Editing Mathematics and architecture

{{short description|Relationship between mathematics and architecture}}
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[[File:30 St Mary Axe from Leadenhall Street.jpg|thumb|upright=1.3|"The Gherkin",<ref name=Freiberger/> [[30 St Mary Axe]], London, completed 2003, is a [[parametric design|parametrically designed]] [[solid of revolution]].]]

[[File:Kandariya mahadeva temple.jpg|thumb|upright=1.2|[[Kandariya Mahadeva Temple]] ({{circa|1030}}), [[Khajuraho]], India, is an example of religious architecture with a [[fractal]]-like structure which has many parts that resemble the whole.<ref name=Rian>{{cite journal |last1=Rian |first1=Iasef Md |last2=Park|first2=Jin-Ho|last3=Ahn |first3=Hyung Uk |last4=Chang |first4=Dongkuk |title=Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho |journal=Building and Environment |date=2007 |volume=42 |issue=12 |pages=4093–4107 |url=https://www.academia.edu/7482254|doi=10.1016/j.buildenv.2007.01.028}}</ref>]]

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

In <!--[[History of Science and Technology in China|ancient China]], Needs section in article before we put this here-->[[Ancient Egyptian architecture|ancient Egypt]], [[Ancient Greek architecture|ancient Greece]], [[Architecture of India|India]], and the [[Islamic architecture|Islamic world]], buildings including [[Egyptian pyramids|pyramids]], temples, mosques, palaces and [[mausoleum]]s were laid out with specific proportions for religious reasons. In Islamic architecture, [[Geometry|geometric]] shapes and [[Islamic geometric patterns|geometric tiling patterns]] are used to decorate buildings, both inside and outside. Some Hindu temples have a [[fractal]]-like structure where parts resemble the whole, conveying a message about the infinite in [[Hindu cosmology]]. In [[Chinese architecture]], the [[tulou]] of [[Fujian province]] are circular, communal defensive structures. In the twenty-first century, mathematical ornamentation is again being used to cover public buildings.

In [[Renaissance architecture]], [[symmetry]] and proportion were deliberately emphasized by architects such as [[Leon Battista Alberti]], [[Sebastiano Serlio]] and [[Andrea Palladio]], influenced by [[Vitruvius]]'s ''[[De architectura]]'' from [[ancient Rome]] and the arithmetic of the Pythagoreans from ancient Greece.
At the end of the nineteenth century, [[Vladimir Shukhov]] in [[Russia]] and [[Antoni Gaudí]] in [[Barcelona]] pioneered the use of [[hyperboloid structures]]; in the [[Sagrada Família]], Gaudí also incorporated [[hyperbola|hyperbolic]] [[paraboloid]]s, tessellations, [[catenary arch]]es, [[catenoid]]s, [[helicoid]]s, and [[ruled surface]]s. In the twentieth century, styles such as [[modern architecture]] and [[Deconstructivism]] explored different geometries to achieve desired effects. [[Minimal surface]]s have been exploited in tent-like roof coverings as at [[Denver International Airport]], while [[Richard Buckminster Fuller]] pioneered the use of the strong [[thin-shell structure]]s known as [[geodesic dome]]s.

==Connected fields==
[[File:CdM, presunto autoritratto di leon battista alberti, white ground.jpg|thumb|upright|In the [[Renaissance]], an [[architect]] like [[Leon Battista Alberti]] was expected to be knowledgeable in many disciplines, including [[arithmetic]] and [[geometry]].]]

The architects Michael Ostwald and [[Kim Williams (architect)|Kim Williams]], considering the relationships between [[architecture]] and [[mathematics]], note that the fields as commonly understood might seem to be only weakly connected, since architecture is a profession concerned with the practical matter of making buildings, while mathematics is the pure [[Number theory|study of number]] and other abstract objects. But, they argue, the two are strongly connected, and have been since [[Classical antiquity|antiquity]]. In ancient Rome, [[Vitruvius]] described an architect as a man who knew enough of a range of other disciplines, primarily [[geometry]], to enable him to oversee skilled artisans in all the other necessary areas, such as masons and carpenters. The same applied in the [[Middle Ages]], where graduates learnt [[arithmetic]], geometry and [[aesthetics]] alongside the basic syllabus of grammar, logic, and rhetoric (the [[trivium]]) in elegant halls made by master builders who had guided many craftsmen. A master builder at the top of his profession was given the title of architect or engineer. In the [[Renaissance]], the [[quadrivium]] of arithmetic, geometry, music and astronomy became an extra syllabus expected of the [[Renaissance man]] such as [[Leon Battista Alberti]]. Similarly in England, Sir [[Christopher Wren]], known today as an architect, was firstly a noted astronomer.<ref name=Ostwald1>{{cite book |editor1=Williams, Kim |editor2=Ostwald, Michael J. |title=Architecture and Mathematics from Antiquity to the Future: Volume I: from Antiquity to the 1500s |publisher=Birkhäuser |year=2015 |isbn=978-3-319-00136-4 |pages=chapter 1. 1–24}}</ref>

Williams and Ostwald, further overviewing the interaction of mathematics and architecture since 1500 according to the approach of the German sociologist [[Theodor W. Adorno|Theodor Adorno]], identify three tendencies among architects, namely: to be ''revolutionary'', introducing wholly new ideas; ''reactionary'', failing to introduce change; or ''[[Revivalism (architecture)|revivalist]]'', actually going backwards. They argue that architects have avoided looking to mathematics for inspiration in revivalist times. This would explain why in revivalist periods, such as the [[Gothic Revival architecture|Gothic Revival]] in 19th century England, architecture had little connection to mathematics. Equally, they note that in reactionary times such as the Italian [[Mannerism]] of about 1520 to 1580, or the 17th century [[Baroque]] and [[Palladian]] movements, mathematics was barely consulted. In contrast, the revolutionary early 20th-century movements such as [[Futurism]] and [[Constructivist architecture|Constructivism]] actively rejected old ideas, embracing mathematics and leading to [[Modernist]] architecture. Towards the end of the 20th century, too, [[fractal]] geometry was quickly seized upon by architects, as was [[aperiodic tiling]], to provide interesting and attractive coverings for buildings.<ref name=Ostwald48>{{cite book |editor1=Williams, Kim |editor2=Ostwald, Michael J. |title=Architecture and Mathematics from Antiquity to the Future: Volume II: The 1500s to the Future |publisher=Birkhäuser |year=2015 |isbn=978-3-319-00142-5 |pages=chapter 48. 1–24}}</ref>

Architects use mathematics for several reasons, leaving aside the necessary use of mathematics in the [[Architectural engineering|engineering of buildings]].<ref>{{cite web |title=Architectural Engineering Overview |url=http://www.careercornerstone.org/pdf/archeng/archeng.pdf |publisher=Sloan Career Cornerstone Center |access-date=11 October 2015 |archive-url=https://web.archive.org/web/20150714164847/http://www.careercornerstone.org/pdf/archeng/archeng.pdf |archive-date=14 July 2015 |url-status=dead }}</ref> Firstly, they [[architectural geometry|use geometry]] because it defines the spatial form of a building.<ref name=Leyton>{{cite book |last=Leyton |first=Michael |title=A Generative Theory of Shape |url=https://archive.org/details/springer_10.1007-3-540-45488-8 |date=2001 |publisher=Springer |isbn=978-3-540-42717-9}}</ref> 
Secondly, they use mathematics to design forms that are [[aesthetic|considered beautiful]] or harmonious.<ref>{{cite book |last1=Stakhov |first1=Alexey |last2=Olsen |first2=Olsen |title=The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science |date=2009 |publisher=World Scientific |isbn=978-981-277-582-5}}</ref> From the time of the [[Pythagoreanism|Pythagoreans]] with their religious philosophy of number,<ref>{{cite book |last1=Smith |first1=William |author-link1=William Smith (lexicographer) |title=[[Dictionary of Greek and Roman Biography and Mythology]] |date=1870 |publisher=Little, Brown |page=620}}</ref> architects in [[Greek architecture|ancient Greece]], [[ancient Rome]], the [[Islamic architecture|Islamic world]] and the [[Renaissance architecture|Italian Renaissance]] have chosen the [[proportion (architecture)|proportion]]s of the built environment – buildings and their designed surroundings – according to mathematical as well as aesthetic and sometimes religious principles.<ref name=Vitruvius2009/><ref name=Tennant/><ref name=Rai/><ref name=StAndrewsArchitecture/> Thirdly, they may use mathematical objects such as [[tessellation]]s to decorate buildings.<ref name=Utrecht>{{cite web |last1=van den Hoeven |first1=Saskia |last2=van der Veen |first2=Maartje |title=Muqarnas: Mathematics in Islamic Arts |url=http://www.jphogendijk.nl/projects/muqarnas2010.pdf |publisher=Utrecht University |access-date=30 September 2015 |date=2010 |archive-url=https://web.archive.org/web/20160304040643/http://www.jphogendijk.nl/projects/muqarnas2010.pdf |archive-date=4 March 2016 |url-status=dead }}</ref><ref name=Cucker103>{{cite book |last=Cucker |first=Felipe |title=Manifold Mirrors: The Crossing Paths of the Arts and Mathematics |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-72876-8 |pages=103–106}}</ref> 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.<ref name=Freiberger/>

==Secular aesthetics==

===Ancient Rome===

[[File:Greekhse1.jpg|thumb|upright=1.3<!--size for low image-->|Plan of a Greek house by [[Vitruvius]]]]

====Vitruvius====

{{further|Vitruvius|Vitruvian module|De architectura}}

[[File:Giovanni Paolo Panini - Interior of the Pantheon, Rome - Google Art Project.jpg|thumb|left|upright|The interior of the [[Pantheon (Rome)|Pantheon]] by [[Giovanni Paolo Panini]], 1758]]

