Mathematics and architecture are related, sincearchitecture,like some other arts, usesmathematics for several reasons. Apart from the mathematics needed when engineeringbuildings, architects usegeometry: to define the spatial form of a building; from thePythagoreans of the sixth century BC onwards, to createarchitectural 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 astessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings.
Inancient Egypt,ancient Greece,India, and theIslamic world, buildings includingpyramids, temples, mosques, palaces andmausoleums were laid out with specific proportions for religious reasons. In Islamic architecture,geometric shapes andgeometric tiling patterns are used to decorate buildings, both inside and outside. Some Hindu temples have afractal-like structure where parts resemble the whole, conveying a message about the infinite inHindu cosmology. InChinese architecture, thetulou ofFujian province are circular, communal defensive structures. In the twenty-first century, mathematical ornamentation is again being used to cover public buildings.
InRenaissance architecture,symmetry and proportion were deliberately emphasized by architects such asLeon Battista Alberti,Sebastiano Serlio andAndrea Palladio, influenced byVitruvius'sDe architectura fromancient Rome and the arithmetic of the Pythagoreans from ancient Greece.At the end of the nineteenth century,Vladimir Shukhov inRussia andAntoni Gaudí inBarcelona pioneered the use ofhyperboloid structures; in theSagrada Família, Gaudí also incorporatedhyperbolicparaboloids, tessellations,catenary arches,catenoids,helicoids, andruled surfaces. In the twentieth century, styles such asmodern architecture andDeconstructivism explored different geometries to achieve desired effects.Minimal surfaces have been exploited in tent-like roof coverings as atDenver International Airport, whileRichard Buckminster Fuller pioneered the use of the strongthin-shell structures known asgeodesic domes.
The architects Michael Ostwald andKim Williams, considering the relationships betweenarchitecture andmathematics, 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 purestudy of number and other abstract objects. But, they argue, the two are strongly connected, and have been sinceantiquity. In ancient Rome,Vitruvius described an architect as a man who knew enough of a range of other disciplines, primarilygeometry, to enable him to oversee skilled artisans in all the other necessary areas, such as masons and carpenters. The same applied in theMiddle Ages, where graduates learntarithmetic, geometry andaesthetics alongside the basic syllabus of grammar, logic, and rhetoric (thetrivium) 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 theRenaissance, thequadrivium of arithmetic, geometry, music and astronomy became an extra syllabus expected of theRenaissance man such asLeon Battista Alberti. Similarly in England, SirChristopher 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 sociologistTheodor Adorno, identify three tendencies among architects, namely: to berevolutionary, introducing wholly new ideas;reactionary, failing to introduce change; orrevivalist, 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 theGothic Revival in 19th century England, architecture had little connection to mathematics. Equally, they note that in reactionary times such as the ItalianMannerism of about 1520 to 1580, or the 17th centuryBaroque andPalladian movements, mathematics was barely consulted. In contrast, the revolutionary early 20th-century movements such asFuturism andConstructivism actively rejected old ideas, embracing mathematics and leading toModernist architecture. Towards the end of the 20th century, too,fractal geometry was quickly seized upon by architects, as wasaperiodic tiling, to provide interesting and attractive coverings for buildings.[4]
Architects use mathematics for several reasons, leaving aside the necessary use of mathematics in theengineering of buildings.[5] Firstly, theyuse geometry because it defines the spatial form of a building.[6] Secondly, they use mathematics to design forms that areconsidered beautiful or harmonious.[7] From the time of thePythagoreans with their religious philosophy of number,[8] architects inancient Greece,ancient Rome, theIslamic world and theItalian Renaissance have chosen theproportions 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 astessellations 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 andsymmetria. 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 termmodularity thanmirror symmetry, as again it relates to the assembling of (modular) parts into the whole building. In his Basilica atFano, he uses ratios of small integers, especially thetriangular 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 hisDe architectura,c. 15 B.C.:firmness, usefulness (or "Commodity" in Henry Wotton's 17th century English), and delight. These can be used as categories for classifying the ways in which mathematics is used in architecture. Firmness encompasses the use of mathematics to ensure a building stands up, hence the mathematical tools used in design and to support construction, for instance to ensure stability and to model performance. Usefulness derives in part from the effective application of mathematics, reasoning about and analysing the spatial and other relationships in a design. Delight is an attribute of the resulting building, resulting from the embodying of mathematical relationships in the building; it includes aesthetic, sensual and intellectual qualities.[16]
ThePantheon in Rome has survived intact, illustrating classical Roman structure, proportion, and decoration. The main structure is a dome, the apex left open as a circularoculus 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 inancient Roman units of measurement: The dome spans 150Roman 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 1450De re aedificatoria (On the Art of Building); it became the first printed book on architecture in 1485. It was partly based on Vitruvius'sDe 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 documentedFilippo Brunelleschi's discovery oflinear perspective, developed to enable the design of buildings which would look beautifully proportioned when viewed from a convenient distance.[12]
