed with uniform curvature in all directions.[51]​.

The deconstructivist movement at the end of the century created deliberate disorder through what Nikos Salingaros calls random forms in his book ,[52]​ employing highly complex forms,[53]​ non-parallel walls, overlapping grids, and complex 2-D surfaces, as in Frank Gehry's Walt Disney Concert Hall and the Guggenheim Museum in Bilbao.[54]​[55]​ Until the end of the century, students of architecture were required to have a considerable mathematical base. Salingaros maintains that, first the "excessively simplistic and politically driven" modernism "Modernism (philosophical and cultural movement)") and then the "anti-scientific" deconstructivism, have in practice meant the separation of architecture and mathematics. He believes that this "renunciation of mathematical values" is harmful, since the "generalized aesthetic" of non-mathematical architecture accustoms people "to rejecting mathematical structuring in the built environment"; and argues that this situation implies negative effects on society.[43]​.

Ancient Egypt

The pyramids "Pyramid (architecture)") of ancient Egypt are funerary monuments built with mathematical proportions, but what they were and whether the Pythagorean theorem was used is debated. The relationship between the inclined height and half the length of the base of the Great Pyramid of Giza differs by approximately 1% from the golden ratio.[56]​ If this were the design method, it would involve the use of Kepler's triangle (face angle 51° 49'),[56][57]​ but according to many historians of science, the golden ratio was not known until the time of the Pythagoreans.[58]​ The Great Pyramid may also have been based on a triangle with a base to hypotenuse ratio of 1:4/π (face angle 51° 50').[59]​.

The proportions of some pyramids may also have been based on the 3:4:5 triangle (face angle 53° 8'), known from the Ahmes papyrus (c. 1650-1550 BC). This possibility was first conjectured by historian Moritz Cantor in 1882.[60] Right angles are known to have been precisely drawn in ancient Egypt using knotted ropes for measuring,[60] which Plutarch recorded in his work Isis and Osiris "Moralia (Works of Morals and Customs)") (c. 100 AD), indicating that the Egyptians admired the triangle 3:4:5.[60]​ Additionally, a scroll from before 1700 BC. C. includes basic formulas for sums of squares "Square (algebra)." C. actually mentions using the theorem to find the length of the sides of a triangle, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain; He assumes that the ancient Egyptians probably knew the Pythagorean theorem, but "there is no evidence that they used it to construct right angles."[60]​.

ancient india

The vastu shastra, the ancient Indian canons of architecture and urban planning, use symmetrical drawings called mandalas. To arrive at the dimensions of a building and its components, complex calculations are used, and the designs are intended to integrate architecture with nature, so that the relative functions of various parts of the structure are related to traditional beliefs involving the use of certain geometric patterns (yantra), conditions of symmetry and directional orientation.[63]​[64]​ However, early builders may have found mathematical proportions by accident. Mathematician Georges Ifrah notes that simple "tricks" with ropes and stakes can be used to design geometric shapes, such as ellipses and right angles.[12]​[65]​.

The mathematics of fractals has been used to demonstrate that the reason existing buildings have universal appeal and are visually satisfying is because they give the viewer a sense of scale at different viewing distances. For example, in the tall towers called gopurams that give entrance to Hindu temples, such as the Virupaksha Temple "Virupaksha (Hampi) Temple") in Hampi built in the 19th century, and others such as the Kandariya Mahadev Temple in Khajuraho, the parts and the whole have the same character, with fractal dimension in the range of 1.7 to 1.8. The group of smaller towers ("shikhara", literally "mountain") around the tallest central tower representing the holy Mount Kailash, abode of Lord Shiva, represents the endless repetition of universes in Hindu cosmology.[2]​[66]​ Religious studies scholar William J. Jackson commented on the pattern of towers grouped between smaller towers, grouped between even smaller towers:

The Meenakshi Amman Temple is a large complex with multiple shrines, with the streets of Madurai arranged concentrically around it according to the shastras. The four entrance gates are tall towers (gopurams) with a repetitive fractal-like structure, as in Hampi. The enclosures around each sanctuary are rectangular and surrounded by high stone walls.[68]​.

Ancient Greece

Pythagoras (c. 569-475 BC) and his followers, the Pythagoreans, held that "all things are numbers." They observed the harmonies produced by musical notes with specific frequency ratios related to small integers, and argued that buildings should also be designed with such ratios. The Greek word symmetry originally denoted the harmony of architectural forms in precise proportions, ranging from the smallest details of a building to its entire design.[12]​.

The Parthenon measures 69.5 meters (228 ft) long, 30.9 meters (101.4 ft) wide, and 13.7 meters (44.9 ft) high to the cornice. This gives a width to length ratio of 4:9, and the same for height to width. Putting these three relationships in a row, we have that height:width:length are in the proportions 16:36:81, or to the delight[69]​ of the Pythagoreans 4:6:9, which implies a module of 0.858 m. A 4:9 rectangle can be constructed as three adjacent rectangles with sides in a 3:4 ratio. Each half-rectangle is then a right triangle with sides in the proportion 3:4:5, which allows the angles and sides to be checked with a properly knotted rope. The interior area (naos) also has proportions of 4:9 (21.44 meters (70.3 ft) wide by 48.3 m long); The ratio between the diameter of the outer columns, 1.905 meters (6.3 feet), and the spacing of their centers, 4.293 meters (14.1 feet), is also 4:9.[12]​.