The influential ancient Roman architect Vitruvius argued that the design of a building such as a temple depends on two qualities, proportion and ''symmetria''. Proportion ensures that each part of a building relates harmoniously to every other part. ''Symmetria'' in Vitruvius's usage means something closer to the English term [[modularity]] than [[Reflection symmetry|mirror symmetry]], as again it relates to the assembling of (modular) parts into the whole building. In his Basilica at [[Fano]], he uses ratios of small integers, especially the [[triangular number]]s (1, 3, 6, 10,&nbsp;...) to proportion the structure into [[Vitruvian module|(Vitruvian) modules]].{{efn|In Book 4, chapter 3 of ''[[De architectura]]'', he discusses modules directly.<ref>{{cite web |last1=Vitruvius |title=VITRUVIUS, BOOK IV, CHAPTER 3 On the Doric order |url=http://www.vitruvius.be/boek4h3.htm |website=Vitruvius.be |access-date=6 October 2015 |archive-date=4 March 2016 |archive-url=https://web.archive.org/web/20160304081906/http://www.vitruvius.be/boek4h3.htm |url-status=dead }}</ref>}} 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.<ref name="Vitruvius2009">{{cite book |author=Vitruvius |title=On Architecture |url=https://books.google.com/books?id=lBLNbOp46CYC&pg=PR9 |year=2009 |publisher=Penguin Books |isbn=978-0-14-193195-1 |pages=8–9}}</ref>

[[File:Dehio 1 Pantheon Floor plan.jpg|thumb|upright|Floor plan of the Pantheon]]

Vitruvius named three qualities required of architecture in his ''[[De architectura]]'', {{circa|15 B.C.}}: [[Firmness, commodity, and delight|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.<ref name="WilliamsOstwald2015">{{cite book |last1=Williams |first1=Kim |last2=Ostwald |first2=Michael J. |title=Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s |url=https://books.google.com/books?id=fWKYBgAAQBAJ&pg=PA42 |date=9 February 2015 |publisher=Birkhäuser |isbn=978-3-319-00137-1 |pages=42, 48}}</ref>

====The Pantheon====
{{main|Pantheon (Rome)}}

The [[Pantheon (Rome)|Pantheon]] in Rome has survived intact, illustrating classical Roman structure, proportion, and decoration. The main structure is a dome, the apex left open as a circular [[Oculus (architecture)|oculus]] to let in light; it is fronted by a short colonnade with a triangular pediment. The height to the oculus and the diameter of the interior circle are the same, {{convert|43.3|m|ft}}, so the whole interior would fit exactly within a cube, and the interior could house a sphere of the same diameter.<ref name=Roth>{{cite book |last=Roth |first=Leland M. |title=Understanding Architecture: Its Elements, History, And Meaning |publisher=Westview Press |location=Boulder |year=1992 |isbn=0-06-438493-4 |page=[https://archive.org/details/understandingarc00roth/page/36 36] |url=https://archive.org/details/understandingarc00roth/page/36 }}</ref> These dimensions make more sense when expressed in [[ancient Roman units of measurement]]: The dome spans 150 [[Roman foot|Roman feet]]{{efn|A [[Roman foot]] was about {{convert|0.296|m|ft}}.}}); the oculus is 30 Roman feet in diameter; the doorway is 40 Roman feet high.<ref name=Claridge>{{cite book |last=Claridge |first=Amanda |title=Rome |series=Oxford Archaeological Guides |publisher=Oxford University Press |location=Oxford Oxfordshire |year=1998 |isbn=0-19-288003-9 |pages=[https://archive.org/details/romeoxfordarchae00aman/page/204 204–5] |url=https://archive.org/details/romeoxfordarchae00aman/page/204 }}</ref> The Pantheon remains the world's largest unreinforced concrete dome.<ref name=Lancaster>{{cite book |last=Lancaster |first=Lynne C. |title=Concrete Vaulted Construction in Imperial Rome: Innovations in Context |url=https://archive.org/details/concretevaultedc00lanc |url-access=limited |publisher=Cambridge University Press |location=Cambridge |year=2005 |isbn=0-521-84202-6 |pages=[https://archive.org/details/concretevaultedc00lanc/page/n67 44]–46}}</ref>

===Renaissance===

{{further|Renaissance architecture}}

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

The first Renaissance treatise on architecture was Leon Battista Alberti's 1450 {{lang|la|[[De re aedificatoria]]}} (On the Art of Building); it became the first printed book on architecture in 1485. It was partly based on Vitruvius's ''De architectura'' and, via Nicomachus, Pythagorean arithmetic. Alberti starts with a cube, and derives ratios from it. Thus the diagonal of a face gives the ratio 1:{{radic|2}}, while the diameter of the sphere which circumscribes the cube gives 1:{{radic|3}}.<ref>{{cite journal |last=March |first=Lionel |title=Renaissance mathematics and architectural proportion in Alberti's De re aedificatoria |journal=Architectural Research Quarterly |date=1996 |volume=2 |issue=1 |pages=54–65 |doi=10.1017/S135913550000110X |s2cid=110346888 }}</ref><ref>{{cite web |title=Sphere circumscribing a cube |url=http://www.mathalino.com/reviewer/solid-mensuration-solid-geometry/013-insciribed-and-circumscribed-sphere-about-cube-volume- |website=Mathalino.com Engineering Math Review |access-date=4 October 2015}}</ref> Alberti also documented [[Filippo Brunelleschi]]'s discovery of [[linear perspective]], developed to enable the design of buildings which would look beautifully proportioned when viewed from a convenient distance.<ref name=StAndrewsArchitecture/>

[[File:Houghton Typ 525.69.781 - Serlio, 69.jpg|thumb|left|Architectural perspective of a stage set by [[Sebastiano Serlio]], 1569<ref>Typ 525.69.781, Houghton Library, Harvard University</ref>]]

The next major text was [[Sebastiano Serlio]]'s ''Regole generali d'architettura'' (General Rules of Architecture); the first volume appeared in Venice in 1537; the 1545 volume (books{{nbsp}}1 and 2) covered geometry and [[perspective (graphical)|perspective]]. Two of Serlio's methods for constructing perspectives were wrong, but this did not stop his work being widely used.<ref name="Andersen2008">{{cite book |last=Andersen |first=Kirsti |title=The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge |pages=[https://books.google.com/books?id=8B_JeMxNUIkC&pg=PA117 117–121]|title-link=The Geometry of an Art |year=2008 |publisher=Springer |isbn=978-0-387-48946-9}}</ref>

[[File:Villa Pisani.jpg|thumb|right|[[Andrea Palladio]]'s plan and elevation of the [[Villa Pisani (Bagnolo)|Villa Pisani]] ]]

In 1570, [[Andrea Palladio]] published the influential ''[[I quattro libri dell'architettura]]'' (The Four Books of Architecture) in [[Venice]]. This widely printed book was largely responsible for spreading the ideas of the [[Italian Renaissance]] throughout Europe, assisted by proponents like the English diplomat Henry Wotton with his 1624 ''The Elements of Architecture''.<ref name=Ruhl>{{cite web |last=Ruhl |first=Carsten |title=Palladianism: From the Italian Villa to International Architecture |url=http://ieg-ego.eu/en/threads/europe-on-the-road/educational-journey-grand-tour/carsten-ruhl-palladianism |publisher=European History Online |access-date=3 October 2015 |date=7 April 2011}}</ref> 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.<ref>{{cite book |last=Copplestone |first=Trewin |year=1963 |title=World Architecture |url=https://archive.org/details/worldarchitectur00copp |url-access=registration |publisher=Hamlyn |page=[https://archive.org/details/worldarchitectur00copp/page/251 251]|isbn=((9780600039549))<!-- isbn is for 1966 reprint --> }}</ref> Palladio permitted a range of ratios in the ''Quattro libri'', stating:<ref>{{cite web |last1=Wassell |first1=Stephen R. |title=The Mathematics Of Palladio's Villas: Workshop '98 |url=http://www.emis.de/journals/NNJ/Wassell.html |publisher=Nexus Network Journal |access-date=3 October 2015}}</ref><ref>{{cite book |last1=Palladio |first1=Andrea |author1-link=Andrea Palladio |title=I quattro libri dell'architettura |author2=Tavernor, Robert; Schofield, Richard (trans.) |publisher=MIT Press |year=1997 |orig-year=1570 |page=book I, chapter xxi, page 57}}</ref>

{{blockquote|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.{{efn|In modern algebraic notation, these ratios are respectively 1:1, {{radic|2}}:1, 4:3, 3:2, 5:3, 2:1.}}<!--end efn-->}}

In 1615, [[Vincenzo Scamozzi]] published the late Renaissance treatise ''L'idea dell'architettura universale'' (The Idea of a Universal Architecture).<ref>{{cite book |last1=Scamozzi |first1=Vincenzo |title=The Idea of a Universal Architecture |orig-year=1615 |year=2003 |author2=Vroom, W. H. M. (trans.) |publisher=Architectura & Natura}}</ref> 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.<ref name="Borys2014">{{cite book |last=Borys |first=Ann Marie |title=Vincenzo Scamozzi and the Chorography of Early Modern Architecture |url=https://books.google.com/books?id=ocGMddu1ZeMC&pg=PA140 |date=28 March 2014 |publisher=Ashgate Publishing |isbn=978-1-4094-5580-6 |pages=140–148 and passim}}</ref>

===Nineteenth century===

[[File:Adziogol hyperboloid Lighthouse by Vladimir Shukhov 1911.jpg|thumb|upright=0.5|[[Hyperboloid structure|Hyperboloid lattice]] [[Adziogol Lighthouse|lighthouse]] by [[Vladimir Shukhov]], [[Ukraine]], 1911]]

