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ArticleArticle MathematicsMathematics and and the the Islamic Islamic Architecture Architecture of of Córdoba Córdoba Felix Arnold Felix Arnold Ḫ ʾ German Archaeological Institute, Calle Serrano 159, 28002 Madrid, Spain; [email protected] German Archaeological Institute, Calle Serrano 159, Madrid 28002, Spain; [email protected]


Received:Received: 29 May 29 May 2018; 2018; Accepted: Accepted: 25 Ju 25ly July2018; 2018; Published: Published: 7 August 8 August 2018 2018


Abstract:Abstract: In 10th-centuryIn 10th-century Córdoba, Córdoba, mathematics—and mathematics—and particularly particularly geometry—was geometry—was applied applied to to architecturalarchitectural design design in new in new ways, ways, constituting constituting a “m aathematical “mathematical turn” turn” of Islamic of Islamic architecture. architecture. In the In the mosquemosque of Córdoba of Córdoba and and in inthe the palaces palaces of of Madīnat al-Zahrāʾ, geometry, geometry was was employed employed in thein the design designof ground of ground plans, plans, elevations, elevations, decorative decorative patterns, patterns, and even and to even measure to measure the human the view. human While view. Roman Whilearchitects Roman likearchitects Vitruvius like had Vitruvius used mathematics had used ma tothematics place each to elementplace each of aelement building of ina building its appropriate in its appropriaterelation to all relation other elements to all ot ofher a building,elements theof a architects building, at the Có rdobaarchitects employed at Córdoba geometry employed to create a geometryspatial to web create in which a spatial all parts web are in equalwhich to all each parts other are and equal part to of aeach single, other unified and space.part of The a single, architects unifiedof C óspace.rdoba The thus architects pointed theof wayCórdoba to new thus possibilities pointed the of designingway to new architecture, possibilities possibilities of designing which architecture,were to be possibilities tested further which by architects were to ofbe Gothic tested and further Renaissance by architects architecture, of Gothic though and toRenaissance different ends. architecture, though to different ends. Keywords: Mezquita de Córdoba; ; architectural proportions; geometry; Vitruvius; Keywords:interlocking Mezquita arches; ribbedde Córdoba; vault; muqarnas Madīnat. al-Zahrāʾ; architectural proportions; geometry; Vitruvius; interlocking arches; ribbed vault; muqarnaṣ

1. Introduction 1. IntroductionThe proclamation of the caliphate at Córdoba in 929 CE marked a turning point not only in politicalThe proclamation and ideological of the terms caliphate but also at inCórdoba the history in 929 of artCE and marked architecture. a turning With point the foundationnot only in of a politicalnew capitaland ideological city— terms but also —inin the 936 histor CE,y the of architectsart and architecture. of the caliphate With initiated the foundation the creation of of a newa novel capital way city—Mad of designing,īnat al-Zahr constructing,āʾ—in 936 and CE, decorating the architects buildings. of the caliphate Among the initiated signature the creation elements to of acome novel out way of of this designing, new “style” constructing, of designing and architecture decorating arebuildings. the interlocking Among the arch signature and the elements ribbed vault. to comeAn innovation out of this of new even “style” greater of consequence designing arch wasitecture the way are mathematics the interlocking was appliedarch and to the the ribbed design of vault.architecture. An innovation Mathematics of even hadgreater been consequence an integral elementwas the ofway architectural mathematics design was sinceapplied ancient to the times. designThe of architects architecture. of the Mathematics caliphate applied had been mathematics—and an integral element geometry of architectural in particular—to design since architecture ancient in times.new The ways, architects however, of which the signifiedcaliphate a mathematicalapplied mathematics—and turn in architectural geometry design. in The particular—to following paper architecturefocuses on in hownew geometryways, however, was applied which insignified different a mathematical ways to the design turn in of architectural architecture design. in 10th-century The followingCórdoba paper and focuses what was on how achieved geometry by doing was applied so. Taken in different together, ways these to innovativethe design of ways architecture of applying in 10th-centurymathematics Córdoba to the design and what of architecture was achieved came by to influencedoing so. notTaken only together, the further these development innovative ofways Islamic of applyingarchitecture mathematics in the region to butthe also design European of archit architectureecture came in general. to influence not only the further developmentSince of the Islamic 8th century, architecture crucial in advances the region had but been also madeEuropean in the architecture field of mathematics in general. at the court ofSince the Abbasidthe 8th century, caliphs incrucial Bagdad advances by combining had been the made knowledge in the field of Classical of mathematics Greek mathematiciansat the court 1 of thelike Abbasid Euclid withcaliphs innovations in Bagdad made by combining in India in thethe 6thknowledge and 7th century.of ClassicalAmong Greek these mathematicians was the use of a like decimalEuclid with numeral innovations system made with zero in India as a in digit, the the6th foundationsand 7th century. of algebra1 Among by al-theseHArabic-Indian These advances numeral quicklysystem reached was introduced the farthest to Cendsórdoba of the already Islamic bythe world, inventor including Ibn Firn Córdoba.as¯ (810–887/888).2 The Arabic-Indian At the court of numeralthe caliph, system the was mathematician introduced to Maslama Córdoba al-Maalreadygrˇ ¯ıt by¯ı (ca. the 950–1007/1008)inventor Ibn Firn editedās (810–887/888). the astronomic At the tables court of the caliph, the mathematician Maslama al-Maǧrītī (ca. 950–1007/1008) edited the astronomic tables of al-Ḫwārizmī, translated Ptolemy’s star chart and improved the translation of his Almagest 1 For an overview see ( Rashed 1984 ). 2 A detailed study of the mathematical knowledge in al-Andalus is still outstanding. 1 For an overview see (Rashed 1984). 2 A detailed study of the mathematical knowledge in al-Andalus is still outstanding.

Arts Arts20182018, 7, x;, 7doi:, 35; FOR doi: 10.3390/arts7030035PEER REVIEW www.mdpi.com/journal/artswww.mdpi.com/journal/arts Arts 2018, 7, x FOR PEER REVIEW 13 of 15 divide the roof of the central nave into segments that are equal in width to the naves of the side aisles. Pointed arches offer the possibility of creating a hierarchy of arches of different widths, independent of their height. Pointed arches and ribbed vaults become part of a logical system. In Islamic architecture, the same elements are employed to a very different end. The architects of Córdoba did not attempt to create a hierarchy but a web in which all parts would be equal to each other. The interlocking arch as employed in Islamic architecture does not create order but rather ambiguity, weaving together elements in a way that is beyond structural logic or even comprehension (cf. Figure 8 again). The ideal was the suspension of differences in space, the ultimate aim the blurring of the distinction between interior and exterior space.

6. Conclusions In 10th-century Córdoba, mathematics—and geometry in particular—was applied to the design of architecture in different ways and at different levels—to organize ground plans and elevations, as a way to measure the human field of view, and as a means of decoration. In all cases, the use of geometry was based on a specific, novel concept of space. Architectural space was conceived in an abstract way as a holistic unity. The difference to how Roman architects had employed mathematics in architecture highlights how revolutionary this new attitude toward architectural design truly was. In Roman architecture mathematics—in the form of arithmetic and more rarely geometric ratios— was employed to place each element in its appropriate relation to all other elements of the building and thus to achieve what Vitruvius called simmetria, or aesthetic harmony. The architects of Islamic Córdoba instead employed geometry to create a spatial web, in which all parts are equal to each other, and part of a single unified space. Both the advances made in the field of mathematics since the 9th century and the application of mathematics to architecture in the 10th century can be viewed as the result of what might be termed a “mathematicalArts 2018, 7, 35 turn” of Islamic culture. The universe was considered in a new, abstract way, as 2 of 15 governedArts 2018by abstract, 7, x FOR rulesPEER REVIEWand, in the case of art and architecture, by a web of regulating lines. This 2 of 15 new world view pointed the way toward new possibilities of designing architecture, possibilities that of al-H