The Parthenon is considered by authors such as John Julius Norwich "the most perfect Doric temple ever built."[70]​ Its elaborate architectural refinements include "a subtle correspondence between the curvature of the stylobate, the tapering of the naos walls, and the entasis of the columns."[70]​ The term entasis refers to the subtle decrease in the diameter of the columns as they rise. The stylobate is the platform on which the columns rest. As in other classical Greek temples,[71] the platform has a slight upward parabolic curvature to evacuate rainwater and reinforce the building against earthquakes. The columns might therefore be supposed to lean outward, but in reality they lean slightly inward, so that if they were extended they would be about a mile above the center of the building; Since all the columns are of the same height, the curvature of the edge of the outer stylobate is transmitted to the architrave and the upper roof: "they all follow the rule of being built with delicate curves."[72]​.

The golden ratio was known in 300 BC. C., when Euclid described the method of its geometric construction.[73] It has been argued that the golden ratio was used in the design of the Parthenon and other ancient Greek buildings, as well as in sculptures, paintings and vases.[74] More recent authors such as Nikos Salingaros, however, doubt all of these claims.[75] Experiments by computer scientist George Markowsky failed to find any preference for the rectangle. golden.[76]​.

Islamic architecture

The Islamic art historian Antonio Fernández-Puertas suggests that the Alhambra, like the Mosque-Cathedral of Córdoba,[77]​ was designed using the Andalusian foot or cubit of approximately 0.62 meters (2 feet). In the Court of the Lions of the palace, the proportions follow a series of surd (square roots). A rectangle with sides 1 and has (by the Pythagorean theorem) a diagonal of , which describes the right triangle formed by the sides of the patio; this series continues with (giving a ratio of 1:2), and so on. The decorative patterns have similar proportions, generates squares within circles and eight-pointed stars, generates six-pointed stars. There is no evidence to support previous claims that the golden ratio was used in the Alhambra.[10][78]​ The Courtyard of the Lions is bounded by the Hall of the Two Sisters and the Hall of the Abencerrajes. A regular hexagon can be drawn from the centers of these two halls and the four interior corners of the Court of the Lions.[79]​.

The Selim Mosque in Edirne, Türkiye, was built by Sinan to provide a space where the mihrab could be seen from anywhere within the building. The large central space is accordingly arranged as an octagon, formed by eight enormous pillars and surmounted by a circular dome 31.25 meters (102.5 feet) in diameter and 43 meters (141.1 feet) high. The octagon is formed into a square with four semi-domes, and externally by four exceptionally tall minarets, 83 meters (272.3 ft) high. The floor plan of the building is, therefore, a circle, inside an octagon, inside a square.[80]​.

Mughal architecture

Mughal architecture, as seen in the abandoned imperial city of Fatehpur Sikri and the Taj Mahal complex, has a distinctive mathematical order and an aesthetic strongly based on symmetry and harmony.[11][81]​.

The Taj Mahal exemplifies Mughal architecture, both representing paradise[82] and displaying the power of the Mughal emperor Shah Jahan through its scale, symmetry and sumptuous decoration. The white marble of the mausoleum, decorated with semi-precious stones, the great gate (Darwaza-i rauza), other buildings, the gardens and the paths, together form a unified hierarchical design. The buildings include a mosque in red sandstone in the west, and an almost identical building, the Jawab or 'answer' in the east to maintain the bilateral symmetry of the complex. The chahar bagh ('quadruple garden') is divided into four parts, symbolizing the four rivers of paradise and offering views and reflections of the mausoleum, and is in turn divided into 16 flower beds.[83]​.

The Taj Mahal complex was laid out on a grid, subdivided into smaller grids. Architectural historians Koch and Barraud agree with traditional accounts that give the width of the complex as 374 Mughal yards or gaz "Gaz (measure)"),[84]​ the main area being composed of three squares of 374 gaz on each side. Each was subdivided into areas such as the bazaar and the caravanserai in modules of 17 gaz; The garden and terraces are in 23 gaz modules and are 368 gaz wide (16x23). The mausoleum, mosque and guest house are laid out on a 7gaz grid. 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,[85]​ which may help explain the choice of proportions in the complex.[86]​.

Christian architecture

The Hagia Sophia in Byzantium (now Istanbul) was built in 537 as the seat of the Patriarch of Constantinople. Rebuilt twice, it held the title of being the largest Christian temple ever built for more than a thousand years.[87] It inspired many later buildings, including the Blue Mosque and other mosques in the Bosphorus city. The layout of its original Byzantine architecture includes a nave surmounted by a circular dome and two semidomes, all of the same diameter (31 meters (101.7 ft)), with another five smaller semidomes forming an apse and four rounded corners delimiting a vast rectangular interior.[88] This configuration was interpreted by medieval architects as a representation of the underworld at the bottom (the square base) and the divine heavens situated above (the elevated spherical dome).[89]​ Emperor Justinian had two geometers as architects, Isidore of Miletus and Anthemius of Trales. Isidore had compiled the works of Archimedes on the geometry of space, receiving the influence of the Greek mathematician.[12]​[90]​.