[[Hyperboloid structure]]s were used starting towards the end of the nineteenth century by [[Vladimir Shukhov]] for masts, lighthouses and cooling towers. Their striking shape is both aesthetically interesting and strong, using structural materials economically. [[Shukhov Tower in Polibino|Shukhov's first hyperboloidal tower]] was exhibited in [[Nizhny Novgorod]] in 1896.<ref name="Beckh2015">{{cite book |last=Beckh |first=Matthias |title=Hyperbolic Structures: Shukhov's Lattice Towers – Forerunners of Modern Lightweight Construction |url=https://books.google.com/books?id=LpyLBgAAQBAJ&pg=PA75 |year=2015 |publisher=John Wiley & Sons |isbn=978-1-118-93268-1 |pages=75 and passim}}</ref><ref>{{cite journal|title=The Nijni-Novgorod exhibition: Water tower, room under construction, springing of 91 feet span |journal=The Engineer |date=19 March 1897 |pages=292–294}}</ref><ref>{{cite book |last=Graefe |first=Rainer |title=Vladimir G. Suchov 1853–1939. Die Kunst der sparsamen Konstruktion |url=https://archive.org/details/isbn_3421029849 |url-access=limited |publisher=Deutsche Verlags-Anstalt |year=1990 |pages=[https://archive.org/details/isbn_3421029849/page/n110 110]–114 |isbn=3-421-02984-9 |display-authors=etal}}</ref>

===Twentieth century===

{{further|Modern architecture|Contemporary architecture}}

[[File:Rietveld Schröderhuis HayKranen-7.JPG|thumb|left|[[De Stijl]]'s sliding, intersecting planes: the [[Rietveld Schröder House]], 1924]]

The early twentieth century movement [[Modern architecture]], pioneered{{efn|Constructivism influenced Bauhaus and Le Corbusier, for example.<ref name=Hatherley/>}} by Russian [[Constructivism (art)|Constructivism]],<ref name=Hatherley>{{cite news |last=Hatherley |first1=Owen |title=The Constructivists and the Russian Revolution in Art and Architecture |url=https://www.theguardian.com/artanddesign/2011/nov/04/russian-avant-garde-constructivists |access-date=6 June 2016 |agency=[[The Guardian]] |date=4 November 2011}}</ref> used rectilinear [[Euclidean geometry|Euclidean]] (also called [[Cartesian coordinate system|Cartesian]]) geometry. In the [[De Stijl]] movement, the horizontal and the vertical were seen as constituting the universal. The architectural form consists of putting these two directional tendencies together, using roof planes, wall planes and balconies, which either slide past or intersect each other, as in the 1924 [[Rietveld Schröder House]] by [[Gerrit Rietveld]].<ref>{{cite web |title=Rietveld Schröderhuis (Rietveld Schröder House) |work=World Heritage Centre |publisher=[[UNESCO]] |url=https://whc.unesco.org/en/list/965 |access-date=13 December 2012}}</ref>

[[File:Raoul Heinrich Francé Poppy and Pepperpot from Die Pflanze als erfinder 1920.jpeg|thumb|[[Raoul Heinrich Francé]]'s [[poppy]] and [[pepper shaker|pepperpot]] ([[biomimetics]]) image from ''Die Pflanze als Erfinder'', 1920]]

Modernist architects were free to make use of curves as well as planes. [[Charles Holden]]'s 1933 [[Arnos Grove tube station|Arnos station]] has a circular ticket hall in brick with a flat concrete roof.<ref>{{NHLE|num=1358981|access-date=5 October 2015}}</ref> In 1938, the [[Bauhaus]] painter [[László Moholy-Nagy]] adopted [[Raoul Heinrich Francé]]'s seven [[biomimetics|biotechnical]] elements, namely the crystal, the sphere, the cone, the plane, the (cuboidal) strip, the (cylindrical) rod, and the spiral, as the supposed basic building blocks of architecture inspired by nature.<ref>{{cite book |last1=Moholy-Nagy |first1=Laszlo |last2=Hoffman |first2=Daphne M. (trans.) |title=The New Vision: Fundamentals of Design, Painting, Sculpture, Architecture |date=1938 |publisher=New Bauhaus Books |page=46}}</ref><ref>{{cite book |last=Gamwell |first=Lynn |author-link=Lynn Gamwell |title=Mathematics and Art: A Cultural History |date=2015 |publisher=Princeton University Press |page=306 |isbn=978-0-691-16528-8}}</ref>

[[Le Corbusier]] proposed an [[anthropometric]] [[scale (ratio)|scale]] of proportions in architecture, the [[Modulor]], based on the supposed height of a man.<ref>{{cite book |author=Le Corbusier |author-link=Le Corbusier |title=The Modulor: A Harmonious Measure to the Human Scale, Universally Applicable to Architecture and Mechanics  |publisher=Birkhäuser |year=2004 |orig-year=1954 and 1958 |isbn=3-7643-6188-3}}</ref> Le Corbusier's 1955 [[Notre-Dame du Haut|Chapelle Notre-Dame du Haut]] uses free-form curves not describable in mathematical formulae.{{efn|Pace Nikos Salingaros, who suggests the contrary,<ref name=SalingarosAPM/> but it is not clear exactly what mathematics may be embodied in the curves of Le Corbusier's chapel.<ref>{{cite web |last1=Greene |first1=Herb |title=Le Corbusier: Notre Dame du Haut at Ronchamp |url=http://www.herbgreene.org/WRITING/Le%20CORBUSIER.html |archive-url=https://web.archive.org/web/20150907204501/http://www.herbgreene.org/WRITING/Le%20CORBUSIER.html |url-status=dead |archive-date=7 September 2015 |access-date=5 October 2015 }}</ref>}} The shapes are said to be evocative of natural forms such as the [[prow]] of a ship or praying hands.<ref name="Hanser2006">{{cite book |last=Hanser |first=David A. |title=Architecture of France |url=https://books.google.com/books?id=zojzUU976h0C&pg=PA211 |year=2006 |publisher=Greenwood Publishing Group |isbn=978-0-313-31902-0 |page=211}}</ref> The design is only at the largest scale: there is no hierarchy of detail at smaller scales, and thus no fractal dimension; the same applies to other famous twentieth-century buildings such as the [[Sydney Opera House]], [[Denver International Airport]], and the [[Guggenheim Museum, Bilbao]].<ref name=SalingarosAPM/>

[[Contemporary architecture]], in the opinion of the 90 leading architects who responded to a 2010 [[World Architecture Survey]], is extremely diverse; the best was judged to be [[Frank Gehry]]'s Guggenheim Museum, Bilbao.<ref name="VF Results">{{cite news |title=Vanity Fair's World Architecture Survey: the Complete Results |url=http://www.vanityfair.com/culture/features/2010/08/architecture-survey-list-201008 |access-date=22 July 2010 |newspaper=[[Vanity Fair (magazine)|Vanity Fair]] |date=30 June 2010 |archive-date=8 November 2014 |archive-url=https://web.archive.org/web/20141108161337/http://www.vanityfair.com/culture/features/2010/08/architecture-survey-list-201008 |url-status=dead }}</ref>

[[File:DIA Airport Roof.jpg|thumb|left|upright=2.5<!--width for very low image-->|The [[minimal surface]]s of the [[fabric structure|fabric roof]] of [[Denver International Airport]], completed in 1995, evoke [[Colorado]]'s snow-capped mountains and the [[teepee]] tents of [[Native Americans in the United States|Native American]]s.]]

Denver International Airport's terminal building, completed in 1995, has a [[fabric structure|fabric roof]] supported as a [[minimal surface]] (i.e., its [[mean curvature]] is zero) by steel cables. It evokes [[Colorado]]'s snow-capped mountains and the [[teepee]] tents of [[Native Americans in the United States|Native American]]s.<ref>{{cite web |title=Denver International Airport Press Kit |url=http://business.flydenver.com/info/news/pressKit.pdf |publisher=Denver International Airport |date=2014 |access-date=5 October 2015 |url-status=dead |archive-url=https://web.archive.org/web/20150412011439/http://business.flydenver.com/info/news/pressKit.pdf |archive-date=12 April 2015 }}</ref><ref>{{cite web |title=Denver International Airport |url=http://www.architonic.com/aisht/denver-international-airport-fentress-architects/5100647 |publisher=Fenstress Architects |access-date=5 October 2015}}</ref>

The architect [[Richard Buckminster Fuller]] is famous for designing strong [[thin-shell structure]]s known as [[geodesic dome]]s. The [[Montréal Biosphère]] dome is {{convert|61|m|ft}} high; its diameter is {{convert|76|m|ft}}.<ref>{{cite web |title=Biosphere |url=http://www.aviewoncities.com/montreal/biosphere.htm |website=A view on cities |access-date=1 October 2015 |archive-date=27 September 2007 |archive-url=https://web.archive.org/web/20070927223117/http://www.aviewoncities.com/montreal/biosphere.htm |url-status=dead }}</ref>

Sydney Opera House has a dramatic roof consisting of soaring white vaults, reminiscent of ship's sails; to make them possible to construct using standardized components, the vaults are all composed of triangular sections of spherical shells with the same radius. These have the required uniform [[curvature]] in every direction.<ref>{{cite web |last=Hahn |first=Alexander J. |title=Mathematical Excursions To Architecture |url=https://www.insidescience.org/content/mathematical-excursions-architecture/927 |publisher=Inside Science |access-date=5 October 2015 |date=4 February 2013}}</ref>