Reference2. From Arithmetic to Geometric Design When ʿAbd al-Raḥmān I (r. 756–788) established himself as the Islamic ruler of the Iberian (Al-ʿUdrī 1965)When Al-ʿAbdUdrī, al-RaAḥmadh. m ibnan¯ ʿ IUmar. (r. 756–788) 1965. Fragmentos established Geográfico-Histórico himself as the de Islamic al-Masālik ruler ilā Ğ ofam theīʿ al- Iberian 3 MamPeninsula,ālik. Edited he by ordered ʿAbd al- theʿAzi constructionz̄ al-Ahwanī .̄ ofMadrid: a congregationalcong Institutoregational de Estudios mosque Islámicos. in Córdoba, Córdoba, his capital city. 3 His (Arias Páramoarchitects 2008) laid Arias out Páramo, the plan Lorenzo. of the 2008. mosque Geometría as a perfectype Proporciónrfect square,square, en la Arquitectura 150 cubitscubits Prerománica wide and 150Asturiana cubitscubits. long Anejos(Figure(Figure de1 1).Archivo). The The surface surface Español area area de Arqueología was was divided divided 49. into intoMadrid two two: parts:Consejo parts: the theSuperior northern northern de Investigaciones half half (75 (75× × 150 Científicas. cubits) was to be (Arnoldleftleft 2012) openopen Arnold, asas aa courtyard, courtyard,Felix. 2012. andArchitek and the thetur southern southern als kulturgeschichtliche half half (75 (75× ×150 150 cubits) cubits)Quelle. was Inwas Politische roofed roofed Räume over over in inin order Vormodernen order to to serve serve as as a Gesellschaften:prayera prayer hall. hall. Gestaltung, The The hall hall was Wahrnehmung,was to to be be divided divided Funktion into into.11 Edited11 naves naves by (10 (10Ortwin naves naves Dally, + + 11 central centralFriedericke nave).nave). Fless DividingDividing and Rudolf 150 cubits Haensch.by 11 would Rahden: have Leidorf, made pp. each 291–300. nave 13 7 cubits wide. To avoid such an awkward number, each nave by 11 would have made each nave 1311 cubits wide. To avoid such an awkward number, each nave (Arnoldwaswas 2017) designeddesigned Arnold, totoFelix. bebe 14142017. cubitscubits Islamic wide, Palace thethe Architecture twotwo outermostoutermos in thet Western naves 11 Mediterranean: cubits, and A the History central. New nave York: 16 cubits Oxford(8(8 × 1414 University + + 16 16 + +2 2× Press. ×11 11= 150). = 150). The The design design of the of theenti entirere mosque mosque waswas thus thus based based on simple on simple arithmetic. arithmetic. The (ArnoldThesame et al. same is2015) true is Arnold, truefor most for Felix, most ancient Alberto ancient Roman Ca Romannto buildings,García, buildings, and andAntonio andfor theVallejo for churches the Triano. churches that 2015. that were Munyat were built ar-Rummaniya: built in Asturias in Asturias at the at Ein Islamischer Landsitz bei Córdoba. 1. Palastanlagen. Madrider Beitrage 34. Mainz: Reichert. thetime time the themosque mosque was was constructed constructed at Córdoba. at Córdoba.4 4 (Arnold, forthcoming) Arnold, Felix. Forthcoming. Félix Hernández y el codo islámico. Cuadernos de Madinat al-Zahra 8. (Belting 2008) Belting, Hans. 2008. Florenz und Bagdad: Eine Westöstliche Geschichte des Blicks. Munich: C. H. Beck. (Binding 1993) Binding, Günther. 1993. Baubetrieb im Mittelalter. Darmstadt: Primus. (Bischoff 1981) Bischoff, Bernhard. 1981. Mittelalterliche Studien. Ausgewählte Aufsätze zur Schriftkunde und Literaturgeschichte 3. Stuttgart: Hiersemann.

Figure 1. Ground plan plan of of the the initial initial phase phase of of the the mosque mosque of of Córdoba, Córdoba, with with measurements measurements in incubits cubits (1 (1cubits cubits = 49.5 = 49.5 cm). cm). Drawing Drawing by byFelix Felix Arnold Arnold,, based based on on the the plan plan of ofFélix Félix Hernández. Hernández.

3 (Hernández Giménez 1961, Figure 3); (Ewert and Wisshak 1981, Figure 35). (Cf. Arnold√ Forthcoming√ ). Fernández-Puertas 3 ( (Hernández2000) has suggested Giménez that the1961, design Figure of the 3); original (Ewert phase and was Wisshak based on 1981, the numbers Figure √35).2 and (Cf.√5. Arnold, The use of forthcoming). a square—and dividingFernández-Puertas the square in (2000) two equal has halves—willsuggested that always the leaddesign todiagonals of the original measuring phase2 was and based5, but on this the does numbers not mean √2 that these geometric properties were taken into consideration (at least in the original phase). Fernández-Puertas(2008) observedand √5. The additional use of geometric a square—and relationships dividing in the the design square of the in original two equal phase halves—will of the mosque. always If true, theylead wouldto diagonals suggest thatmeasuring geometry √2 was and applied √5, but already this does two centuriesnot mean earlier that thanthese suggested geometric here. properties were taken into consideration 4 ((atArias least Pá ramoin the 2008 original). phase). Fernández-Puertas (2008) observed additional geometric relationships in the design of the original phase of the mosque. If true, they would suggest that geometry was applied already two centuries earlier than suggested here. 4 (Arias Páramo 2008).

Article Mathematics and the Islamic Architecture of Córdoba

Felix Arnold Ḫ ʾ

German Archaeological Institute, Calle Serrano 159, Madrid 28002, Spain; [email protected]

Arts 2018, 7, 35 Received: 29 May 2018; Accepted: 25 July 2018; Published: 7 August 2018 3 of 15 Arts 2018, 7, x FOR PEER REVIEW 3 of 15 Abstract: In 10th-century Córdoba, mathematics—and particularly geometry—was applied to In thethe 10th10th century, century, architects architecturalarchitects began began design to use to in geometryuse new ge ways,ometry in addition constituting in addition to arithmetic a “mto athematicalarithmetic to design turn”to buildings. design of Islamic architecture. In the buildings.An early example An early is theexample so calledmosque is Salthe ofó son Córdoba basilical called Salón inand the inbasilical caliphal the palaces residencein the of caliphal Madīnat residence al-Zahr āMadʾ, identified, īgeometrynat al- was employed in the Zahrby someāʾ, identified as the Dar¯ by al- someGundˇ design as “House the ofD ā ofgroundr al- theǦ Army”und plans, “House and elevations, dated of the to Army” ca.decorative 950 and CE. Thepatterns,dated building to ca.and 950 comprises even CE. to The measure a the human view. buildingthree-aisled comprises hall with a sidethree-aisledWhile chambers Roman hall as with well architects side as a courtyardchambers like Vitruvius withas well porticoshad as used a courtyard on ma threethematics sideswith 5 to.porticos The place width each on element of a building in threeof the sides hall5 was. The again width set of out theits tohallappropriate measure was again an relation evenset out number: to to allmeasure ot 50her cubits anelements even of46.3 number: of cma building, each 50 cubits (23.12–23.16 the of architects46.3 cm m). at Córdoba employed eachThehall (23.12–23.16 was divided m). intoThe threehallgeometry was naves, divided to each create 16into2 /a3 threespatialcubits naves, wideweb (7.76eachin which m).16⅔ The cubitsall hallparts wide is are not (7.76 equal square, m). to howeverThe each hall other and part of a single, is(Figure not square,2). The however depth measures (Figureunified only2). Thespace. 20.03–20.07 depth The measuresarchitects m, or about only of Córdoba 43.320.03–20.07 cubits. thus Them, po or designinted about the of 43.3 theway cubits. floor to new plan The possibilities of designing 6 was in fact based on an equilateralarchitecture, triangle, possibilities 50 cubits which wide were on eachto be side. testedThe further base by of architects the triangle6 of Gothic and Renaissance design of the floor plan was in fact based on an equilateral triangle, 50 cubits √wide on each side. The baseequals of thethe widthtriangle of theequals hall architecture,the (50 width cubits), of the thoughthe height hall to(50 of different cubits), the triangle ends.the height its depth—25 of the triangle3 cubits its ( ≈depth—2543.3 cubits),√3 cubitsan irrational (≈43.3 number.cubits), Thean irrational design of number. the hall is The thus design based notof the on simplehall is arithmetic,thus based but not on on geometry. simple arithmetic,The same is but true on for geometry. anotherKeywords: famousThe same hallMezquita is oftrue for de another Córdoba; famous—the Mad hall Salīnat óof n al-Zahr Rico,Madīnat whichāʾ ; al-Zahrarchitectural is 44āʾ cubits—the proportions; geometry; √ 7 Salón(20.26–20.31 Rico, which m) wide is 44 and cubits 22Vitruvius; (20.26–20.313 cubits interlocking (17.53 m) m) wide deep. arches; and 22 ribbed√3 cubits vault; (17.53 muqarna m) deep.ṣ 7