The importance in Christianity of water in the baptismal rite was reflected in the scale of the architecture of the baptisteries. The oldest, the Lateran Baptistery in Rome, built in 440,[91] marked the trend of designing octagonal buildings. In fact, the baptismal font located within these buildings was often octagonal, although in the Baptistery of Pisa, the largest in Italy (built between 1152 and 1363), it is circular, although it has an octagonal font. It is 54.86 meters (180 feet) high, with a diameter of 34.13 meters (112 feet) (a ratio of 8:5).[92] Ambrose of Milan wrote that the fonts and baptisteries were octagonal "because on the eighth day,[94]​ by rising from the dead, Christ ends the bondage of death and receives the dead from their tombs."[93]​[95]​.

St. Augustine similarly described the eighth day as "eternal...sanctified by the resurrection of Christ."[95]​[96]​ The Baptistery of St. John (Florence) (built between 1059 and 1128) is also octagonal. It is one of the oldest buildings in the city, and one of the last in the direct tradition of classical antiquity. It became an extremely influential example in the later Florentine Renaissance, as leading architects, including Francesco Talenti, Alberti and Brunelleschi, used it as a model for classical architecture.[97]​.

The number five is used "exuberantly"[98]​ in the Pilgrimage Church of St. John of Nepomuk in Zelená (built 1721), 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 that alternate with pointed apses. Furthermore, the church has five doors, five chapels, five altars and five stars; One legend states that when John Nepomucenus was martyred, five stars appeared above his head.[98][99] The repetition of groups of five elements may also symbolize the five wounds of Jesus Christ and the five letters of "Tacui" (Latin: "I kept silent" about the secrets of the [confessional]).[100]​.

Islamic architectural decoration

Islamic buildings are often decorated with geometric patterns, which generally use various mathematical tessellations, formed by ceramic tiles (called zellige, and whose style is known as girih) that can be plain or decorated with stripes.[12] Symmetries, such as stars with six, eight, or multiples of eight points, are used in Islamic decoration patterns. Some of these are based on the motif of the seal of 'Khatem Sulemani' or Solomon, which is an eight-pointed star formed by two squares, one rotated at 45 degrees to the other and with the same center.[112] Islamic patterns exploit many of the 17 possible wallpaper groups. Already in 1944, Edith Müller demonstrated that the Alhambra made use of 11 of these groups in its decorations, while in 1986 Branko Grünbaum claimed to have found up to 13 groups in the Alhambra, controversially stating that the remaining four groups (up to a total of 17) are not found anywhere in Islamic ornamentation.[112]

Modern architectural decoration

Towards the end of the century, architects turned to new mathematical constructions such as fractal geometry and aperiodic mosaic in order to obtain novel and attractive coverings for their buildings.[4]​ In 1913, the modernist architect Adolf Loos had declared that "Ornament is a crime", influencing[113]​ architectural thinking for the rest of the century. In the 20th century, architects are again beginning to explore the use of ornamentation, which has become an extremely diverse element. The Harpa building "Harpa (Reykjavík)"), built in Reykjavík in 2011 by Henning Larsen, includes what appears to be a rock crystal wall composed of large blocks of glass.[113] A London building, Ravensbourne College by the firm Foreign Office Architects (2010), is decoratively tessellated with 28,000 anodized aluminum tiles in red, white and brown, giving way to interlocking circular windows of different sizes. The tessellation uses three types of mosaics, an equilateral triangle and two irregular pentagons.[114]​[115]​[116]​ The Umimirai Library in Kanazawa, designed by Kazumi Kudo, creates a decorative grid made of small circular glass blocks placed on smooth concrete walls.[113]​.

Europe

The architecture of the fortifications evolved from the medieval castle (defended by high masonry walls), to the concept of the so-called Italian trace (conceived based on moats and low, symmetrical walls, capable of resisting artillery bombardment), an idea developed between the middle of the century and the 19th century. The geometry of the star shapes was dictated by the need to avoid dead zones where attacking infantry could protect themselves from defensive fire; the sides of the salient points were angled to allow such fire to sweep the ground and provide crossfire (from both sides) beyond each salient point. Among the well-known architects who at some point designed such defenses are Michelangelo, Baldassarre Peruzzi, Vincenzo Scamozzi and Vauban.[117]​[118]​.

The architectural historian Sigfried Giedion argued that the star-shaped fortification had a formative influence on the Renaissance pattern of the ideal city: "The Renaissance was hypnotized by a type of city that for a century and a half, from Filarete to Scamozzi, was etched in all utopian schemes: this scheme is the star-shaped city."[119]​.

China

In Chinese architecture, the tulou of Fujian, with some examples dating back to the 17th century, are circular communal defensive structures with smooth walls and a single iron-plated wooden door. The walls are topped with roofs that slope gently both outwards and inwards, forming a ring. The center of the circle is an open cobblestone courtyard, often with a well, surrounded by wooden galleries up to five stories high.[120]

Architects may also select the shape of a building to meet some environmental objectives.[98] For example, Foster and Partners designed the skyscraper at 30 St Mary Ax in London, known as "The Gherkin" for its cucumber-like shape. The building is a solid of revolution conceived through parametric design. Its geometry was not only chosen for purely aesthetic reasons, but also to minimize eddy air currents that could form at its base. Despite the building's apparently curved surface, all the glass panels that make up its covering are flat, except for the roof at the top. Most panels are quadrilateral, since they can be obtained from rectangular glass with less waste than triangular panels.[1]​.