The late twentieth century movement [[Deconstructivism]] creates deliberate disorder with what [[Nikos Salingaros]] in ''[[A Theory of Architecture]]'' calls random forms<ref>{{cite book |title=A Theory of Architecture |publisher=Umbau |author=Salingaros, Nikos |year=2006 |url=https://books.google.com/books?id=FV_0_RHD4cQC |pages=139–141 |isbn=9783937954073 }}</ref> of high complexity<ref>{{cite book |title=A Theory of Architecture |publisher=Umbau |author=Salingaros, Nikos |year=2006 |url=https://books.google.com/books?id=FV_0_RHD4cQC |pages=124–125|isbn=9783937954073 }}</ref> by using non-parallel walls, superimposed grids and complex 2-D surfaces, as in Frank Gehry's [[Disney Concert Hall]] and Guggenheim Museum, Bilbao.<ref>{{cite book |last1=Gehry |first1=Frank O. |author1-link=Frank Gehry |author2=Mudford, Grant |author3=Koshalek, Richar |url=https://books.google.com/books?id=ezxDPgAACAAJ |title=Symphony: Frank Gehry's Walt Disney Concert Hall |publisher=Five Tiesd |year=2009 |isbn=9780979472749 }}</ref><ref>{{cite book |last=Garcetti |first=Gil |title=Iron: Erecting the Walt Disney Concert Hall |publisher=Princeton Architectural Press |year=2004 |isbn=9781890449285  |url=https://books.google.com/books?id=TrH5I3Ure1EC}}</ref> 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.<ref name=SalingarosAPM>{{cite web |last=Salingaros |first=Nikos |author-link=Nikos Salingaros |title=Architecture, Patterns, and Mathematics |url=http://www.emis.de/journals/NNJ/Salingaros.html |publisher=Nexus Network Journal |access-date=9 October 2015}} Updated version of {{cite journal |last1=Salingaros |first1=Nikos|title=Architecture, Patterns, and Mathematics |journal=Nexus Network Journal |date=April 1999 |volume=1 |issue=2 |pages=75–86 |doi=10.1007/s00004-998-0006-0 |s2cid=120544101|doi-access=free}}</ref>

<gallery>
File:Bauhaus-Dessau Wohnheim Balkone.jpg|[[New Objectivity (architecture)|New Objectivity]]: [[Walter Gropius]]'s [[Bauhaus]], [[Dessau]], 1925
File:Arnos Grove underground station 16 November 2012.jpg|[[cylinder (geometry)|Cylinder]]: [[Charles Holden]]'s [[Arnos Grove tube station]], 1933
File:RonchampCorbu.jpg|[[Modern Architecture|Modernism]]: [[Le Corbusier]]'s [[Chapelle Notre Dame du Haut]], 1955
File:Mtl. Biosphere in Sept. 2004.jpg|[[Geodesic dome]]: the [[Montréal Biosphère]] by [[R. Buckminster Fuller]], 1967
File:Sydney Opera House Sails.jpg|Uniform [[curvature]]: [[Sydney Opera House]], 1973
File:Image-Disney Concert Hall by Carol Highsmith edit-2.jpg|[[Deconstructivism]]: [[Disney Concert Hall]], Los Angeles, 2003
</gallery>

==Religious principles==

===Ancient Egypt===

{{see also|Golden ratio#Egyptian pyramids}}

[[File:Mathematical Pyramid.svg|thumb|Base:hypotenuse (b:a) ratios for pyramids like the [[Great Pyramid of Giza]] could be: 1:φ ([[Kepler triangle]]), 3:5 ([[Special right triangle|3:4:5 triangle]]), or 1:4/π]]

The [[pyramid]]s of [[ancient Egypt]] are [[Ancient Egyptian funerary practices|tombs]] constructed with mathematical proportions, but which these were, and whether the [[Pythagorean theorem]] was used, are debated. The ratio of the slant height to half the base length of the [[Great Pyramid of Giza]] is less than 1% from the [[golden ratio]].<ref name=nexus/> If this was the design method, it would imply the use of [[Kepler triangle|Kepler's triangle]] (face angle 51°49'),<ref name=nexus>{{cite journal |last=Bartlett |first=Christopher |title=The Design of The Great Pyramid of Khufu |journal=Nexus Network Journal |year=2014 |volume=16 |issue=2 |pages=299–311 |doi=10.1007/s00004-014-0193-9 |doi-access=free }}</ref><ref name=Markowsky1992>{{cite journal |last=Markowsky |first=George |url=http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf |title=Misconceptions About the Golden Ratio |journal=The College Mathematics Journal |date=January 1992 |volume=23 |issue=1 |pages=2–19 |doi=10.1080/07468342.1992.11973428 |access-date=2015-10-01 |archive-url=https://web.archive.org/web/20080408200850/http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf |archive-date=2008-04-08 |url-status=dead }}</ref> but according to many [[History of science|historians of science]], the golden ratio was not known until the time of the [[Pythagoreans]].<ref>{{cite book |last=Livio |first=Mario |author-link=Mario Livio |title=The Golden Ratio: The Story of Phi, the World's Most Astonishing Number|url=https://books.google.com/books?id=bUARfgWRH14C|orig-year=2002 |edition=First trade paperback |year=2003 |publisher=[[Random House|Broadway Books]] |location=New York City |isbn=0-7679-0816-3 |page=61}}</ref>

The proportions of some pyramids may have also been based on the [[Special right triangles|3:4:5 triangle]] (face angle 53°8'), known from the [[Rhind Mathematical Papyrus]] (c.&nbsp;1650–1550&nbsp;BC); this was first conjectured by historian [[Moritz Cantor]] in 1882.<ref name=Cooke2011>{{cite book |last=Cooke |first=Roger L. |title=The History of Mathematics: A Brief Course |url=https://books.google.com/books?id=wOGh7XPowAMC&pg=PA237 |edition=2nd |year=2011 |publisher=John Wiley & Sons |isbn=978-1-118-03024-0 |pages=237–238}}</ref> It is known that right angles were laid out accurately in ancient Egypt using [[knotted cord]]s for measurement,<ref name=Cooke2011/> that [[Plutarch]] recorded in ''[[De Iside et Osiride|Isis and Osiris]]'' (c.&nbsp;100 AD) that the Egyptians admired the 3:4:5 triangle,<ref name=Cooke2011/> and that a scroll from before 1700&nbsp;BC demonstrated basic [[Square (algebra)|square]] formulas.<ref>{{cite book |last=Gillings |first=Richard J. |title=Mathematics in the Time of the Pharaohs |url=https://archive.org/details/mathematicsintim0000gill |url-access=registration |publisher=Dover |date=1982 |page=[https://archive.org/details/mathematicsintim0000gill/page/161 161]}}</ref>{{efn|[[Berlin Papyrus 6619]] from the [[Middle Kingdom of Egypt|Middle Kingdom]] stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other."}} 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&nbsp;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."<ref name=Cooke2011/>

===Ancient India===

{{further|Architecture of India|Vaastu Shastra}}

[[File:Virupaksha Temple,Hampi.JPG|thumb|upright|left|[[Gopuram]] of the [[Hindu]] [[Virupaksha Temple, Hampi|Virupaksha Temple]] has a [[fractal]]-like structure where the parts resemble the whole.]]

[[Vaastu Shastra]], the ancient [[India]]n canons of architecture and town planning, employs symmetrical drawings called [[mandala]]s. Complex calculations are used to arrive at the dimensions of a building and its components. The designs are intended to integrate architecture with nature, the relative functions of various parts of the structure, and ancient beliefs utilizing geometric patterns ([[yantra]]), symmetry and [[Direction (geometry, geography)|directional]] alignments.<ref>Kramrisch, Stella (1976), The Hindu Temple Volume 1 & 2, {{isbn|81-208-0223-3}}</ref><ref>{{cite book |last1=Vibhuti Sachdev, [[Giles Tillotson]] |title=Building Jaipur: The Making of an Indian City |date=2004 |isbn=978-1-86189-137-2 |pages=155–160}}</ref> 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.<ref name=StAndrewsArchitecture/><ref>{{cite book |last=Ifrah |first=Georges |title=A Universal History of Numbers |publisher=Penguin |year=1998 }}</ref>

[[File:Plan of Meenakshi Amman Temple Madurai India.jpg|thumb|upright|Plan of [[Meenakshi Amman Temple]], [[Madurai]], from the 7th century onwards. The four gateways (numbered I-IV) are tall [[gopuram]]s.]]

The mathematics of [[fractals]] has been used to show that the reason why existing buildings have universal appeal and are visually satisfying is because they provide the viewer with a sense of scale at different viewing distances. For example, in the tall [[gopuram]] gatehouses of [[Hindu]] temples such as the [[Virupaksha Temple, Hampi|Virupaksha Temple]] at [[Hampi]] built in the seventh century, and others such as the [[Kandariya Mahadev Temple]] at [[Khajuraho]], the parts and the whole have the same character, with [[fractal dimension]] in the range 1.7 to 1.8. The cluster of smaller towers (''shikhara'', lit. 'mountain') about the tallest, central, tower which represents the holy [[Mount Kailash]], abode of Lord [[Shiva]], depicts the endless repetition of universes in [[Hindu cosmology]].<ref name=Rian/><ref name=Yale>{{cite web |title=Fractals in Indian Architecture |url=https://classes.yale.edu/fractals/Panorama/Architecture/IndianArch/IndianArch.html |publisher=Yale University |access-date=1 October 2015 |archive-url=https://web.archive.org/web/20120206011944/http://classes.yale.edu/Fractals/Panorama/Architecture/IndianArch/IndianArch.html |archive-date=6 February 2012 |url-status=dead }}</ref> The religious studies scholar William J. Jackson observed of the pattern of towers grouped among smaller towers, themselves grouped among still smaller towers, that:

{{blockquote|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.<ref name=Yale/><ref name=Jackson>{{cite web |last1=Jackson |first1=William J. |title=For All Fractal Purposes&nbsp;... an introduction |url=http://liberalarts.iupui.edu/rel/OLD_SITE/Fractals/COVERPAGE.HTM |publisher=Indiana University-Purdue University Indianapolis |access-date=1 October 2015 |archive-url=https://web.archive.org/web/20150914170816/http://liberalarts.iupui.edu/rel/OLD_SITE/Fractals/COVERPAGE.HTM |archive-date=14 September 2015}}</ref>}}

The [[Meenakshi Amman Temple]] is a large complex with multiple shrines, with the streets of [[Madurai]] laid out concentrically around it according to the shastras. The four gateways are tall towers ([[gopuram]]s) with fractal-like repetitive structure as at Hampi. The enclosures around each shrine are rectangular and surrounded by high stone walls.<ref>{{cite book |last=King |first=Anthony D. |title=Buildings and Society: Essays on the Social Development of the Built Environment |year=2005 |publisher=Taylor & Francis |isbn=0-203-48075-9 |url=https://books.google.com/books?id=1HtVU6D2LOUC&q=meenakshi |page=72}}</ref>

===Ancient Greece===

{{further|Greek architecture|golden ratio|Pythagoreanism|Euclidean geometry}}

[[File:The Parthenon Athens.jpg|thumb|left|The [[Parthenon]] was designed using [[Pythagoreanism|Pythagorean]] ratios.]]