1. Introduction The proclamation of the caliphate at Córdoba in 929 CE marked a turning point not only in political and ideological terms but also in the history of art and architecture. With the foundation of a new capital city—Madīnat al-Zahrāʾ—in 936 CE, the architects of the caliphate initiated the creation of a novel way of designing, constructing, and decorating buildings. Among the signature elements to come out of this new “style” of designing architecture are the interlocking arch and the ribbed vault. An innovation of even greater consequence was the way mathematics was applied to the design of architecture. Mathematics had been an integral element of architectural design since ancient times. The architects of the caliphate applied mathematics—and geometry in particular—to architecture in new ways, however, which signified a mathematical turn in architectural design. The following paper focuses on how geometry was applied in different ways to the design of architecture in 10th-century Córdoba and what was achieved by doing so. Taken together, these innovative ways of applying mathematics to the design of architecture came to influence not only the further development of Islamic architecture in the region but also European architecture in general.

Since the 8th century, crucial advances had been made in the field of mathematics at the court Ǧ Figure 2. 2. ApplicationApplication ofof ofequilateralthe equilateral Abbasid triangles trianglescaliphs in thein in Bagdaddesign the design of by the combiningof hall the of hallthe D oftheār theal- knowledge Dundar¯ al-at GundMadˇ ofī natClassical at Greek mathematicians ʾ al-Zahrā , with measurements, withlike Euclid measurements in cubits with (1innovations incubit cubits = 46.3 (1 made cubitcm). Drawing =in 46.3 India cm). by in Felixthe Drawing 6th Arnold, and by 7th Felixbased century. Arnold, on the1 Among these was the use of planbased by on Ana the Zamorano. plan by Anaa decimal Zamorano. numeral system with zero as a digit, the foundations of algebra by al-Ḫwārizmī (Latinized Algoritmi, ca. 780–846) and the use of the trigonometric functions sine and cosine. These advances The use of equilateral triangles had a practical advantage. One of the challenges in setting out a The use of equilateralquickly triangles reached had athe practical farthest advantage. ends of the One Islamic of the challengesworld, including in setting Córdoba. out 2 The Arabic-Indian floor plan at the building site is establishing right angles. In order to set out a right angle Roman a floor plan at the buildingnumeral site is system establishing was introduced right angles. to Córd In orderoba toalready set out by a the right inventor angle RomanIbn Firnās (810–887/888). At the architects—and the builders of 9th- and 10th-century Asturias—made use of a Pythagorean triangle architects—and the builderscourt of of 9th- the and caliph, 10th-century the mathematician Asturias—made Maslama use al-Ma of a Pythagoreanǧrītī (ca. 950–1007/1008) triangle edited the astronomic (known as norma) with sides measuring 3, 4, and 5 units.8 Such a triangle can be set out easily using (known as norma) with sidestables measuring of al-Ḫwā 3,rizm 4, andī, translated 5 units.8 SuchPtolemy’s a triangle star chart can be and set improved out easily usingthe translation of his Almagest sticks or ropes marked in corresponding units. This procedure has two disadvantages, however. The sticks or ropes marked in corresponding units. This procedure has two disadvantages, however. triangle has to be set out at each corner of the building, thus increasing potential inaccuracies. A The triangle has to be set1 out For at an each overview corner see of (Rashed the building, 1984). thus increasing potential inaccuracies. Pythagorean triangle that is 3 units wide and 4 units long furthermore often does not coincide with A Pythagorean triangle that2 isA 3detailed units wide study and of the 4 unitsmathematical long furthermore knowledge oftenin al-Andalus does not is coincide still outstanding. with the proportions of the whole floor plan and the directions gained thus need to be extrapolated to the proportions of the wholeArts 2018 floor, 7, x; plan doi: FOR and PEER the REVIEW directions gained thus need to be extrapolated to www.mdpi.com/journal/arts greater distances, creating an additional potential source of error. Equilateral triangles can be set out on the building site even more easily than a Pythagorean triangle by stretching out three ropes of

5 ( Vel á zquez Bosco 1923 ; Vallejo Triano 2010, pp. 485–501, Figure 42; Arnold 2017, pp. 73–83). 56 (VelázquezRiobóo Camacho Bosco(2016 1923;) correctly Vallejo points Triano out 2010, that the pp. proportions 485–501, Figure are equal 42; to Arnold those of 2017, a hexagon. pp. 73–83). 67 Riobóo(Vallejo TrianoCamacho 2010 ,(2016) p. 486, correctly Figure 57; points Arnold out 2012 that; Arnold the proportions 2017, p. 89, Figure are equal 2.34. to Cf. those Riobó ofo Camachoa hexagon. 2015 , Figure 17; 7 (VallejoRiobóo Camacho Triano 20162010,). Thep. 486, equilateral Figure triangle 57; Arnold also became 2012; important Arnold 2017, in Romanesque p. 89, Figure and Gothic 2.34. architecture,Cf. Riobóo though Camacho how far it was used for determining the proportion of buildings (ad triangulum)—particularly in the elevation of churches—is 2015,still being Figure debated. 17; Riobóo (Cf. Hecht Camacho 1969; Wu 2016). 2002 ).The equilateral triangle also became important in Romanesque and 8 GothicVitruvius architecture, IX, prologue; though (Binding how 1993 far, p. it 344; was Arias used P áforramo determining 2008, pp. 34–36). the proportion of buildings (ad triangulum)— particularly in the elevation of churches—is still being debated. (Cf. Hecht 1969; Wu 2002). 8 Vitruvius IX, prologue; (Binding 1993, p. 344; Arias Páramo 2008, pp. 34–36).