The traditional yakhchal (ice well) of Persia functions as an evaporative cooling system. Above ground, the structure is dome-shaped, but it has underground storage space for ice and sometimes food as well. Underground space and thick heat-resistant construction insulates storage space all year round. The internal space is often cooled further by capturing wind. Ice was available in summer to make the frozen dessert known as faloodeh.[121]​.

• - Burning Man.

• - Mathematics and art.

• - Patterns in nature.

• - What is the concept of architecture?

• - Nexus Network Journal: Architecture and mathematics online.

• - The International Society of Arts, Mathematics and Architecture Archived March 4, 2016 at the Wayback Machine.

• - University of St Andrews: Mathematics and Architecture.

• - overlapping grids.

• - National University of Singapore: Mathematics in art and Architecture.

• - the tapering of the walls of the.

• - Dartmouth College: Geometry in Art and Architecture Archived February 24, 2020 at the Wayback Machine.

Architects Michael Ostwald and Kim Williams ("Kim Williams (architect)"), considering the relationships between architecture and mathematics, point out that both fields, as commonly understood, may appear to be loosely connected, since architecture is a profession that deals with the practical question of designing and constructing buildings, while mathematics is the pure study of numbers and other abstract objects. However, they argue, both disciplines are strongly connected, and have been since classical antiquity.

In ancient Rome, Vitruvius described an architect as a man who knew enough of a variety of other disciplines, primarily geometry, to enable him to supervise craftsmen skilled in all other necessary areas, such as bricklayers and carpenters. The same applied in the Middle Ages, when university-trained architects learned arithmetic, geometry, and aesthetics along with the basic curriculum of grammar, logic, and rhetoric (the liberal arts) in elegant classrooms built by master builders who had mentored many craftsmen. A master builder at the top of his profession received the title of architect or engineer. In the Renaissance, the liberal arts (arithmetic, geometry, music and astronomy) became an extra program of polymaths like Leon Battista Alberti. Similarly, in England, Sir Christopher Wren, known today as an architect, was first a noted astronomer.[3]​.

Williams and Ostwald, reviewing the interaction between 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 completely new ideas; be "reactionary", not introducing changes; or be revivalist "Revivalism (architecture)"), drawing on styles from the past. They argue that architects have avoided seeking inspiration from mathematics in retrospect, which would explain why in revivalist periods, such as neo-Gothic architecture in 19th-century England, architecture had little connection with mathematics. Likewise, they point out that in reactionary periods such as Italian Mannerism from around 1520 to 1580, or the Baroque and Palladian movements of the 19th century, mathematics was hardly taken into consideration. In contrast, revolutionary movements at the turn of the century such as futurism and constructivism actively rejected old ideas, adopting mathematics as a conceptual basis and leading to modern architecture "Modernism (philosophical and cultural movement)"). Also towards the end of the century, architects quickly took advantage of fractal geometry, as well as aperiodic tessellations, to provide buildings with attractive coatings.[4]​.

Architects use mathematics for several reasons, leaving aside the necessary use of mathematics in building engineering.[5]​ Firstly, they use geometry because it defines the spatial form of a building.[6]​ Secondly, they use mathematics to design shapes that can be considered beautiful or harmonious.[7]​ Since the time of the Pythagoreans with their religious philosophy of number, architects[8]​ of classical Greece, ancient Rome, the Islamic world and the Italian Renaissance They were concerned with providing their creations with a sense of proportion "Proportion (architecture)") in accordance with mathematical, aesthetic and, sometimes, religious principles. the structure of tall buildings.[1]​.

Ancient Rome

Vitruvius, the influential architect of ancient Rome, argued that the design of a building such as a temple depends on two qualities, proportion and symmetry. Proportion ensures that each part of a building relates harmoniously to all the others. symmetry, according to Vitruvius, means something closer to modularity than mirror symmetry, since the concept relates to the assembly of the parts (modules) in relation to the entire building. In his Basilica of Fano, he uses proportions of small integers, especially triangular numbers (1, 3, 6, 10, ...) to provide the structure with the so-called Vitruvian modulation.[16]​ Therefore, the ratio between the width and length of the Basilica is 1:2; the corridor around it is as tall as it is wide, 1:1; the columns are five feet thick and fifty feet high, 1:10.[9]​.

Vitruvius mentioned three fundamental qualities of architecture in his work De architectura (c. 15 BC): firmness, usefulness and beauty, a sentence that would become a famous phrase. These three principles can be used as categories to classify the ways in which mathematics is used in architecture. Firmness encompasses the use of mathematics to ensure that a building remains standing, hence the mathematical tools used in design and to foundation the construction, for example, to ensure the stability and behavior of the structure. Utility is derived in part from the effective application of mathematics, reasoning and analyzing spatial and other relationships in a design. Beauty is an attribute of the resulting building, a product of the incorporation of mathematical relationships in the proportions of the building; includes aesthetic, sensitive and intellectual qualities.[17]​.

The Pantheon of Rome has survived intact, illustrating classical Roman structure, proportion and decoration. The main structure is a dome, the crown of which was left open like a circular oculus to let in light. It has a short colonnade, topped with a triangular pediment. The height of the oculus and the diameter of the inner circle are the same, 43.3 meters (142.1 ft), so the entire interior would fit exactly inside a cube, and the interior could hold a sphere of the same diameter.[18] These dimensions make more sense when expressed in ancient Roman measurements: the dome reaches 150 Roman feet in diameter;[19] the oculus measures 30 Roman feet in diameter; and the entrance is 40 Roman feet high.[20]​ The Pantheon remains the largest mass concrete dome in the world.[21]​.