[[Pythagoras]] (c. 569 – c. 475 B.C.) and his followers, the Pythagoreans, held that "all things are numbers". They observed the harmonies produced by notes with specific small-integer ratios of frequency, and argued that buildings too should be designed with such ratios. The Greek word ''symmetria'' originally denoted the harmony of architectural shapes in precise ratios from a building's smallest details right up to its entire design.<ref name=StAndrewsArchitecture>{{cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |title=Mathematics and Architecture |url=http://www-history.mcs.st-and.ac.uk/HistTopics/Architecture.html |publisher=University of St Andrews |access-date=4 October 2015 |date=February 2002}}</ref>

The [[Parthenon]] is {{convert|69.5|m|ft}} long, {{convert|30.9|m|ft}} wide and {{convert|13.7|m|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<ref name="Maor2007">{{cite book |last=Maor |first=Eli |title=The Pythagorean Theorem: A 4,000-year History |url=https://books.google.com/books?id=Z5VoBGy3AoAC&pg=PA19|year=2007 |publisher=Princeton University Press |isbn=978-0-691-12526-8 |page=19}}</ref> of the Pythagoreans 4<sup>2</sup>:6<sup>2</sup>:9<sup>2</sup>. This sets the module as 0.858&nbsp;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 ({{convert|21.44|m|ft}} wide by 48.3&nbsp;m long); the ratio between the diameter of the outer columns, {{convert|1.905|m|ft}}, and the spacing of their centres, {{convert|4.293|m|ft}}, is also 4:9.<ref name=StAndrewsArchitecture/>

[[File:Parthenon-top-view.svg|thumb|upright=0.6<!--ratio for v. tall img-->|Floor plan of the Parthenon]]

The Parthenon is considered by authors such as [[John Julius Norwich]] "the most perfect Doric temple ever built".<ref name=Norwich/> Its elaborate architectural refinements include "a subtle correspondence between the curvature of the stylobate, the taper of the [[naos (architecture)|naos]] walls and the ''entasis'' of the columns".<ref name=Norwich>{{cite book |last=Norwich |first=John Julius |author-link=John Julius Norwich |title=Great Architecture of the World |publisher=Artists House |year=2001 |page=63}}</ref> ''[[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,<ref>{{cite book |last=Penrose |first=Francis |title=Principles of Athenian Architecture |publisher=Society of Dilettanti |year=1973 |orig-year=1851 |page=ch. II.3, plate 9}}</ref> the platform has a slight parabolic upward curvature to shed rainwater and reinforce the building against earthquakes. The columns might therefore be supposed to lean outwards, but they actually lean slightly inwards so that if they carried on, they would meet about a kilometre and a half above the centre of the building; since they are all the same height, the curvature of the outer stylobate edge is transmitted to the [[architrave]] and roof above: "all follow the rule of being built to delicate curves".<ref>{{cite journal |last=Stevens |first=Gorham P. |title=Concerning the Impressiveness of the Parthenon |journal=American Journal of Archaeology |volume=66 |issue=3 |date=July 1962 |pages=337–338 |doi=10.2307/501468|jstor=501468 |s2cid=192963601 }}</ref>

The golden ratio was known in 300 B.C., when [[Euclid]] described the method of geometric construction.<ref>[[Euclid's Elements|Euclid. ''Elements'']]. Book 6, Proposition 30.</ref> 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.<ref>{{cite web |last=Archibald |first=R. C. |url=http://www.spirasolaris.ca/hambidge1a.html |title=Notes on the Logarithmic Spiral, Golden Section and the Fibonacci Series |access-date=1 October 2015}}</ref> More recent authors such as [[Nikos Salingaros]], however, doubt all these claims.<ref>{{cite web |last=Salingaros |first=Nikos |author-link=Nikos Salingaros |title=Applications of the Golden Mean to Architecture |date=22 February 2012 |url=http://meandering-through-mathematics.blogspot.com/2012/02/applications-of-golden-mean-to.html}}</ref> Experiments by the computer scientist George Markowsky failed to find any preference for the [[golden rectangle]].<ref name=Markowsky>{{cite journal |last=Markowsky |first=George |url=http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf |title=Misconceptions about the Golden Ratio |journal=The College Mathematics Journal |volume=23 |issue=1 |date=January 1992 |pages=2–19 |doi=10.1080/07468342.1992.11973428 |access-date=2015-10-01 |archive-url=https://web.archive.org/web/20080408200850/http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf |archive-date=2008-04-08 |url-status=dead }}</ref>

===Islamic architecture===

{{further|Islamic architecture|Golden ratio#Architecture}}

[[File:Selimiye... - panoramio.jpg|thumb|right|[[Selimiye Mosque, Edirne]], 1569–1575]]

The historian of Islamic art Antonio Fernandez-Puertas suggests that the [[Alhambra]], like the [[Great Mosque of Cordoba]],<ref>{{cite web |last1=Gedal |first1=Najib |title=The Great Mosque of Cordoba: Geometric Analysis |url=http://islamic-arts.org/2011/the-great-mosque-of-cordoba-geometric-analysis/ |publisher=Islamic Art & Architecture |access-date=16 October 2015 |archive-url=https://web.archive.org/web/20151002022611/http://islamic-arts.org/2011/the-great-mosque-of-cordoba-geometric-analysis/ |archive-date=2 October 2015 |url-status=dead }}</ref> was designed using the [[Al-Andalus|Hispano-Muslim]] foot or ''codo'' of about {{convert|0.62|m|ft}}. In the palace's [[Court of the Lions]], the proportions follow a series of [[surd (mathematics)|surd]]s. A rectangle with sides 1{{nbsp}}and {{radic|2}} has (by [[Pythagorean theorem|Pythagoras's theorem]]) a diagonal of {{radic|3}}, which describes the right triangle made by the sides of the court; the series continues with {{radic|4}} (giving a 1:2 ratio), {{radic|5}} and so on. The decorative patterns are similarly proportioned, {{radic|2}} generating squares inside circles and eight-pointed stars, {{radic|3}} generating six-pointed stars. There is no evidence to support earlier claims that the golden ratio was used in the Alhambra.<ref name=Tennant>{{cite journal |last1=Tennant |first1=Raymond |title=International Joint Conference of ISAMA, the International Society of the Arts, Mathematics, and Architecture, and BRIDGES. Mathematical Connections in Art Music, and Science, University of Granada, Spain, July, 2003. Islamic Constructions: The Geometry Needed by Craftsmen. |journal=International Joint Conference of ISAMA, the International Society of the Arts, Mathematics, and Architecture, and BRIDGES, Mathematical Connections in Art Music, and Science |date=July 2003 |url=http://www2.kenyon.edu/Depts/Math/Aydin/Teach/Fall13/128/GeometryNeededbyCraftsmen.pdf}}</ref><ref name="Irwin2011">{{cite book |last=Irwin |first=Robert |title=The Alhambra |url=https://books.google.com/books?id=dS_cIXaGKI4C&pg=PA109 |date=26 May 2011 |publisher=Profile Books |isbn=978-1-84765-098-6 |pages=109–112}}</ref> The [[Court of the Lions]] is bracketed by the Hall of Two Sisters and the Hall of the Abencerrajes; a regular [[hexagon]] can be drawn from the centres of these two halls and the four inside corners of the Court of the Lions.<ref>{{cite web |last1=Robertson |first1=Ann |title=Revisiting the Geometry of the Sala de Dos Hermanas |url=http://archive.bridgesmathart.org/2007/bridges2007-303.pdf |publisher=BRIDGES |access-date=11 October 2015|date=2007}}</ref>

The [[Selimiye Mosque, Edirne|Selimiye Mosque]] in [[Edirne]], Turkey, was built by [[Mimar Sinan]] to provide a space where the [[mihrab]] 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 {{convert|31.25|m|ft}} diameter and {{convert|43|m|ft}} high. The octagon is formed into a square with four semidomes, and externally by four exceptionally tall minarets, {{convert|83|m|ft}} tall. The building's plan is thus a circle, inside an octagon, inside a square.<ref>{{cite book |first1=Sheila |last1=Blair |first2=Jonathan M. |last2=Bloom |title=The Art and Architecture of Islam 1250–1800 |url=https://books.google.com/books?id=-mhIgewDtNkC&pg=PA226 |publisher=Yale University Press |year=1995 |isbn=0-300-06465-9}}</ref>

===Mughal architecture===

{{main|Mughal architecture|Fatehpur Sikri|Origins and architecture of the Taj Mahal}}