Article Mathematics and the Islamic Architecture of Córdoba Arts 2018, 7, 35 4 of 15 Felix Arnold Ḫ ʾ German Archaeological Institute,greaterArts Calle 2018 Serrano distances,, 7, x FOR 159, PEER creating Madrid REVIEW 28002, an additional Spain; [email protected] potential source of error. Equilateral triangles can be set4 of out 15 Received: 29 May 2018; Accepted:on 25 the Ju buildingly 2018; Published: site even 7 moreAugust easily 2018 than a Pythagorean triangle by stretching out three ropes of equalequal length.length. If the midpoints of two sides are marked and a triangle of equal kind is flippedflipped around thesethese points,points, eacheach cornercorner marksmarks thethe limitslimits ofof aa prefectprefect rectangle.rectangle. TheThe onlyonly disadvantagedisadvantage is thatthat thethe Abstract: In 10th-century Córdoba, mathematics—and particularly geometry—was applied√ to√ width of the rectangle is not equal to its length, but measures only (about3 0.866) of the length. The architectural design in new ways,width constituting of the rectangle a “m isathematical not equal turn” to its length,of Islamic but architecture. measures only In the2 (about 0.866) of the length. mosque of Córdoba and in Thearchitectsthe architectspalaces of ofMad of Madīnatī natal-Zahr al-Zahrāʾ appearāʾ appear , geometry to have to have accepted was accepted employed this this disadvantage, in disadvantage, the as the as thehall hall of the ofthe Dār D al-ar¯ design of ground plans, elevations,al-ǦundGundˇ shows. shows.decorative9 9 patterns, and even to measure the human view. While Roman architects like VitruviusTheThe architect had used of ma the thematics D Dāar¯r al- al-ǦGundˇund to place applied applied each the theelement equilateral equilateral of a building triangle triangle into to other other parts parts of ofthe the building building as well (Figure 3). The hall, including its side chambers, its vestibule and the platform in front of the its appropriate relation to allas wellother (Figure elements3). Theof a hall, building, including the itsarchitects side chambers, at Córdoba its vestibule employed√ and the platform in front of the √3 geometry to create a spatialvestibule web in which also occupies all parts a surfaceare equal area to witheach the other proportion and part of of 1: a single,. The same holds true for the surface vestibule also occupies a surface area with the proportion of 1: 2 . The same holds true for the surface unified space. The architectsareaarea of ofCórdobaof thethe adjoiningadjoining thus po courtyard. courtyard.inted the The wayThe sizes sizesto new of of the possibilitiesthe three three areas areas of are aredesigning in in fact fact related related to eachto each other other and and are are all architecture, possibilities whichbased were on anto overarchingbe tested further design. by The architects architect of apparentlyGothic and first Renaissance set out a large equilateral triangle, each all based on an overarching design. The √architect apparently first set out a large equilateral triangle, architecture, though to differentsideeach ends. measuringside measuring 200 cubits, 200 itscubits, height its 100 height3 cubits. 100√ The3 cubits. base length The base was thenlength divided was arithmeticallythen divided intoarithmetically 12 units (12a), into each12 units 162 /(12a),3 cubits each wide. 16⅔ Thecubits naves wide. of theThe hallnaves and ofits the side hall chambers and its side each chambers occupy Keywords: Mezquita de Córdoba; Madīnat al-Zahrāʾ; architectural proportions; geometry; oneeach such occupy unit one (1a such = 162 unit/3 cubits). (1a = 16 The⅔ cubits). hall at theThe tip hall of at the the triangle tip of the occupies triangle an occupies area that an is threearea that units is Vitruvius; interlocking arches; ribbed vault; muqarnaṣ widethree (3aunits = 50wide cubits), (3a = the50 cubits), hall including the hall its including side chambers, its side itschambers, vestibule, its and vestibule, the platform and the in platform front of 1 thein front vestibule of the an vestibule area that an is fivearea units that is wide five (5a units = 83 wide/3 cubits). (5a = 83 The⅓ cubits). courtyard The occupies courtyard an occupies area that an is sevenarea that units is seven wide (7aunits = 116wide2/3 (7acubits). = 116⅔ cubits).

1. Introduction The proclamation of the caliphate at Córdoba in 929 CE marked a turning point not only in political and ideological terms but also in the history of art and architecture. With the foundation of a new capital city—Madīnat al-Zahrāʾ—in 936 CE, the architects of the caliphate initiated the creation of a novel way of designing, constructing, and decorating buildings. Among the signature elements to come out of this new “style” of designing architecture are the interlocking arch and the ribbed vault. An innovation of even greater consequence was the way mathematics was applied to the design of architecture. Mathematics had been an integral element of architectural design since ancient times. The architects of the caliphate applied mathematics—and geometry in particular—to architecture in new ways, however, which signified a mathematical turn in architectural design. The following paper focuses on how geometry was applied in different ways to the design of architecture in 10th-century Córdoba and what was achieved by doing so. Taken together, these innovative ways of applying mathematics to the design of architecture came to influence not only the further development of Islamic architecture in the region but also European architecture in general. Since the 8th century, crucial advances had been made in the field of mathematics at the court of the Abbasid caliphs in Bagdad by combining the knowledge of Classical Greek mathematicians like Euclid with innovations made in India in the 6th and 7th century.1 Among these was the use of Figure 3. Design scheme of the Dār al-Ǧund, based on the use of equilateral triangles. Drawing by a decimal numeral system with zeroFigure as a 3.digit,Design the schemefoundations of the Dofar¯ algebra al-Gund,ˇ by based al-Ḫw onārizm the useī (Latinized of equilateral triangles. Drawing by Algoritmi, ca. 780–846) and the useFelixFelix of Arnold,theArnold, trigonom basedbased onetricon thethe functions planplan byby AnaAna sine Zamorano.Zamorano. and cosine. These advances quickly reached the farthest ends of the Islamic world, including Córdoba. 2 The Arabic-Indian The design of the Dār al-Ǧund is remarkable not only for the use of geometry but also for the numeral system was introduced to Córdoba already by¯ the inventorˇ Ibn Firnās (810–887/888). At the way Thegeometry design is of used the in Dar the al- designGund isprocess. remarkable The Romans not only had for established the use of geometrythe proportions but also of forspaces the court of the caliph, the mathematician Maslama al-Maǧrītī (ca. 950–1007/1008) edited the astronomic wayusing geometry arithmetic is usedand inat thetimes design geometric process. means The Romans for aesthetic had established purposes the (what proportions Vitruvius of spacescalled tables of al-Ḫwārizmī, translatedusing Ptolemy’s arithmetic star and chart at times and geometricimproved means the translation for aesthetic of his purposes Almagest (what Vitruvius called symmetria symmetria and eurythmia)—calculating the proportion of each individual space at a time.10 In the Dār eurythmia 10 ¯ ˇ andal-Ǧund, the proportion)—calculating resulting the proportion from the ofuse each of eq individualuilateral triangles space at aapply time. notIn only the Dtoar individual al-Gund, 1 For an overview see (Rashed 1984). thespaces proportion or structures resulting but to from the thebuilding use of complex equilateral as a triangles whole. The apply architect not only applied to individual the proportions spaces orto 2 A detailed study of the mathematical knowledge in al-Andalus is still outstanding. spaces that can hardly be appreciated by the user of the building—the equilateral triangles on which Arts 2018, 7, x; doi: FOR PEER REVIEWthe design is based are not recognizable once www.mdpi.com/journal/artsthe building has been constructed. Geometry was 9 neitherThe examples used for given an aesthetic here were allpurpose set out on nor the with horizontal regard surface to construction of a terrace. On but a slope to create the architects order in would the haveground had additional problems. √ 10 Vitruvius I, 2 and III, 1; ( Wilson Jones 2000 ). For the use of the geometric proportion 1: 2 (diagonal of a square) see Vitruvius VI, 3. 9 The examples given here were all set out on the horizontal surface of a terrace. On a slope the architects would have had additional problems. 10 Vitruvius I, 2 and III, 1; (Wilson Jones 2000). For the use of the geometric proportion 1:√2 (diagonal of a square) see Vitruvius VI, 3.