Renaissance

The first Renaissance treatise on architecture, "De re aedificatoria" (On the Art of Building) by Leon Battista Alberti, written in 1450, became the first printed book on architecture in 1485. It was based in part on Vitruvius's "De architectura" and, through Nicomanus, on Pythagorean arithmetic. Alberti begins with a cube, from which he deduces various proportions. Thus, the diagonal of a face gives the ratio 1:, while the diameter of the sphere circumscribing the cube gives 1:.[22]​[23]​ It also documented Filippo Brunelleschi's discovery of perspective, developed to allow the design of buildings that would look beautifully proportioned when viewed from a suitable distance.[12]​.

The next important text was "Regole generali d'architettura" (General Rules of Architecture) by Sebastiano Serlio; 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 incorrect, but this did not prevent his work from being widely used.[25]​.

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, with the help of some followers such as the English diplomat Henry Wotton, with his 1624 text entitled *"The Elements of Architecture".[26] The proportions of each room within the villa were calculated in simple mathematical proportions such as 3:4 and 4:5, and the different rooms within the house were interrelated by these proportions. Previous architects had used these formulas to balance the main façade; However, Palladio's designs related to the entire villa, generally square.[27] Palladio allowed a range of proportions in the Quattro libri, indicating that:[28]​[29]​.

In 1615, Vincenzo Scamozzi published the late Renaissance treatise titled *"L'idea dell'architettura universale" (The Idea of ​​a Universal Architecture).[31] Scamozzi 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.[32]

19th century

Vladimir Shukhov began using hyperboloids at the end of the century, designing masts, lighthouses and cooling towers with this shape. Its striking configuration is both aesthetically interesting and strong, using structural materials economically. The Shukhov Tower at Polybino, his first hyperbolic structure, was exhibited in Nizhny Novgorod in 1896.[33]​[34]​[35]​.

20th century

The early-century movement called modern architecture, presaged[37]​ by Russian constructivism "Constructivism (art)"),[36]​ used Euclidean (also called Cartesian) rectilinear geometry. In the movement called De Stijl, the horizontal and the vertical were considered universal elements. The architectural form consists of uniting these two directional tendencies, using roof planes, wall planes and balconies, which slide or intersect, as in Gerrit Rietveld's Schröder House (1924).[38]​.

Modernist architects were free to use curves and planes. The Arnos Grove station of the London Underground, designed by Charles Holden in 1933, has a circular brick access atrium, with a flat concrete roof. and the spiral), as the supposed basic building blocks of architecture inspired by nature.[40]​[41]​.

Le Corbusier proposed anthropometry as a scale relationship "Scale (relationship)") to establish proportions in architecture. He devised the system called Modulor, based on the estimated height of a man.[42]​ Le Corbusier's 1955 Notre Dame du Haut Chapel uses free-form curves, which cannot be described with mathematical formulas.[45]​ The curves are said to evoke natural shapes such as the bow of a ship or hands in prayer.[46]​ The configuration only applies to the largest scale: there is no hierarchy of details at smaller scales. and, therefore, a fractal dimension is not incorporated; The same applies to other famous buildings of the century, such as the Sydney Opera House, Denver International Airport or the Guggenheim Museum Bilbao.[43]​.

Contemporary architecture, in the opinion of the 90 leading architects who responded to the World Architecture Survey organized by the magazine Vanity Fair "Vanity Fair (magazine)") in 2010, is extremely diverse, with Frank Gehry's Guggenheim Museum in Bilbao being the highest-rated building.[47]​.

The Denver International Airport terminal building, completed in 1995, has a suspended sheet roof configured as a minimum surface (i.e., its mean curvature is zero) using steel cables. It evokes the snow-capped mountains of Colorado and the tents (teepees) of the Native Americans.[48]​[49]​.

Architect Richard Buckminster Fuller is famous for designing rigid shell structures, known as geodesic domes. The Biosphère de Montreal dome has a height of 61 meters (200.1 feet) and its diameter is 76 meters (249.3 feet).[50]​.

The Sydney Opera House has a spectacular roof, consisting of high white vaults, reminiscent of ship sails. In order to be built with standardized components, they were composed of triangular sections of spherical shells with the same radius, designed with uniform curvature in all directions.[51]​.

The deconstructivist movement at the end of the century created deliberate disorder through what Nikos Salingaros calls random forms in his book ,[52]​ employing highly complex forms,[53]​ non-parallel walls, overlapping grids, and complex 2-D surfaces, as in Frank Gehry's Walt Disney Concert Hall and the Guggenheim Museum in Bilbao.[54]​[55]​ Until the end of the century, students of architecture were required to have a considerable mathematical base. Salingaros maintains that, first the "excessively simplistic and politically driven" modernism "Modernism (philosophical and cultural movement)") and then the "anti-scientific" deconstructivism, have in practice meant the separation of architecture and mathematics. He believes that this "renunciation of mathematical values" is harmful, since the "generalized aesthetic" of non-mathematical architecture accustoms people "to rejecting mathematical structuring in the built environment"; and argues that this situation implies negative effects on society.[43]​.