[[File:TajMahalbyAmalMongia.jpg|thumb|left|The [[Taj Mahal]] mausoleum with part of the complex's gardens at [[Agra]]]]

[[Mughal architecture]], as seen in the abandoned imperial city of [[Fatehpur Sikri]] and the [[Taj Mahal]] complex, has a distinctive mathematical order and a strong aesthetic based on symmetry and harmony.<ref name=Rai>{{cite journal |last1=Rai |first1=Jaswant |title=Mathematics and Aesthetics in Islamic Architecture: Reference to Fatehpur Sikri |journal=Journal of King Saud University, Architecture & Planning |volume=5 |issue=1 |pages=19–48 |date=1993 |url=https://cap.ksu.edu.sa/sites/cap.ksu.edu.sa/files/imce_images/5_5e.pdf }}</ref><ref>{{cite book |last1=Michell |first1=George |last2=Pasricha |first2=Amit |title=Mughal Architecture & Gardens |date=2011 |publisher=Antique Collectors Club |isbn=978-1-85149-670-9}}</ref>

The Taj Mahal exemplifies Mughal architecture, both representing [[paradise]]<ref name="Parker2010">{{cite book |last=Parker |first=Philip |title=World History |url=https://books.google.com/books?id=uAUkCOFGgDoC&pg=PA224 |year=2010 |publisher=Dorling Kindersley |isbn=978-1-4053-4124-0 |page=224}}</ref> and displaying the [[list of Mughal emperors|Mughal Emperor]] [[Shah Jahan]]'s power through its scale, symmetry and costly decoration. The white marble [[mausoleum]], decorated with [[pietra dura]], the great gate (''Darwaza-i rauza''), other buildings, the gardens and paths together form a unified hierarchical design. The buildings include a [[mosque]] in red sandstone on the west, and an almost identical building, the Jawab or 'answer' on the east to maintain the bilateral symmetry of the complex. The formal [[charbagh]] ('fourfold garden') is in four parts, symbolising the four [[rivers of Paradise]], and offering views and reflections of the mausoleum. These are divided in turn into 16 parterres.<ref>{{cite book |last=Koch |first=Ebba |title=The Complete Taj Mahal: And the Riverfront Gardens of Agra |url=https://archive.org/details/completetajmahal0000koch/page/24 |url-access=registration |edition=1st |publisher=Thames & Hudson |isbn=0-500-34209-1 |pages=[https://archive.org/details/completetajmahal0000koch/page/24 24 and passim] |date=2006 }}</ref>

[[File:Taj site plan.png|thumb|upright=1.2|Site plan of the [[Taj Mahal]] complex. The great gate is at the right, the mausoleum in the centre, bracketed by the mosque (below) and the jawab. The plan includes squares and [[octagon]]s.]]

The Taj Mahal complex was laid out on a grid, subdivided into smaller grids. The historians of architecture Koch and Barraud agree with the traditional accounts that give the width of the complex as 374 Mughal yards or [[gaz (measure)|gaz]],{{efn|1 gaz is about {{convert|0.86|m|ft}}.}} 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{{nbsp}}gaz. Koch and Barraud observe that if an octagon, used repeatedly in the complex, is given sides of 7{{nbsp}}units, then it has a width of 17 units,{{efn|A square drawn around the octagon by prolonging alternate sides adds four right angle triangles with hypotenuse of 7{{nbsp}}and the other two sides of {{radic|49/2}} or 4.9497..., nearly{{nbsp}}5. The side of the square is thus 5+7+5, which is 17.}} which may help to explain the choice of ratios in the complex.<ref>{{cite book|last=Koch |first=Ebba |title=The Complete Taj Mahal: And the Riverfront Gardens of Agra|url=https://archive.org/details/completetajmahal0000koch |url-access=registration |edition=1st|publisher=Thames & Hudson |isbn=0-500-34209-1 |pages=[https://archive.org/details/completetajmahal0000koch/page/104 104–109] |date=2006}}</ref>

===Christian architecture===

{{further|Church architecture}}

The [[Christianity|Christian]] [[Ecumenical Patriarch of Constantinople|patriarchal]] [[basilica]] of [[Haghia Sophia]] in [[Byzantium]] (now [[Istanbul]]), first constructed in 537 (and twice rebuilt), was for a thousand years{{efn|Until [[Seville Cathedral]] was completed in 1520.}} the largest cathedral ever built. It inspired many later buildings including [[Sultan Ahmed Mosque|Sultan Ahmed]] and other mosques in the city. The [[Byzantine architecture]] includes a nave crowned by a circular dome and two half-domes, all of the same diameter ({{convert|31|m|ft}}), with a further five smaller half-domes forming an [[apse]] and four rounded corners of a vast rectangular interior.<ref>{{cite book |last1=Fazio |first1=Michael |last2=Moffett |first2=Marian |last3=Wodehouse |first3=Lawrence |title=Buildings Across Time |edition=3rd |isbn=978-0-07-305304-2 |publisher=McGraw-Hill Higher Education |year=2009}}</ref> This was interpreted by mediaeval architects as representing the mundane below (the square base) and the divine heavens above (the soaring spherical dome).<ref>{{cite book |last1=Gamwell |first1=Lynn|author1-link= Lynn Gamwell |title=Mathematics and Art: A Cultural History |date=2015 |publisher=Princeton University Press |page=48 |isbn=978-0-691-16528-8}}</ref> The emperor [[Justinian]] used two geometers, [[Isidore of Miletus]] and [[Anthemius of Tralles]] as architects; Isidore compiled the works of [[Archimedes]] on [[solid geometry]], and was influenced by him.<ref name=StAndrewsArchitecture/><ref>{{cite book |last=Kleiner |first=Fred S. |author2=Mamiya, Christin J. |title=[[Gardner's Art Through the Ages|Gardner's Art Through the Ages: Volume I, Chapters 1–18]] |publisher=Wadsworth |edition=12th |year = 2008 |page=329 |isbn =978-0-495-46740-3}}</ref>

[[File:Hagia-Sophia-Grundriss.jpg|thumb|upright=1.4<!--size for detailed plan-->|left|[[Haghia Sophia]], Istanbul<br/>a) Plan of gallery (upper half)<br/>b) Plan of the ground floor (lower half)]]

The importance of water [[baptism]] in Christianity was reflected in the scale of [[baptistry]] architecture. The oldest, the [[Lateran Baptistry]] in Rome, built in 440,<ref>{{cite web |last1=Menander |first1=Hanna |last2=Brandt |first2=Olof |last3=Appetechia |first3=Agostina |last4=Thorén |first4=Håkan |title=The Lateran Baptistery in Three Dimensions |url=http://samla.raa.se/xmlui/bitstream/handle/raa/5009/ro2010_18.pdf |publisher=Swedish National Heritage Board |access-date=30 October 2015 |date=2010 |archive-date=16 May 2017 |archive-url=https://web.archive.org/web/20170516164637/http://samla.raa.se/xmlui/bitstream/handle/raa/5009/ro2010_18.pdf |url-status=dead }}</ref> set a trend for octagonal baptistries; the [[baptismal font]] inside these buildings was often octagonal, though Italy's largest [[Pisa Baptistry|baptistry, at Pisa]], built between 1152 and 1363, is circular, with an octagonal font. It is {{convert|54.86|m|ft}} high, with a diameter of {{convert|34.13|m|ft}} (a ratio of 8:5).<ref>{{cite web |title=The Baptistery |url=http://www.leaningtowerofpisa.net/pisa-baptistery.html |website=The Leaning Tower of Pisa |access-date=30 October 2015 |archive-date=11 October 2015 |archive-url=https://web.archive.org/web/20151011133643/http://www.leaningtowerofpisa.net/pisa-baptistery.html |url-status=dead }}</ref> [[Saint Ambrose]] wrote that fonts and baptistries were octagonal "because on the eighth day,{{efn|The sixth day of [[Holy Week]] was [[Good Friday]]; the following Sunday (of the [[resurrection]]) was thus the eighth day.<ref name=Huyser-Konig/>}} by rising, Christ loosens the bondage of death and receives the dead from their graves."<ref name=Huyser-Konig>{{cite web |last1=Huyser-Konig |first1=Joan |title=Theological Reasons for Baptistry Shapes |url=http://worship.calvin.edu/resources/resource-library/theological-reasons-for-baptistry-shapes/|publisher=Calvin Institute of Christian Worship |access-date=30 October 2015}}</ref><ref name=Kuehn1992>{{cite book |last=Kuehn |first=Regina |title=A Place for Baptism |url=https://books.google.com/books?id=Sf2hmbe1J30C&pg=PA53 |year=1992 |publisher=Liturgy Training Publications |isbn=978-0-929650-00-5 |pages=53–60}}</ref>
[[Saint Augustine]] similarly described the eighth day as "everlasting&nbsp;... hallowed by the [[resurrection]] of Christ".<ref name=Kuehn1992/><ref>{{cite book |author=Augustine of Hippo |author-link=Augustine of Hippo |title=The City of God |title-link=The City of God (book) |year=426 |page=Book 22, Chapter 30}}</ref> The octagonal [[Florence Baptistery|Baptistry of Saint John, Florence]], built between 1059 and 1128, is one of the oldest buildings in that city, and one of the last in the direct tradition of classical antiquity; it was extremely influential in the subsequent Florentine Renaissance, as major architects including [[Francesco Talenti]], Alberti and Brunelleschi used it as the model of classical architecture.<ref name=Kleiner2012>{{cite book |last=Kleiner |first=Fred |title=Gardner's Art through the Ages: A Global History |url=https://books.google.com/books?id=PR4KAAAAQBAJ&pg=PT389 |date=2012 |publisher=Cengage Learning |isbn=978-1-133-71116-2 |pages=355–356}}</ref>