Arts 2018, 7, 35 5 of 15 structures but to the building complex as a whole. The architect applied the proportions to spaces that can hardly be appreciated by the user of the building—the equilateral triangles on which the design is based are not recognizableArticle once the building has been constructed. Geometry was neither used for an aesthetic purpose nor with regard to construction but to create order in the ground plan. The order was applied to spacesMathematics that Vitruvius would have and treated the as separateIslamic entities—an Architecture interior of Córdoba hall and a courtyard. The architect of the Dar¯ al-Gundˇ not only treated these two spatial elements as Felix Arnold Ḫ ʾ one single space, but used geometry to bind them together, and in a way overcoming the difference between interior and exterior space—theGerman Archaeological approach that Institute, would become Calle Serrano of particular 159, Madrid importance 28002, Spain; in the [email protected] 11 architecture of the subsequent 11thReceived: century. 29 May 2018; Accepted: 25 July 2018; Published: 7 August 2018 3. Toward an Abstract Notion of Space Abstract: In 10th-century Córdoba, mathematics—and particularly geometry—was applied to In the Dar¯ al-Gund,ˇ equilateralarchitectural triangles design were applied in new alsoways, to constituting the design of a arcades,“mathematical and this turn” is of Islamic architecture. In the where the difference with Romanmosque design methodsof Córdoba becomes and in most the apparent.palaces of At Madīnat al-Zahrāʾ, the, geometry was employed in the design of the columns themselvesdesign actually of adheres ground rather plans, closely elevations, to Roman decorative prototypes. patterns,12 The and columns even to measure the human view. are composed of a base, a shaft andWhile a capital, Roman and architects the capital like is Vitruvius either of Corinthian had used ma orthematics Composite to type, place each element of a building in though executed in a contemporary,its appropriate often highly relation intricate to style. all ot Theher proportions elements of of a thebuilding, columns the and architects at Córdoba employed their parts furthermore often resemblegeometry those to create of the a Corinthian spatial web order in which of Roman all parts architecture. are equal Thus, to each other and part of a single, the height of the capital usually equalsunified the space. width The of architects the abacus, of like Córdoba in Roman thus prototypes. pointed the The way ratio to new possibilities of designing between the height of the columnarchitecture, and that of its possibilities shaft is about which 6:5 (1.2:1).were to And be tested the columns further are by usually architects of Gothic and Renaissance 9.5 times as high as they are thickarchitecture, (see Table1 ).though The architects to different of ends. may either have copied these proportions from examples of the Roman period preserved in the city of Córdoba or may actually have read the text of VitruviusKeywords: regarding Mezquita the classic de Córdoba; orders13. TheMad generalīnat al-Zahr characteristicsāʾ; architectural of proportions; geometry; columns remained the same throughoutVitruvius; the interlocking 10th century arches; and ribbed were changed vault; muqarna only graduallyṣ in the ensuing 11th century, with capitals becoming twice as high, and columns more slender, with a ratio between thickness and height of 1:11–12 instead of 1:9.5. In contrast to the principles expounded by Vitruvius, the design of the arcades of the Dar¯ al-Gundˇ was not derived from the proportions1. Introduction of columns but from overarching considerations (Figure4). The opening to the lateral naves of the hall is 10 cubits wide and is divided by columns into two bays, The proclamation of the caliphate at Córdoba in 929 CE marked a turning point not only in each 5 cubits wide. The height of the arcade was determined by an equilateral triangle laid on its political and ideological terms but also in the history of art and architecture. With the foundation of side: the center of the triangle marks the height of the columns including base, shaft, capital, and a new capital city—Madīnat al-Zahrāʾ—in 936 CE, the architects of the caliphate initiated the creation impost ( √10 cubits), the tip the height of the alfiz—the rectangle surrounding the arches ( √20 cubits). 3 of a novel way of designing, constructing, and decorating buildings.3 Among the signature elements The entrance to the central naveto ofcome the hallout isof dividedthis new into “style” three of bays. designing The columns architecture have are the the same interlocking arch and the ribbed √10 height as in the lateral arcades: vault.3 cubits. An innovation The spacing of of even the columns greater isconsequence determined bywas an the equilateral way mathematics was applied to the √10 triangle whose height correspondsdesign to the of heightarchitecture. of the columnsMathematics ( 3 cubits)had been and an whose integral base element spans of two architectural design since ancient times.1 The architects of the caliphate applied mathematics—and geometry in particular—to bays (62/3 cubits). Each bay is thus 3 /3 cubits wide instead of 5 cubits, although the column height is the same in both arcades. architecture in new ways, however, which signified a mathematical turn in architectural design. The following paper focuses on how geometry was applied in different ways to the design of architecture in 10th-century Córdoba and what was achieved by doing so. Taken together, these innovative ways of applying mathematics to the design of architecture came to influence not only the further development of Islamic architecture in the region but also European architecture in general. Since the 8th century, crucial advances had been made in the field of mathematics at the court of the Abbasid caliphs in Bagdad by combining the knowledge of Classical Greek mathematicians like Euclid with innovations made in India in the 6th and 7th century.1 Among these was the use of a decimal numeral system with zero as a digit, the foundations of algebra by al-Ḫwārizmī (Latinized Algoritmi, ca. 780–846) and the use of the trigonometric functions sine and cosine. These advances quickly reached the farthest ends of the Islamic world, including Córdoba. 2 The Arabic-Indian numeral system was introduced to Córdoba already by the inventor Ibn Firnās (810–887/888). At the court of the caliph, the mathematician Maslama al-Maǧrītī (ca. 950–1007/1008) edited the astronomic 11 For example in the design of the Aljafertablesía ( Ewertof al- 1978Ḫw,ā p.rizm 22, n.ī, 141,translated plan 1; Arnold Ptolemy’s 2017, pp. star 169–72, chart Figure and 3.27–28). improved the translation of his Almagest 12 For the Roman prototypes see (Wilson Jones 2000, pp. 135–57). 13 Vitruvius was known and read already at the court of Charlemagne (r. 768–814). (Bischoff 1981, pp. 277–97). 1 For an overview see (Rashed 1984). 2 A detailed study of the mathematical knowledge in al-Andalus is still outstanding.

Arts 2018, 7, x; doi: FOR PEER REVIEW www.mdpi.com/journal/arts Arts 2018, 7, x FOR PEER REVIEW 13 of 15 divide the roof of the central nave into segments that are equal in width to the naves of the side aisles. Pointed arches offer the possibility of creating a hierarchy of arches of different widths, independent of their height. Pointed arches and ribbed vaults become part of a logical system. In Islamic architecture, the same elements are employed to a very different end. The architects of Córdoba did not attempt to create a hierarchy but a web in which all parts would be equal to each other. The interlocking arch as employed in Islamic architecture does not create order but rather ambiguity, weaving together elements in a way that is beyond structural logic or even comprehension (cf. Figure 8 again). The ideal was the suspension of differences in space, the ultimate aim the blurring of the distinction between interior and exterior space.

6. Conclusions In 10th-century Córdoba, mathematics—and geometry in particular—was applied to the design of architecture in different ways and at different levels—to organize ground plans and elevations, as a way to measure the human field of view, and as a means of decoration. In all cases, the use of geometry was based on a specific, novel concept of space. Architectural space was conceived in an abstract way as a holistic unity. The difference to how Roman architects had employed mathematics in architecture highlights how revolutionary this new attitude toward architectural design truly was. In Roman architecture mathematics—in the form of arithmetic and more rarely geometric ratios— was employed to place each element in its appropriate relation to all other elements of the building and thus to achieve what Vitruvius called simmetria, or aesthetic harmony. The architects of Islamic Córdoba instead employed geometry to create a spatial web, in which all parts are equal to each Artsother,2018 and, 7part, 35 of a single unified space. 6 of 15 Both the advances made in the field of mathematics since the 9th century and the application of mathematics to architecture in the 10th century can be viewed as the result of what might be termed 14 a “mathematical turn” of Islamic culture. The universeTable was 1. Measurements considered in a andnew, proportionsabstract way, of as columns of the 10th and 11th century. governed by abstract rules and, in the case of art and architecture, by a web of regulating lines. This Capital Capital Column Column Column new worldBuilding view pointed the way Date toward CE new possibilities of designingAbaco Widtharchitecture, possibilities that Shaft Height Shaft Diameter were to be tested further by the architects of GoHeightthic architecture and eventuallyHeight/Abaco the Renaissance, Width Height Height/Shaft Height Height/Diameter thoughDar¯ in al-Mulk, different entrance ways. 940 21.5 21.5 1.0 156 121.5 1.28 18 8.7 Dar¯ al-Mulk, hall 940 21 22.5 0.9 182.5 148.5 1.23 19 9.6 Funding:Casa This de research la Alberca received no external c. 940 funding. 33.5 33.5 1.0 235.5 185 1.27 26 9.1 Dar¯ al-Gundˇ c. 950 42 41.5 1.0 263 202.5 1.30 33.5 7.9 ConflictsPavillon of Interest: of the portico The author declares c. 950 no conflict of interest. 27 27 1.0 237.5 191 1.24 24.5 9.7 Salón Rico, entrance 953–957 37 37 1.0 232.5 175 1.33 28 8.3 ReferenceSalón Rico, naves 953–957 43 42 1.0 257 191 1.35 33 7.8 Salón Rico, portico 953–957 44.5 42.5 1.1 287 218 1.32 32 9.0 (Al-ʿUdrCasaī 1965) de Al-Gaˇ ʿfarUdrī, Aḥmad ibnc. 961ʿUmar. 1965. Fragmentos 40.5 Geográfico-Histórico 38.5 de al-Masālik il 1.1ā Ğamīʿ al- 243.5 182.5 1.33 29.5 8.3 MamMosque,ālik. Edited extension by ʿAbd al-ʿAzi 961–971z̄ al-Ahwanī .̄ Madrid: 52.5 Instituto de Estudios 55 Islámicos. 1.0 400 301 1.33 42.5 9.4 (Arias DPáramoar¯ al-Mulk, 2008) annex Arias Páramo, Lorenzo. 972 2008. Geometría 22 y Proporción en la 21.5 Arquitectura Prerománica 1.0 Asturiana. 183.5 148 1.24 19 9.7 Málaga, entrance 1023 or later 19 22 0.9 200 181 1.10 17.5 11.4 Anejos de Archivo Español de Arqueología 49. Madrid: Consejo Superior de Investigaciones Científicas. Málaga, pavillon 1023 or later 22.5 27 0.8 217 195 1.11 18 12.1 (ArnoldAljafer 2012)í Arnold,a, entrance Felix. 2012. 1046Architek or latertur als kulturgeschichtliche 59 Quelle. 30 In Politische Räume in Vormodernen 2.0 286 179 1.60 25 11.4 Gesellschaften:Aljafería, portico Gestaltung, Wahrnehmung, 1046 or later Funktion. Edited 47 by Ortwin Dally, 27 Friedericke Fless 1.7and Rudolf 280 201 1.39 23 12.2 Haensch. Rahden: Leidorf, pp. 291–300. (Arnold 2017) Arnold, Felix. 2017. Islamic Palace Architecture in the Western Mediterranean: A History. New York: Oxford University Press. (Arnold et al. 2015) Arnold, Felix, Alberto Canto García, and Antonio Vallejo Triano. 2015. Munyat ar-Rummaniya: Ein Islamischer Landsitz bei Córdoba. 1. Palastanlagen. Madrider Beitrage 34. Mainz: Reichert. (Arnold, forthcoming) Arnold, Felix. Forthcoming. Félix Hernández y el codo islámico. Cuadernos de Madinat al-Zahra 8. (Belting 2008) Belting, Hans. 2008. Florenz und Bagdad: Eine Westöstliche Geschichte des Blicks. Munich: C. H. Beck. (Binding 1993) Binding, Günther. 1993. Baubetrieb im Mittelalter. Darmstadt: Primus. (Bischoff 1981) Bischoff, Bernhard. 1981. Mittelalterliche Studien. Ausgewählte Aufsätze zur Schriftkunde und Literaturgeschichte 3. Stuttgart: Hiersemann.