The next major text wasSebastiano Serlio'sRegole generali d'architettura (General Rules of Architecture); the first volume appeared in Venice in 1537; the 1545 volume (books 1 and 2) covered geometry andperspective. 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 influentialI quattro libri dell'architettura (The Four Books of Architecture) inVenice. This widely printed book was largely responsible for spreading the ideas of theItalian Renaissance throughout Europe, assisted by proponents like the English diplomat Henry Wotton with his 1624The 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 theQuattro 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 treatiseL'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 byVladimir 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 inNizhny Novgorod in 1896.[30][31][32]
The early twentieth century movementModern architecture, pioneered[d] by RussianConstructivism,[33] used rectilinearEuclidean (also calledCartesian) geometry. In theDe 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 1924Rietveld Schröder House byGerrit Rietveld.[34]
Modernist architects were free to make use of curves as well as planes.Charles Holden's 1933Arnos station has a circular ticket hall in brick with a flat concrete roof.[35] In 1938, theBauhaus painterLászló Moholy-Nagy adoptedRaoul Heinrich Francé's sevenbiotechnical 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 ananthropometricscale of proportions in architecture, theModulor, based on the supposed height of a man.[38] Le Corbusier's 1955Chapelle 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 theprow 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 theSydney Opera House,Denver International Airport, and theGuggenheim Museum, Bilbao.[39]
Contemporary architecture, in the opinion of the 90 leading architects who responded to a 2010World Architecture Survey, is extremely diverse; the best was judged to beFrank Gehry's Guggenheim Museum, Bilbao.[42]
Denver International Airport's terminal building, completed in 1995, has afabric roof supported as aminimal surface (i.e., itsmean curvature is zero) by steel cables. It evokesColorado's snow-capped mountains and theteepee tents ofNative Americans.[43][44]
The architectRichard Buckminster Fuller is famous for designing strongthin-shell structures known asgeodesic domes. TheMontréal 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 uniformcurvature in every direction.[46]
The late twentieth century movementDeconstructivism creates deliberate disorder with whatNikos Salingaros inA 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'sDisney 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]
Thepyramids ofancient Egypt aretombs constructed with mathematical proportions, but which these were, and whether thePythagorean theorem was used, are debated. The ratio of the slant height to half the base length of theGreat Pyramid of Giza is less than 1% from thegolden ratio.[51] If this was the design method, it would imply the use ofKepler's triangle (face angle 51°49'),[51][52] but according to manyhistorians of science, the golden ratio was not known until the time of thePythagoreans.[53]
The proportions of some pyramids may have also been based on the3:4:5 triangle (face angle 53°8'), known from theRhind Mathematical Papyrus (c. 1650–1550 BC); this was first conjectured by historianMoritz Cantor in 1882.[54] It is known that right angles were laid out accurately in ancient Egypt usingknotted cords for measurement,[54] thatPlutarch recorded inIsis and Osiris (c. 100 AD) that the Egyptians admired the 3:4:5 triangle,[54] and that a scroll from before 1700 BC demonstrated basicsquare formulas.[55][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."[54]
Vaastu Shastra, the ancientIndian canons of architecture and town planning, employs symmetrical drawings calledmandalas. 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 anddirectional alignments.[56][57] 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][58]
The mathematics offractals 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 tallgopuram gatehouses ofHindu temples such as theVirupaksha Temple atHampi built in the seventh century, and others such as theKandariya Mahadev Temple atKhajuraho, the parts and the whole have the same character, withfractal 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 holyMount Kailash, abode of LordShiva, depicts the endless repetition of universes inHindu cosmology.[2][59] 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.[59][60]
TheMeenakshi Amman Temple is a large complex with multiple shrines, with the streets ofMadurai 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.[61]
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 wordsymmetria originally denoted the harmony of architectural shapes in precise ratios from a building's smallest details right up to its entire design.[12]
TheParthenon 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[62] of the Pythagoreans 42:62:92. 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 asJohn Julius Norwich "the most perfect Doric temple ever built".[63] Its elaborate architectural refinements include "a subtle correspondence between the curvature of the stylobate, the taper of thenaos walls and theentasis of the columns".[63]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,[64] 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 thearchitrave and roof above: "all follow the rule of being built to delicate curves".[65]
The golden ratio was known in 300 B.C., whenEuclid described the method of geometric construction.[66] 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.[67] More recent authors such asNikos Salingaros, however, doubt all these claims.[68] Experiments by the computer scientist George Markowsky failed to find any preference for thegolden rectangle.[69]
The historian of Islamic art Antonio Fernandez-Puertas suggests that theAlhambra, like theGreat Mosque of Cordoba,[70] was designed using theHispano-Muslim foot orcodo of about 0.62 metres (2.0 ft). In the palace'sCourt of the Lions, the proportions follow a series ofsurds. A rectangle with sides 1 and√2 has (byPythagoras'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][71] TheCourt of the Lions is bracketed by the Hall of Two Sisters and the Hall of the Abencerrajes; a regularhexagon can be drawn from the centres of these two halls and the four inside corners of the Court of the Lions.[72]
TheSelimiye Mosque inEdirne, Turkey, was built byMimar Sinan to provide a space where themihrab could be seen 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.[73]