The number five is used "exuberantly"<ref name="SimitchWarke2014"/> in the 1721 [[Pilgrimage Church of St John of Nepomuk]] at Zelená hora, near [[Žďár nad Sázavou]] in the Czech republic, designed by [[Jan Santini Aichel|Jan Blažej Santini Aichel]]. The nave is circular, surrounded by five pairs of columns and five oval domes alternating with ogival apses. The church further has five gates, five chapels, five altars and five stars; a legend claims that when [[Saint John of Nepomuk]] was martyred, five stars appeared over his head.<ref name="SimitchWarke2014"/><ref>{{cite web |title=Zelená hora near Žďár nad Sázavou |url=http://www.czechtourism.com/c/zdar-nad-sazavou-unesco-zelena-hora-pilgrimage-church-of-st-john-nepomuk/ |publisher=Czech Tourism |access-date=10 November 2015}}</ref> The fivefold architecture may also symbolise the [[five wounds of Christ]] and the five letters of "Tacui" (Latin: "I kept silence" [about secrets of the [[confessional]]]).<ref>{{cite web |title=Attributes of Saint John of Nepomuk |url=http://www.sjn.cz/eng/attributes.htm |website=Saint John of Nepomuk |access-date=10 November 2015 |url-status=dead |archive-url=https://web.archive.org/web/20160304064831/http://www.sjn.cz/eng/attributes.htm |archive-date=4 March 2016 }}</ref>

[[Antoni Gaudí]] used a wide variety of geometric structures, some being minimal surfaces, in the [[Sagrada Família]], [[Barcelona]], started in 1882 (and not completed as of 2023). These include hyperbolic [[paraboloid]]s and [[hyperboloid|hyperboloids of revolution]],<ref name=Burry/> tessellations, [[catenary arch]]es, [[catenoid]]s, [[helicoid]]s, and [[ruled surface]]s. This varied mix of geometries is creatively combined in different ways around the church. For example, in the Passion Façade of Sagrada Família, Gaudí assembled stone "branches" in the form of hyperbolic paraboloids, which overlap at their tops (directrices) without, therefore, meeting at a point. In contrast, in the colonnade there are hyperbolic paraboloidal surfaces that smoothly join other structures to form unbounded surfaces. Further, Gaudí exploits [[patterns in nature|natural patterns]], themselves mathematical, with [[column]]s derived from the shapes of [[tree]]s, and [[lintel]]s made from unmodified [[basalt]] naturally cracked (by cooling from molten rock) into [[List of places with columnar jointed volcanics|hexagonal columns]].<ref>{{cite web |title=The Geometry of Antoni Gaudi |url=http://euler.slu.edu/escher/index.php/The_Geometry_of_Antoni_Gaudi |website=Math & the Art of MC Escher |publisher=Saint Louis University Mathematics and Computer Science |access-date=4 October 2015}}</ref><ref>{{cite web |last=Usvat |first=Liliana |title=Antony Gaudi and Mathematics |url=http://www.mathematicsmagazine.com/Articles/AntonyGaudiandMathematics.php#.VhFZ7WtUWHg |publisher=Mathematics Magazine |access-date=4 October 2015}}</ref><ref name=Burry>{{cite book |author=M.C. Burry |author2=J.R. Burry |author3=G.M. Dunlop |author4=A. Maher |contribution=Drawing Together Euclidean and Topological Threads |contribution-url=http://www.business.otago.ac.nz/SIRC05/conferences/2001/05_burry.pdf |year=2001 |title=The 13th Annual Colloquium of the Spatial Information Research Centr|publisher= University of Otago, Dunedin, New Zealand |access-date=5 August 2008 |archive-url = https://web.archive.org/web/20080625010337/http://www.business.otago.ac.nz/SIRC05/conferences/2001/05_burry.pdf |archive-date = 25 June 2008}}</ref>

The 1971 [[Cathedral of Saint Mary of the Assumption (San Francisco, California)|Cathedral of Saint Mary of the Assumption, San Francisco]] has a [[saddle roof]] composed of eight segments of hyperbolic paraboloids, arranged so that the bottom horizontal cross section of the roof is a square and the top cross section is a [[Christian cross]]. The building is a square {{convert|77.7|m|ft}} on a side, and {{convert|57.9|m|ft}} high.<ref>{{cite web |last1=Nervi |first1=Pier Luigi |title=Cathedral of Saint Mary of the Assumption |url=http://architectuul.com/architecture/cathedral-of-saint-mary-of-the-assumption |website=Architectuul |access-date=12 October 2015}}</ref> The 1970 [[Cathedral of Brasília]] by [[Oscar Niemeyer]] makes a different use of a hyperboloid structure; it is constructed from 16 identical concrete beams, each weighing 90 tonnes,{{efn|This is {{convert|90|t|ton}}.}} 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.<ref>{{cite web |url=http://www.aboutbrasilia.com/travel/brasilia-cathedral.html |title=Brasilia Cathedral |publisher=About Brasilia |access-date=13 November 2015}}</ref><ref name="BehrendsCrato2012">{{cite book |last1=Behrends |first1=Ehrhard |last2=Crato |first2=Nuno |last3=Rodrigues |first3=Jose Francisco |title=Raising Public Awareness of Mathematics |url=https://books.google.com/books?id=b3aIukYk6ccC&pg=PA143 |year=2012 |publisher=Springer Science & Business Media |isbn=978-3-642-25710-0 |page=143}}</ref><ref name="Emmer2012">{{cite book|last=Emmer|first=Michele |title=Imagine Math: Between Culture and Mathematics|url=https://books.google.com/books?id=qcaeLynFUzgC&pg=PA111 |year=2012 |publisher=Springer Science & Business Media |isbn=978-88-470-2427-4 |page=111}}</ref><ref>{{cite web |last1=Mkrtchyan |first1=Ruzanna|title=Cathedral of Brasilia |url=http://www.building.am/page.php?id=149 |publisher=Building.AM |access-date=13 November 2015 |date=2013}}</ref>

Several medieval [[Nordic round churches|churches in Scandinavia are circular]], including four on the Danish island of [[Bornholm]]. One of the oldest of these, [[Østerlars Church]] from {{circa|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.<ref>{{cite web |url=http://www.nordenskirker.dk/Tidligere/oesterlars_kirke/oesterlars_kirke.htm |title=Østerlars kirke |publisher=Nordens kirker |language=da |access-date=2 December 2016 |archive-date=2 April 2016 |archive-url=https://web.archive.org/web/20160402105645/http://nordenskirker.dk/Tidligere/oesterlars_kirke/oesterlars_kirke.htm |url-status=dead }}</ref>
<ref>{{cite web |url=http://www.naturbornholm.dk/default.asp?m=373 |title=Østerlars kirke |publisher=Natur Bornholm |language=da |access-date=2 December 2016 |url-status=dead |archive-url=https://web.archive.org/web/20110719142238/http://www.naturbornholm.dk/default.asp?m=373 |archive-date=19 July 2011 }}</ref>
<gallery>
File:Istanbul 036 (6498284165).jpg|The vaulting of the nave of [[Haghia Sophia]], Istanbul ''([[Commons:File:Istanbul 036 (6498284165).jpg|annotations]]''), 562
File:Battistero Firenze.jpg|The octagonal [[Florence Baptistery|Baptistry of Saint John, Florence]], completed in 1128
File:Jan Santini Aichel - Zelená Hora ground plan 2.jpg|Fivefold symmetries: [[Jan Santini Aichel]]'s [[Pilgrimage Church of St John of Nepomuk]] at Zelená hora, 1721
File:Sagfampassion.jpg|Passion façade of [[Antoni Gaudí]]'s [[Sagrada Família]], [[Barcelona]], started 1882
File:Catedral1 Rodrigo Marfan.jpg|[[Oscar Niemeyer]]'s [[Cathedral of Brasília]], 1970
File:St Mary's Cathedral - San Francisco.jpg|The [[Cathedral of Saint Mary of the Assumption (San Francisco, California)|Cathedral of Saint Mary of the Assumption, San Francisco]], 1971
File:Oesterlarsfresco.jpg|Central column of [[Østerlars Church|Østerlars]] [[Nordic round church]] in [[Bornholm]], Denmark
</gallery>

==Mathematical decoration==

===Islamic architectural decoration===
{{main|Islamic geometric patterns}}

Islamic buildings are often decorated with [[Islamic geometric patterns|geometric patterns]] which typically make use of several mathematical [[tessellations]], formed of ceramic tiles ([[girih]], [[zellige]]) that may themselves be plain or decorated with stripes.<ref name=StAndrewsArchitecture/> 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.<ref name=Ronning>{{cite web |url=http://people.exeter.ac.uk/PErnest/pome24/ronning%20_geometry_and_Islamic_patterns.pdf |title=Islamic Patterns And Symmetry Groups |publisher=University of Exeter |access-date=18 April 2014 |author=Rønning, Frode}}</ref> Islamic patterns exploit many of the 17 possible [[wallpaper groups]]; as early as 1944, Edith Müller showed that the Alhambra made use of 11 wallpaper groups in its decorations, while in 1986 [[Branko Grünbaum]] claimed to have found 13 wallpaper groups in the Alhambra, asserting controversially that the remaining four groups are not found anywhere in Islamic ornament.<ref name=Ronning/>

<gallery>
File:Mezquita Shah, Isfahán, Irán, 2016-09-20, DD 64 (cropped).jpg|The complex geometry and tilings of the [[muqarnas]] vaulting in the [[Sheikh Lotfollah Mosque]], [[Isfahan]], 1603–1619
File:Louvre Abu Dhabi under construction (cropped).jpg|[[Louvre Abu Dhabi]] under construction in 2015, its dome built up of layers of stars made of octagons, triangles, and squares