Arts 2018, 7, x FOR PEER REVIEW 5 of 15

plan. The order was applied to spaces that Vitruvius would have treated as separate entities—an interior hall and a courtyard. The architect of the Dār al-Ǧund not only treated these two spatial elements as one single space, but used geometry to bind them together, and in a way overcoming the difference between interior and exterior space—the approach that would become of particular importance in the architecture of the subsequent 11th century.11

3. Toward an Abstract Notion of Space In the Dār al-Ǧund, equilateral triangles were applied also to the design of arcades, and this is where the difference with Roman design methods becomes most apparent. At Madīnat al-Zahrāʾ, the design of the columns themselves actually adheres rather closely to Roman prototypes. 12 The columns are composed of a base, a shaft and a capital, and the capital is either of Corinthian or Composite type, though executed in a contemporary, often highly intricate style. The proportions of the columns and their parts furthermore often resemble those of the Corinthian order of Roman architecture. Thus, the height of the capital usually equals the width of the abacus, like in Roman prototypes. The ratio between the height of the column and that of its shaft is about 6:5 (1.2:1). And the columns are usually 9.5 times as high as they are thick (see Table 1). The architects of Madīnat al- Zahrāʾ may either have copied these proportions from examples of the Roman period preserved in the city of Córdoba or may actually have read the text of Vitruvius regarding the classic orders13. The general characteristics of columns remained the same throughout the 10th century and were changed only gradually in the ensuing 11th century, with capitals becoming twice as high, and columns more slender, with a ratio between thickness and height of 1:11–12 instead of 1:9.5. In contrast to the principles expounded by Vitruvius, the design of the arcades of the Dār al- Ǧund was not derived from the proportions of columns but from overarching considerations (Figure 4). The opening to the lateral naves of the hall is 10 cubits wide and is divided by columns into two bays, each 5 cubits wide. The height of the arcade was determined by an equilateral triangle laid on its side: the center of the triangle marks the height of the columns including base, shaft, capital, and impost ( cubits), the tip the height of the alfiz—the rectangle surrounding the arches ( cubits). √ √ The entrance to the central nave of the hall is divided into three bays. The columns have the same height as in the lateral arcades: cubits. The spacing of the columns is determined by an equilateral √ triangle whose height corresponds to the height of the columns ( cubits) and whose base spans two √

Artsbays2018 (6⅔, 7 ,cubits). 35 Each bay is thus 3⅓ cubits wide instead of 5 cubits, although the column height7 of 15is the same in both arcades.

Article Mathematics and the Islamic Architecture of Córdoba

Felix Arnold Ḫ ʾ

German Archaeological Institute, Calle Serrano 159, Madrid 28002, Spain; [email protected]

Received: 29 May 2018; Accepted: 25 July 2018; Published: 7 August 2018

Abstract: In 10th-century Córdoba, mathematics—and particularly geometry—was applied to architectural design in new ways, constitutingFigure a “m athematical4. 4. DesignDesign scheme schemeturn” of ofthe of Islamic arcades the arcades architecture. at the atentrance the In entrance tothe the hall to of the the hall Dār of al- theǦund D ar¯at Mad al-Gundˇ īnat al- at mosque of Córdoba and in the palaces of MadZahrāīnatʾ, with al-Zahr measurementsāʾ, with , geometry measurements in cubits was (1 cubit employed in cubits = 46.3 (1cm). cubitin theDrawings = 46.3 cm). by Felix Drawings Arnold. by Felix Arnold. design of ground plans, elevations, decorative patterns, and even to measure the human view.