Ancient Egypt

The pyramids "Pyramid (architecture)") of ancient Egypt are funerary monuments built with mathematical proportions, but what they were and whether the Pythagorean theorem was used is debated. The relationship between the inclined height and half the length of the base of the Great Pyramid of Giza differs by approximately 1% from the golden ratio.[56]​ If this were the design method, it would involve the use of Kepler's triangle (face angle 51° 49'),[56][57]​ but according to many historians of science, the golden ratio was not known until the time of the Pythagoreans.[58]​ The Great Pyramid may also have been based on a triangle with a base to hypotenuse ratio of 1:4/π (face angle 51° 50').[59]​.

The proportions of some pyramids may also have been based on the 3:4:5 triangle (face angle 53° 8'), known from the Ahmes papyrus (c. 1650-1550 BC). This possibility was first conjectured by historian Moritz Cantor in 1882.[60] Right angles are known to have been precisely drawn in ancient Egypt using knotted ropes for measuring,[60] which Plutarch recorded in his work Isis and Osiris "Moralia (Works of Morals and Customs)") (c. 100 AD), indicating that the Egyptians admired the triangle 3:4:5.[60]​ Additionally, a scroll from before 1700 BC. C. includes basic formulas for sums of squares "Square (algebra)." C. actually mentions using the theorem to find the length of the sides of a triangle, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain; He assumes that the ancient Egyptians probably knew the Pythagorean theorem, but "there is no evidence that they used it to construct right angles."[60]​.

ancient india

The vastu shastra, the ancient Indian canons of architecture and urban planning, use symmetrical drawings called mandalas. To arrive at the dimensions of a building and its components, complex calculations are used, and the designs are intended to integrate architecture with nature, so that the relative functions of various parts of the structure are related to traditional beliefs involving the use of certain geometric patterns (yantra), conditions of symmetry and directional orientation.[63]​[64]​ However, early builders may have found mathematical proportions by accident. Mathematician Georges Ifrah notes that simple "tricks" with ropes and stakes can be used to design geometric shapes, such as ellipses and right angles.[12]​[65]​.

The mathematics of fractals has been used to demonstrate that the reason existing buildings have universal appeal and are visually satisfying is because they give the viewer a sense of scale at different viewing distances. For example, in the tall towers called gopurams that give entrance to Hindu temples, such as the Virupaksha Temple "Virupaksha (Hampi) Temple") in Hampi built in the 19th century, and others such as the Kandariya Mahadev Temple in Khajuraho, the parts and the whole have the same character, with fractal dimension in the range of 1.7 to 1.8. The group of smaller towers ("shikhara", literally "mountain") around the tallest central tower representing the holy Mount Kailash, abode of Lord Shiva, represents the endless repetition of universes in Hindu cosmology.[2]​[66]​ Religious studies scholar William J. Jackson commented on the pattern of towers grouped between smaller towers, grouped between even smaller towers:

The Meenakshi Amman Temple is a large complex with multiple shrines, with the streets of Madurai arranged concentrically around it according to the shastras. The four entrance gates are tall towers (gopurams) with a repetitive fractal-like structure, as in Hampi. The enclosures around each sanctuary are rectangular and surrounded by high stone walls.[68]​.

Ancient Greece

Pythagoras (c. 569-475 BC) and his followers, the Pythagoreans, held that "all things are numbers." They observed the harmonies produced by musical notes with specific frequency ratios related to small integers, and argued that buildings should also be designed with such ratios. The Greek word symmetry originally denoted the harmony of architectural forms in precise proportions, ranging from the smallest details of a building to its entire design.[12]​.

The Parthenon measures 69.5 meters (228 ft) long, 30.9 meters (101.4 ft) wide, and 13.7 meters (44.9 ft) high to the cornice. This gives a width to length ratio of 4:9, and the same for height to width. Putting these three relationships in a row, we have that height:width:length are in the proportions 16:36:81, or to the delight[69]​ of the Pythagoreans 4:6:9, which implies a module of 0.858 m. A 4:9 rectangle can be constructed as three adjacent rectangles with sides in a 3:4 ratio. Each half-rectangle is then a right triangle with sides in the proportion 3:4:5, which allows the angles and sides to be checked with a properly knotted rope. The interior area (naos) also has proportions of 4:9 (21.44 meters (70.3 ft) wide by 48.3 m long); The ratio between the diameter of the outer columns, 1.905 meters (6.3 feet), and the spacing of their centers, 4.293 meters (14.1 feet), is also 4:9.[12]​.

The Parthenon is considered by authors such as John Julius Norwich "the most perfect Doric temple ever built."[70]​ Its elaborate architectural refinements include "a subtle correspondence between the curvature of the stylobate, the tapering of the naos walls, and the entasis of the columns."[70]​ The term entasis refers to the subtle decrease in the diameter of the columns as they rise. The stylobate is the platform on which the columns rest. As in other classical Greek temples,[71] the platform has a slight upward parabolic curvature to evacuate rainwater and reinforce the building against earthquakes. The columns might therefore be supposed to lean outward, but in reality they lean slightly inward, so that if they were extended they would be about a mile above the center of the building; Since all the columns are of the same height, the curvature of the edge of the outer stylobate is transmitted to the architrave and the upper roof: "they all follow the rule of being built with delicate curves."[72]​.