</gallery>

===Modern architectural decoration===

{{further|Ornament (art)|Contemporary architecture}}

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.<ref name=Ostwald48/> In 1913, the Modernist architect [[Adolf Loos]] had declared that "Ornament is a crime",<ref name=Gibberd>{{cite news |last1=Gibberd |first1=Matt |last2=Hill |first2=Albert |title=The Return of Ornamentation |url=https://www.telegraph.co.uk/luxury/property-and-architecture/7279/the-return-of-ornamentation.html |archive-url=https://web.archive.org/web/20141018120901/http://www.telegraph.co.uk/luxury/property-and-architecture/7279/the-return-of-ornamentation.html |url-status=dead |archive-date=18 October 2014 |access-date=12 October 2015 |work=The Telegraph |date=20 August 2013}}</ref> influencing architectural thinking for the rest of the 20th century. In the 21st century, architects are again starting to explore the use of [[Ornament (art)|ornament]]. 21st century ornamentation is extremely diverse. Henning Larsen's 2011 [[Harpa Concert and Conference Centre]], Reykjavik has what looks like a crystal wall of rock made of large blocks of glass.<ref name=Gibberd/> Foreign Office Architects' 2010 [[Ravensbourne College]], London is tessellated decoratively with 28,000 anodised aluminium tiles in red, white and brown, interlinking circular windows of differing sizes. The tessellation uses three types of tile, an equilateral triangle and two irregular pentagons.<ref>{{cite web|title=Ravensbourne College by Foreign Office Architects|url=http://www.dezeen.com/2010/09/13/ravensbourne-college-by-foreign-office-architects/|publisher=de zeen magazine|access-date=12 October 2015|date=13 September 2010}}</ref><ref name=Bizley/>{{efn|An aperiodic tiling was considered, to avoid the rhythm of a structural grid, but in practice a Penrose tiling was too complex, so a grid of 2.625m horizontally and 4.55m vertically was chosen.<ref name=Bizley>{{cite web |last1=Bizley |first1=Graham |title=FOA's peninsula patterns for Ravensbourne College |url=http://www.bdonline.co.uk/foa%E2%80%99s-peninsula-patterns-for-ravensbourne-college/3144928.article |website=bdonline.co.uk |access-date=16 October 2015}}</ref>}} Kazumi Kudo's [[Kanazawa Umimirai Library]] creates a decorative grid made of small circular blocks of glass set into plain concrete walls.<ref name=Gibberd/>

<gallery>
File:London MMB «T1 Ravensbourne College.jpg|[[Ravensbourne College]], London, 2010
File:Harpa.JPG|[[Harpa Concert and Conference Centre]], Iceland, 2011
File:Umimirai Library.jpg|[[Kanazawa Umimirai Library]], Japan, 2011
File:Museo Soumaya Plaza Carso V.jpg|[[Museo Soumaya]], México, 2011
</gallery>

==Defence==

===Europe===

{{further|Star fort}}

The architecture of [[fortification]]s evolved from [[medieval fortification|medieval fortresses]], which had high masonry walls, to low, symmetrical [[star fort]]s able to resist [[artillery]] bombardment between the mid-fifteenth and nineteenth centuries. The geometry of the star shapes was dictated by the need to avoid dead zones where attacking infantry could shelter from defensive fire; the sides of the projecting points were angled to permit such fire to sweep the ground, and to provide crossfire (from both sides) beyond each projecting point. Well-known architects who designed such defences include [[Michelangelo]], [[Baldassare Peruzzi]], [[Vincenzo Scamozzi]] and [[Sébastien Le Prestre de Vauban]].<ref>{{cite book |author=Duffy, C. |year=1975 |title=Fire & Stone, The Science of Fortress Warfare 1660–1860 |publisher=Booksales |isbn=978-0-7858-2109-0}}</ref><ref>{{cite book|last1=Chandler |first1=David |title=The Art of Warfare in the Age of Marlborough |date=1990 |publisher=Spellmount |isbn=978-0-946771-42-4}}</ref>

The architectural historian [[Siegfried Giedion]] argued that the star-shaped fortification had a formative influence on the patterning of the Renaissance [[Urban planning|ideal city]]: "The Renaissance was hypnotized by one city type which for a century and a half—from Filarete to Scamozzi—was impressed upon all utopian schemes: this is the star-shaped city."<ref>{{cite book |author=Giedion, Siegfried |author-link=Siegfried Giedion |title=Space, Time and Architecture |publisher=Harvard University Press |orig-year=1941 |date=1962 |page=43}}</ref>

<gallery>
File:Coevorden.jpg|[[Coevorden]] fortification plan. 17th century
File:Palmanova1600.jpg|[[Palmanova]], [[Italy]], a [[Republic of Venice|Venetian]] city within a [[star fort]]. 17th century
File:Neuf-Brisach 007 850.jpg|[[Neuf-Brisach]], [[Alsace]], one of the [[Fortifications of Vauban]]
</gallery>

===China===

[[File:HakkaYongding.jpg|thumb|A [[tulou]] in [[Yongding County]], [[Fujian province]]]]

In [[Chinese architecture]], the [[tulou]] of [[Fujian province]] are circular, communal defensive structures with mainly blank walls and a single iron-plated wooden door, some dating back to the sixteenth century. The walls are topped with roofs that slope gently both outwards and inwards, forming a ring. The centre of the circle is an open cobbled courtyard, often with a well, surrounded by timbered galleries up to five stories high.<ref>{{cite magazine |last=O'Neill |first=Tom |title=China's Remote Fortresses Lose Residents, Gain Tourists |url=http://news.nationalgeographic.com/news/2015/01/150102-hakka-china-tulou-fujian-world-heritage-culture-housing/ |archive-url=https://web.archive.org/web/20150102235850/http://news.nationalgeographic.com/news/2015/01/150102-hakka-china-tulou-fujian-world-heritage-culture-housing/ |url-status=dead |archive-date=2 January 2015 |magazine=[[National Geographic (magazine)|National Geographic]] |access-date=6 January 2017 |date=4 January 2015}}</ref>

==Environmental goals==

[[File:Yakhchal of Yazd province.jpg|thumb|left|[[Yakhchal]] in [[Yazd]], Iran]]

Architects may also select the form of a building to meet environmental goals.<ref name="SimitchWarke2014">{{cite book |last1=Simitch |first1=Andrea |last2=Warke |first2=Val |title=The Language of Architecture: 26 Principles Every Architect Should Know|url=https://books.google.com/books?id=2v-1AwAAQBAJ&pg=PT191 |year=2014 |publisher=Rockport Publishers |isbn=978-1-62788-048-0 |page=191}}</ref> For example, [[Foster and Partners]]' [[30 St Mary Axe]], London, known as "[[#Top|The Gherkin]]"<!--links to top (lead image)--> for its [[cucumber]]-like shape, is a [[solid of revolution]] designed using [[Computer aided design|parametric modelling]]. Its geometry was chosen not purely for aesthetic reasons, but to minimise whirling air currents at its base. Despite the building's apparently curved surface, all the panels of glass forming its skin are flat, except for the lens at the top. Most of the panels are [[quadrilateral]]s, as they can be cut from rectangular glass with less wastage than triangular panels.<ref name=Freiberger>{{cite web |last1=Freiberger |first1=Marianne |title=Perfect buildings: the maths of modern architecture |url=https://plus.maths.org/content/perfect-buildings-maths-modern-architecture|publisher=Plus magazine|access-date=5 October 2015|date=1 March 2007}}</ref>

The traditional [[yakhchal]] (ice pit) of [[Iran|Persia]] functioned as an [[evaporative cooler]]. Above ground, the structure had a domed shape, but had a subterranean storage space for ice and sometimes food as well. The subterranean space and the thick heat-resistant construction insulated the storage space year round. The internal space was often further cooled with [[windcatcher]]s.<ref>{{Cite web |last=Ayre |first=James |url=https://cleantechnica.com/2018/04/28/yakhchals-ab-anbars-wind-catchers-passive-cooling-refrigeration-technologies-of-greater-iran-persia/|title=Yakhchāls, Āb Anbārs, & Wind Catchers — Passive Cooling & Refrigeration Technologies Of Greater Iran (Persia) |date=28 April 2018 |website=[[CleanTechnica]] |access-date=25 February 2022 |archive-date=1 May 2018 |url-status=live |archive-url=https://web.archive.org/web/20180501174422/https://cleantechnica.com/2018/04/28/yakhchals-ab-anbars-wind-catchers-passive-cooling-refrigeration-technologies-of-greater-iran-persia/}}</ref>{{-}}

==See also==

* [[Black Rock City]]
* [[Mathematics and art]]
* [[Patterns in nature]]

==Notes==

{{notelist}}

==References==

{{reflist|28em}}

==External links==

* [http://www.nexusjournal.com Nexus Network Journal: Architecture and Mathematics Online]
* [http://www.isama.org The International Society of the Arts, Mathematics, and Architecture] {{Webarchive|url=https://web.archive.org/web/20160304082745/http://www.isama.org/ |date=4 March 2016 }}
* [https://web.archive.org/web/20040722075506/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Architecture.html University of St Andrews: Mathematics and Architecture]
* [https://web.archive.org/web/20150507151115/http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml National University of Singapore: Mathematics in Art and Architecture]
* [http://www.dartmouth.edu/~matc/math5.geometry/unit1/INTRO.html Dartmouth College: Geometry in Art & Architecture] {{Webarchive|url=https://web.archive.org/web/20200224094601/http://www.dartmouth.edu/%7Ematc/math5.geometry/unit1/INTRO.html |date=24 February 2020 }}

{{Mathematical art}}

[[Category:Mathematics and culture]]
[[Category:Architectural theory]]