While Roman architects like Vitruvius had usedThe ma spacingthematics of the to columnsplace each was element thus established of a building by in geometric means on the basis of the overall 11 For example in the design of the Aljafería (Ewert 1978, p. 22, n. 141, plan 1; Arnold 2017, pp. 169–72, Figure its appropriate relation to all other elementsproportions of a building, of the arcades. the architects Such a procedureat Córdoba would employed have been unthinkable for Roman architects. 3.27–28). geometry to create a spatial web in which all parts are equal to each other and part of a single, 12For For Vitruvius, the Roman the prototypes proportional see (Wilson relationship Jones 2000, of each pp. element135–57). of a building to the building as a whole unified space. The architects of Córdoba thus pointed the way to new possibilities of designing was13 Vitruvius fixed and was could known not and be read changed. already In at histhe cour treatiest of CharlemagneDe architectura (r. 768–814).(Vitruvius (Bischoff), he used 1981, the pp. analogy 277–97). architecture, possibilities which were to ofbe the tested human further body, by inarchitects which the of Gothic size of and each Renaissance limb has a specific relation to the size of the whole architecture, though to different ends. body15. Vitruvius insisted on the use of a basic unit (modulus) for designing buildings, equivalent to the thickness of a column at its lower end. He thus proceeded from the individual element—the Keywords: Mezquita de Córdoba; Madcolumn—toīnat al-Zahr theā designʾ; architectural of the whole proportions; building. Thisgeometry; is not the case in the Dar¯ al-Gund:ˇ the architect Vitruvius; interlocking arches; ribbed vault;started muqarna fromṣ the design of the complete building complex, subdividing it by geometric means until he arrived at the dimensions of smaller elements (Figure4). The size and proportion of individual columns was thus not related in a fixed way to the design of the whole building and not even to the design of the arcade of which it was part. 1. Introduction Vitruvius worked from the premise of the dignity of each individual element, which was to be ˇ The proclamation of the caliphate atachieved Córdoba by in keeping 929 CE all marked other parts a turning of the buildingpoint not in only proportion. in The architect of the Dar¯ al-Gund had political and ideological terms but also ina the more histor abstracty of art approach and architecture. to architecture With andthe architecturalfoundation of space. In an innovative way, he proceeded 16 a new capital city—Madīnat al-Zahrāʾ—infrom 936 CE, the the idea architects of the unity of the of caliphate space, to whichinitiated he the subordinated creation the role of individual parts. The way ˇ of a novel way of designing, constructing,the and D ar¯de al-coratingGund buildings. was designed Among liberated the signature the architect elements from the constraints of Roman architecture as to come out of this new “style” of designingdefined arch byitecture Vitruvius are andthe pointedinterlocking the way arch to and how the architectural ribbed space could be organized by geometric vault. An innovation of even greater consequencemeans, going was beyond the way the mathematics concept of the was module applied as the to basicthe element of design. design of architecture. Mathematics had been an integral element of architectural design since ancient 4. Mathematics of Vision times. The architects of the caliphate applied mathematics—and geometry in particular—to architecture in new ways, however, which signifiedThe arcades a mathematical of the Dar¯ turn al-Gundˇ in architectural differ from Roman design. architecture The in another important aspect. The size following paper focuses on how geometryof was the applied columns in isdifferent much smaller ways to compared the design to of the architecture height of the opening than in any Roman building in 10th-century Córdoba and what was achieved(Figure5 by). Placing doing so. arches Taken instead together, of horizontal these innovative architraves ways above columns had already reduced the height of applying mathematics to the design ofof columns architecture significantly—the came to influence first example not only being the the further church of Santa Costanza in Rome (ca. 340–345 development of Islamic architecture in theCE). region The but use also of horseshoe-shaped European architecture arches, in introduced general. to the architecture of the Iberian Peninsula in the Since the 8th century, crucial advancesVisigothic had been period, made reduced in the field the height of mathematics of the columns at the further. court In the 6th century horseshoe arches were of the Abbasid caliphs in Bagdad by combiningabout 133% the higherknowledge than regularof Classical arches, Greek in the mathematicians late 10th century 150% higher, and in the 11th century even like Euclid with innovations made in India175% in the higher—an 6th and 7th increase century. to be1 surpassed Among these only was by pointed the use arches. of 17 As a result, columns were only about a decimal numeral system with zero as a digit,two-thirds the foundations as high as the of algebra opening by of theal-Ḫ arcade,wārizm eventuallyī (Latinized even only half as high (see Table2). Algoritmi, ca. 780–846) and the use of the trigonometric functions sine and cosine. These advances quickly reached the farthest ends of the Islamic world, including Córdoba. 2 The Arabic-Indian numeral system was introduced to Córdoba15 alreadyVitruvius III,by 1.the inventor Ibn Firnās (810–887/888). At the court of the caliph, the mathematician Maslama16 The differenceal-Maǧrī betweentī (ca. 950–1007/1008) the design methods edited of Vitruvius the astronomic and the architect of the Dar¯ al-Gundˇ may reflect to some degree tables of al-Ḫwārizmī, translated Ptolemy’s starbasic chart differences and in improved how the role the of the translation individual within of his society Almagest was conceived—an aspect that would warrant further study. 17 (Camps Cazorla 1953). Pointed arches are first found in the extension of the mosque of Córdoba by al-Hakam II and become more and more frequent in the 11th century. 1 For an overview see (Rashed 1984). 2 A detailed study of the mathematical knowledge in al-Andalus is still outstanding.

Arts 2018, 7, x; doi: FOR PEER REVIEW www.mdpi.com/journal/arts Arts 2018, 7, x FOR PEER REVIEW 13 of 15

divide the roof of the central nave into segments that are equal in width to the naves of the side aisles. Pointed arches offer the possibility of creating a hierarchy of arches of different widths, independent of their height. Pointed arches and ribbed vaults become part of a logical system. In Islamic architecture, the same elements are employed to a very different end. The architects of Córdoba did not attempt to create a hierarchy but a web in which all parts would be equal to each other. The interlocking arch as employed in Islamic architecture does not create order but rather ambiguity, weaving together elements in a way that is beyond structural logic or even comprehension (cf. Figure 8 again). The ideal was the suspension of differences in space, the ultimate aim the blurring of the distinction between interior and exterior space.

6. Conclusions In 10th-century Córdoba, mathematics—and geometry in particular—was applied to the design of architecture in different ways and at different levels—to organize ground plans and elevations, as a way to measure the human field of view, and as a means of decoration. In all cases, the use of geometry was based on a specific, novel concept of space. Architectural space was conceived in an abstract way as a holistic unity. The difference to how Roman architects had employed mathematics in architecture highlights how revolutionary this new attitude toward architectural design truly was. In Roman architecture mathematics—in the form of arithmetic and more rarely geometric ratios— was employed to place each element in its appropriate relation to all other elements of the building and thus to achieve what Vitruvius called simmetria, or aesthetic harmony. The architects of Islamic ArtsCórdoba2018, 7, 35instead employed geometry to create a spatial web, in which all parts are equal to each 8 of 15 other, and part of a single unified space. Both the advances made in the field of mathematics since the 9th century and the application of mathematics to architecture in the 10th century canTable be viewed 2. Measurements as the result andof what proportions might be termed of arcades of the 10th and 11th century.18 a “mathematical turn” of Islamic culture. The universe was considered in a new, abstract way, as governed by abstract rules and, in the case of art and architecture, by a web of regulating lines. This Column Opening Shaft Diameter Opening Column Bay Width/Shaft Building Date CE No. of Bays Bay Width in cm Height/Opening Height/Column new world view pointed the way toward new possibilities of designing architecture, possibilitiesin cm thatHeight in cm Height in cm Diameter were to be tested further by the architects of Gothic architecture and eventually the Renaissance, Height Diameter thoughDar¯ in al-Mulk, different entrance ways. 940 1 146 18 268 156 8.1 0.58 14.9 Dar¯ al-Mulk, hall 940 1 146 19 298 182.5 7.7 0.61 15.7 Funding:Casa This de research la Alberca received no external c. 940 funding. 3 165 26 367 235.5 6.4 0.64 14.1 Dar¯ al-Gund,ˇ entrance c. 950 3 165 33.5 383.5 263 4.9 0.69 11.4 ConflictsSaló ofn Rico,Interest: entrance The author declares 953–957 no conflict of interest. 3 180.5 28 369 233 6.4 0.63 13.2 Salón Rico, side 953–957 2 189 28 374 232 6.8 0.62 13.4 ReferenceSaló n Rico, nave 953–957 6 238 33 432 257 7.2 0.59 13.1 Salón Rico, portico 953–957 5 239 32 459 287 7.5 0.63 14.3 (Al-ʿUdrCasaī 1965) de Al-Gaˇ ʿfarUdrī, Aḥmad ibnc. 961ʿUmar. 1965. Fragmentos 3 Geográfico-Histórico 169 de al-Masālik 29.5 ilā Ğamīʿ al- 373 243.5 5.7 0.65 12.6 MamDar¯ ā al-Mulklik. Edited annex by ʿAbd al-ʿAziz̄ 972 al-Ahwanī .̄ Madrid: 3 Instituto de Estudios 149 Islámicos. 19 293.5 183.5 7.8 0.63 15.4 (Arias PáramoMálaga, 2008) entrance Arias Páramo, 1023 Lorenzo. or later 2008. Geometría 3 y Proporción en la 102Arquitectura Prerománica 17.5 Asturiana. 285 200 5.8 0.70 16.3 Málaga, pavillon 1023 or later 2 118 18 322 217 6.6 0.67 17.9 Anejos de Archivo Español de Arqueología 49. Madrid: Consejo Superior de Investigaciones Científicas. Aljafería, entrance 1046 or later 4 167 25 445 286 6.7 0.64 17.8 (ArnoldAljafer 2012) Arnold,ía, portico Felix. 2012. 1046Architek or latertur als kulturgeschichtliche 4 Quelle. 330 In Politische Räume in 23 Vormodernen 585 280 14.4 0.48 25.4 Gesellschaften: Gestaltung, Wahrnehmung, Funktion. Edited by Ortwin Dally, Friedericke Fless and Rudolf Haensch. Rahden: Leidorf, pp. 291–300. (Arnold 2017) Arnold, Felix. 2017. Islamic Palace Architecture in the Western Mediterranean: A History. New York: Oxford University Press. (Arnold et al. 2015) Arnold, Felix, Alberto Canto García, and Antonio Vallejo Triano. 2015. Munyat ar-Rummaniya: Ein Islamischer Landsitz bei Córdoba. 1. Palastanlagen. Madrider Beitrage 34. Mainz: Reichert. (Arnold, forthcoming) Arnold, Felix. Forthcoming. Félix Hernández y el codo islámico. Cuadernos de Madinat al-Zahra 8. (Belting 2008) Belting, Hans. 2008. Florenz und Bagdad: Eine Westöstliche Geschichte des Blicks. Munich: C. H. Beck. (Binding 1993) Binding, Günther. 1993. Baubetrieb im Mittelalter. Darmstadt: Primus. (Bischoff 1981) Bischoff, Bernhard. 1981. Mittelalterliche Studien. Ausgewählte Aufsätze zur Schriftkunde und Literaturgeschichte 3. Stuttgart: Hiersemann.