The golden ratio was known in 300 BC. C., when Euclid described the method of its geometric construction.[73] It has been argued that the golden ratio was used in the design of the Parthenon and other ancient Greek buildings, as well as in sculptures, paintings and vases.[74] More recent authors such as Nikos Salingaros, however, doubt all of these claims.[75] Experiments by computer scientist George Markowsky failed to find any preference for the rectangle. golden.[76]​.

Islamic architecture

The Islamic art historian Antonio Fernández-Puertas suggests that the Alhambra, like the Mosque-Cathedral of Córdoba,[77]​ was designed using the Andalusian foot or cubit of approximately 0.62 meters (2 feet). In the Court of the Lions of the palace, the proportions follow a series of surd (square roots). A rectangle with sides 1 and has (by the Pythagorean theorem) a diagonal of , which describes the right triangle formed by the sides of the patio; this series continues with (giving a ratio of 1:2), and so on. The decorative patterns have similar proportions, generates squares within circles and eight-pointed stars, generates six-pointed stars. There is no evidence to support previous claims that the golden ratio was used in the Alhambra.[10][78]​ The Courtyard of the Lions is bounded by the Hall of the Two Sisters and the Hall of the Abencerrajes. A regular hexagon can be drawn from the centers of these two halls and the four interior corners of the Court of the Lions.[79]​.

The Selim Mosque in Edirne, Türkiye, was built by Sinan to provide a space where the mihrab could be seen from anywhere within the building. The large central space is accordingly arranged as an octagon, formed by eight enormous pillars and surmounted by a circular dome 31.25 meters (102.5 feet) in diameter and 43 meters (141.1 feet) high. The octagon is formed into a square with four semi-domes, and externally by four exceptionally tall minarets, 83 meters (272.3 ft) high. The floor plan of the building is, therefore, a circle, inside an octagon, inside a square.[80]​.

Mughal architecture

Mughal architecture, as seen in the abandoned imperial city of Fatehpur Sikri and the Taj Mahal complex, has a distinctive mathematical order and an aesthetic strongly based on symmetry and harmony.[11][81]​.

The Taj Mahal exemplifies Mughal architecture, both representing paradise[82] and displaying the power of the Mughal emperor Shah Jahan through its scale, symmetry and sumptuous decoration. The white marble of the mausoleum, decorated with semi-precious stones, the great gate (Darwaza-i rauza), other buildings, the gardens and the paths, together form a unified hierarchical design. The buildings include a mosque in red sandstone in the west, and an almost identical building, the Jawab or 'answer' in the east to maintain the bilateral symmetry of the complex. The chahar bagh ('quadruple garden') is divided into four parts, symbolizing the four rivers of paradise and offering views and reflections of the mausoleum, and is in turn divided into 16 flower beds.[83]​.

The Taj Mahal complex was laid out on a grid, subdivided into smaller grids. Architectural historians Koch and Barraud agree with traditional accounts that give the width of the complex as 374 Mughal yards or gaz "Gaz (measure)"),[84]​ the main area being composed of three squares of 374 gaz on each side. Each was subdivided into areas such as the bazaar and the caravanserai in modules of 17 gaz; The garden and terraces are in 23 gaz modules and are 368 gaz wide (16x23). The mausoleum, mosque and guest house are laid out on a 7gaz grid. 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,[85]​ which may help explain the choice of proportions in the complex.[86]​.

Christian architecture

The Hagia Sophia in Byzantium (now Istanbul) was built in 537 as the seat of the Patriarch of Constantinople. Rebuilt twice, it held the title of being the largest Christian temple ever built for more than a thousand years.[87] It inspired many later buildings, including the Blue Mosque and other mosques in the Bosphorus city. The layout of its original Byzantine architecture includes a nave surmounted by a circular dome and two semidomes, all of the same diameter (31 meters (101.7 ft)), with another five smaller semidomes forming an apse and four rounded corners delimiting a vast rectangular interior.[88] This configuration was interpreted by medieval architects as a representation of the underworld at the bottom (the square base) and the divine heavens situated above (the elevated spherical dome).[89]​ Emperor Justinian had two geometers as architects, Isidore of Miletus and Anthemius of Trales. Isidore had compiled the works of Archimedes on the geometry of space, receiving the influence of the Greek mathematician.[12]​[90]​.

The importance in Christianity of water in the baptismal rite was reflected in the scale of the architecture of the baptisteries. The oldest, the Lateran Baptistery in Rome, built in 440,[91] marked the trend of designing octagonal buildings. In fact, the baptismal font located within these buildings was often octagonal, although in the Baptistery of Pisa, the largest in Italy (built between 1152 and 1363), it is circular, although it has an octagonal font. It is 54.86 meters (180 feet) high, with a diameter of 34.13 meters (112 feet) (a ratio of 8:5).[92] Ambrose of Milan wrote that the fonts and baptisteries were octagonal "because on the eighth day,[94]​ by rising from the dead, Christ ends the bondage of death and receives the dead from their tombs."[93]​[95]​.

St. Augustine similarly described the eighth day as "eternal...sanctified by the resurrection of Christ."[95]​[96]​ The Baptistery of St. John (Florence) (built between 1059 and 1128) is also octagonal. It is one of the oldest buildings in the city, and one of the last in the direct tradition of classical antiquity. It became an extremely influential example in the later Florentine Renaissance, as leading architects, including Francesco Talenti, Alberti and Brunelleschi, used it as a model for classical architecture.[97]​.