Mughal architecture, as seen in the abandoned imperial city ofFatehpur Sikri and theTaj Mahal complex, has a distinctive mathematical order and a strong aesthetic based on symmetry and harmony.[11][74]
The Taj Mahal exemplifies Mughal architecture, both representingparadise[75] and displaying theMughal EmperorShah Jahan's power through its scale, symmetry and costly decoration. The white marblemausoleum, decorated withpietra dura, the great gate (Darwaza-i rauza), other buildings, the gardens and paths together form a unified hierarchical design. The buildings include amosque 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 formalcharbagh ('fourfold garden') is in four parts, symbolising the fourrivers of Paradise, and offering views and reflections of the mausoleum. These are divided in turn into 16 parterres.[76]
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 orgaz,[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.[77]
TheChristianpatriarchalbasilica ofHaghia Sophia inByzantium (nowIstanbul), first constructed in 537 (and twice rebuilt), was for a thousand years[i] the largest cathedral ever built. It inspired many later buildings includingSultan Ahmed and other mosques in the city. TheByzantine 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 anapse and four rounded corners of a vast rectangular interior.[78] This was interpreted by mediaeval architects as representing the mundane below (the square base) and the divine heavens above (the soaring spherical dome).[79] The emperorJustinian used two geometers,Isidore of Miletus andAnthemius of Tralles as architects; Isidore compiled the works ofArchimedes onsolid geometry, and was influenced by him.[12][80]
The importance of waterbaptism in Christianity was reflected in the scale ofbaptistry architecture. The oldest, theLateran Baptistry in Rome, built in 440,[81] set a trend for octagonal baptistries; thebaptismal font inside these buildings was often octagonal, though Italy's largestbaptistry, 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).[82]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."[83][84]Saint Augustine similarly described the eighth day as "everlasting ... hallowed by theresurrection of Christ".[84][85] The octagonalBaptistry 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 includingFrancesco Talenti, Alberti and Brunelleschi used it as the model of classical architecture.[86]
The number five is used "exuberantly"[87] in the 1721Pilgrimage Church of St John of Nepomuk at Zelená hora, nearŽďár nad Sázavou in the Czech republic, designed byJan 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 whenSaint John of Nepomuk was martyred, five stars appeared over his head.[87][88] The fivefold architecture may also symbolise thefive wounds of Christ and the five letters of "Tacui" (Latin: "I kept silence" [about secrets of theconfessional]).[89]
Antoni Gaudí used a wide variety of geometric structures, some being minimal surfaces, in theSagrada Família,Barcelona, started in 1882 (and not completed as of 2023). These include hyperbolicparaboloids andhyperboloids of revolution,[90] tessellations,catenary arches,catenoids,helicoids, andruled 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 exploitsnatural patterns, themselves mathematical, withcolumns derived from the shapes oftrees, andlintels made from unmodifiedbasalt naturally cracked (by cooling from molten rock) intohexagonal columns.[91][92][90]
The 1971Cathedral of Saint Mary of the Assumption, San Francisco has asaddle 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 aChristian cross. The building is a square 77.7 metres (255 ft) on a side, and 57.9 metres (190 ft) high.[93] The 1970Cathedral of Brasília byOscar 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.[94][95][96][97]
Several medievalchurches in Scandinavia are circular, including four on the Danish island ofBornholm. One of the oldest of these,Østerlars Church fromc. 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.[98][99]
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Islamic buildings are often decorated withgeometric patterns which typically make use of several mathematicaltessellations, 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.[100] Islamic patterns exploit many of the 17 possiblewallpaper groups; as early as 1944, Edith Müller showed that the Alhambra made use of 11 wallpaper groups in its decorations, while in 1986Branko 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.[100]
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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 architectAdolf Loos had declared that "Ornament is a crime",[101] influencing architectural thinking for the rest of the 20th century. In the 21st century, architects are again starting to explore the use ofornament. 21st century ornamentation is extremely diverse. Henning Larsen's 2011Harpa Concert and Conference Centre, Reykjavik has what looks like a crystal wall of rock made of large blocks of glass.[101] Foreign Office Architects' 2010Ravensbourne 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.[102][103][l] Kazumi Kudo'sKanazawa Umimirai Library creates a decorative grid made of small circular blocks of glass set into plain concrete walls.[101]
The architecture offortifications evolved frommedieval fortresses, which had high masonry walls, to low, symmetricalstar forts able to resistartillery 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 includeMichelangelo,Baldassare Peruzzi,Vincenzo Scamozzi andSébastien Le Prestre de Vauban.[104][105]
The architectural historianSiegfried Giedion argued that the star-shaped fortification had a formative influence on the patterning of the Renaissanceideal 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."[106]
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InChinese architecture, thetulou ofFujian 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.[107]
Architects may also select the form of a building to meet environmental goals.[87] For example,Foster and Partners'30 St Mary Axe, London, known as "The Gherkin" for itscucumber-like shape, is asolid of revolution designed usingparametric 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 arequadrilaterals, as they can be cut from rectangular glass with less wastage than triangular panels.[1]
The traditionalyakhchal (ice pit) ofPersia functioned as anevaporative 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 withwindcatchers.[108]
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