The number five is used "exuberantly"[98]​ in the Pilgrimage Church of St. John of Nepomuk in Zelená (built 1721), 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 that alternate with pointed apses. Furthermore, the church has five doors, five chapels, five altars and five stars; One legend states that when John Nepomucenus was martyred, five stars appeared above his head.[98][99] The repetition of groups of five elements may also symbolize the five wounds of Jesus Christ and the five letters of "Tacui" (Latin: "I kept silent" about the secrets of the [confessional]).[100]​.

Islamic architectural decoration

Islamic buildings are often decorated with geometric patterns, which generally use various mathematical tessellations, formed by ceramic tiles (called zellige, and whose style is known as girih) that can be plain or decorated with stripes.[12] Symmetries, such as stars with six, eight, or multiples of eight points, are used in Islamic decoration patterns. Some of these are based on the motif of the seal of 'Khatem Sulemani' or Solomon, which is an eight-pointed star formed by two squares, one rotated at 45 degrees to the other and with the same center.[112] Islamic patterns exploit many of the 17 possible wallpaper groups. Already in 1944, Edith Müller demonstrated that the Alhambra made use of 11 of these groups in its decorations, while in 1986 Branko Grünbaum claimed to have found up to 13 groups in the Alhambra, controversially stating that the remaining four groups (up to a total of 17) are not found anywhere in Islamic ornamentation.[112]

Modern architectural decoration

Towards the end of the century, architects turned to new mathematical constructions such as fractal geometry and aperiodic mosaic in order to obtain novel and attractive coverings for their buildings.[4]​ In 1913, the modernist architect Adolf Loos had declared that "Ornament is a crime", influencing[113]​ architectural thinking for the rest of the century. In the 20th century, architects are again beginning to explore the use of ornamentation, which has become an extremely diverse element. The Harpa building "Harpa (Reykjavík)"), built in Reykjavík in 2011 by Henning Larsen, includes what appears to be a rock crystal wall composed of large blocks of glass.[113] A London building, Ravensbourne College by the firm Foreign Office Architects (2010), is decoratively tessellated with 28,000 anodized aluminum tiles in red, white and brown, giving way to interlocking circular windows of different sizes. The tessellation uses three types of mosaics, an equilateral triangle and two irregular pentagons.[114]​[115]​[116]​ The Umimirai Library in Kanazawa, designed by Kazumi Kudo, creates a decorative grid made of small circular glass blocks placed on smooth concrete walls.[113]​.

Europe

The architecture of the fortifications evolved from the medieval castle (defended by high masonry walls), to the concept of the so-called Italian trace (conceived based on moats and low, symmetrical walls, capable of resisting artillery bombardment), an idea developed between the middle of the century and the 19th century. The geometry of the star shapes was dictated by the need to avoid dead zones where attacking infantry could protect themselves from defensive fire; the sides of the salient points were angled to allow such fire to sweep the ground and provide crossfire (from both sides) beyond each salient point. Among the well-known architects who at some point designed such defenses are Michelangelo, Baldassarre Peruzzi, Vincenzo Scamozzi and Vauban.[117]​[118]​.

The architectural historian Sigfried Giedion argued that the star-shaped fortification had a formative influence on the Renaissance pattern of the ideal city: "The Renaissance was hypnotized by a type of city that for a century and a half, from Filarete to Scamozzi, was etched in all utopian schemes: this scheme is the star-shaped city."[119]​.

China

In Chinese architecture, the tulou of Fujian, with some examples dating back to the 17th century, are circular communal defensive structures with smooth walls and a single iron-plated wooden door. The walls are topped with roofs that slope gently both outwards and inwards, forming a ring. The center of the circle is an open cobblestone courtyard, often with a well, surrounded by wooden galleries up to five stories high.[120]

Architects may also select the shape of a building to meet some environmental objectives.[98] For example, Foster and Partners designed the skyscraper at 30 St Mary Ax in London, known as "The Gherkin" for its cucumber-like shape. The building is a solid of revolution conceived through parametric design. Its geometry was not only chosen for purely aesthetic reasons, but also to minimize eddy air currents that could form at its base. Despite the building's apparently curved surface, all the glass panels that make up its covering are flat, except for the roof at the top. Most panels are quadrilateral, since they can be obtained from rectangular glass with less waste than triangular panels.[1]​.

The traditional yakhchal (ice well) of Persia functions as an evaporative cooling system. Above ground, the structure is dome-shaped, but it has underground storage space for ice and sometimes food as well. Underground space and thick heat-resistant construction insulates storage space all year round. The internal space is often cooled further by capturing wind. Ice was available in summer to make the frozen dessert known as faloodeh.[121]​.

• - Burning Man.

• - Mathematics and art.

• - Patterns in nature.

• - What is the concept of architecture?

• - Nexus Network Journal: Architecture and mathematics online.

• - The International Society of Arts, Mathematics and Architecture Archived March 4, 2016 at the Wayback Machine.

• - University of St Andrews: Mathematics and Architecture.

• - overlapping grids.

• - National University of Singapore: Mathematics in art and Architecture.

• - the tapering of the walls of the.

• - Dartmouth College: Geometry in Art and Architecture Archived February 24, 2020 at the Wayback Machine.