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The spacing of the columns was thus established by geometric means on the basis of the overall proportions of the arcades. Such a procedure would have been unthinkable for Roman architects. For Vitruvius, the proportional relationship of each element of a building to the building as a whole was fixed and could not be changed. In his treaties De architectura (Vitruvius ca.25 BCE), he used the analogy of the human body, in which the size of each limb has a specific relation to the size of the whole body15. Vitruvius insisted on the use of a basic unit (modulus) for designing buildings, equivalent to the thickness of a column at its lower end. He thus proceeded from the individual element—the column—to the design of the whole building. This is not the case in the Dār al-Ǧund: the architect started from the design of the complete building complex, subdividing it by geometric means until he arrived at the dimensions of smaller elements (Figure 4). The size and proportion of individual columns was thus not related in a fixed way to the design of the whole building and not even to the design of the arcade of which it was part. Vitruvius worked from the premise of the dignity of each individual element, which was to be achieved by keeping all other parts of the building in proportion. The architect of the Dār al-Ǧund had a more abstract approach to architecture and architectural space. In an innovative way, he proceeded from the idea of the unity of space, to which he subordinated the role of individual parts.16 The way the Dār al-Ǧund was designed liberated the architect from the constraints of Roman architecture as defined by Vitruvius and pointed the way to how architectural space could be organized by geometric means, going beyond the concept of the module as the basic element of design.

4. Mathematics of Vision The arcades of the Dār al-Ǧund differ from Roman architecture in another important aspect. The size of the columns is much smaller compared to the height of the opening than in any Roman building (Figure 5). Placing arches instead of horizontal architraves above columns had already reduced the height of columns significantly—the first example being the church of Santa Costanza in Rome (ca. 340–345 CE). The use of horseshoe-shaped arches, introduced to the architecture of the Iberian Peninsula in the Visigothic period, reduced the height of the columns further. In the 6th century horseshoe arches were about 133% higher than regular arches, in the late 10th century 150% higher, and in the 11th century even 175% higher—an increase to be surpassed only by pointed arches.17 As a result, columns were only about two-thirds as high as the opening of the arcade, Arts 2018, 7, 35 9 of 15 eventually even only half as high (see Table 2).

Article Mathematics and the Islamic Architecture of Córdoba

Felix Arnold Ḫ ʾ

German Archaeological Institute, Calle Serrano 159, Madrid 28002, Spain; [email protected]

Received: 29 May 2018; Accepted: 25 July 2018; Published: 7 August 2018

Abstract: In 10th-century Córdoba, mathematics—and particularly geometry—was applied to architecturalFigure design 5. Schematic in new ways, comparison constituting of of the design a “mathematical of arca arcades:des: Corinthian Corinthianturn” of Islamic order order of ofarchitecture. Roman architecture; In the mosque ofarcade Córdoba of horseshoe-shapedhorseshoe-shaped and in the palaces archesarches of atat MadMadīīnat al-Zahrāāʾ ʾ(mid-10th (mid-10th , geometry century); century); was arcadeemployed arcade of of multi-foil, multi-foil, in the design of pointedground arches plans, at elevations, the AljaferAljaferíaía (11thdecorative(11th century). patterns, Drawing and by Felixeven Arnold. to measure the human view. While Roman architects like Vitruvius had used mathematics to place each element of a building in its appropriate These relation changes notto all only ot her reduced elements significantly of a building, the size the of the architects column shafts—aat Córdoba concern employed for builders 15 Vitruvius III, 1. geometrytrying to to create secure a suitablespatial web materials in which of hard all parts stone. are They equal reduced to each the other thickness and ofpart the of columns a single, needed 16 The difference between the design methods of Vitruvius and the architect of the Dār al-Ǧund may reflect to unifiedto supportspace. The an architects arcade. The of Córdoba ratio between thus po bayinted width the and way column to new width possibilities was increased of designing from 3:1 in some degree basic differences in how the role of the individual within society was conceived—an aspect that architecture, possibilities which were to be tested further by architects of Gothic and Renaissance thewould Corinthian warrant order further up study. to 8:1 in the 10th century, and even to 14:1 in the 11th century (see Table2). architecture,17The (Camps change though Cazorla becomes to 1953).different apparent Pointed ends. also arches when are first the found ratio betweenin the extension the thickness of the mosque of the of columns Córdoba and by al-Hakam the total heightII and of become the opening more and of themore arcade frequent is in compared. the 11th century. In the Corinthian order, this ratio had been 1:10. Keywords: In the 10thMezquita century, de the ratioCórdoba; was increasedMadīnat toal-Zahr 1:11–15,āʾ in; thearchitectural 11th century proportions; up to 1:16–25. geometry; In other words, Vitruvius;the thickness interlocking of the arches; supports ribbed needed vault; to muqarna carry anṣ arcade was eventually cut in half compared to the Roman period. The aim of the architects at Córdoba appears to have been to reduce the physical impact of the arcade on the ground. The reduction of the column thickness reduced how much they impeded 1. Introductionvisibility across a space and increased the potential use of the floor surface. This was a major Theconcern proclamation especially of inthe mosque caliphate architecture, at Córdoba in in which 929 CE the marked surface a area turning available point fornot prayer only in needed politicalto and be maximized. ideological Thoughterms but the also faithful in the direct histor theiry of prayers art and towardarchitecture. Mecca With (qibla the), it foundation would have of been an a new capitaladded valuecity—Mad to seeīnat the al-Zahr prayerāʾ leader—in 936 (im CE,am¯ ).the Particularly architects of during the caliphate the sermon, initiated the the columned creation prayer of a novelhall way of the of mosquedesigning, was constructing, viewed by the and congregation decorating buildings. not only inAmong the direction the signature of the elements naves, but also to comein out a concentric of this new manner, “style” in theof designing direction ofarch theitecture preacher are on the his interlocking pulpit (minbar arch). Inand the the extension ribbed of the vault. mosqueAn innovation of Córdoba of even executed greater by consequence al-H. akam II inwas 961–971, the way diagonal mathematics views acrosswas applied the prayer to the hall were designhighlighted of architecture. by the Mathematics distribution had of been gray an and integral pink column element shafts of architectural along diagonal design lines, since directed ancient toward 19 times. theThe central architects axis of of the the mosque caliphate (Figure applied6). The mathematics—and same attention to concentricgeometry views in particular—to was also paid to the architecturethrone in halls new of ways, however, which, in signified which a largea mathematical number of turn people in architectural gathered to greet design. the The caliph. following paperVisibility focuses became on how amajor geometry concern was applied to architects in different of Islamic ways C toórdoba. the design Literary of architecture descriptions of in 10th-centurygarden mansions Córdoba (andmunya what) in was the achieved suburbs of by C doingórdoba so. show Taken that together, the view these (na¯ z.innovativeir) onto the ways landscape 20 of applyingwas appreciated mathematics already to the in thedesign 8th century.of architectureBy the came late 10th to influence century,the not framing only the of thefurther view from developmentpalatial of halls Islamic became architecture an important in the motive region in but palace also design.European The architecture earliest example in general. is the garden hall of 21 Sinceal-Rumm the 8than¯ century,¯ıya, a palace crucial built advances by the financehad been minister made ofin al-theH .fieldakam of II mathematics in 965/966 CE. at theThe court hall was of the builtAbbasid on a caliphs dam that in separated Bagdad by a large combining water reservoir the knowledge from an of extensive Classical garden. Greek Arcadesmathematicians opened in both like Eucliddirections, with innovations thus giving made the hall in aIndia transparency in the 6th not and found 7th century. in other1 buildings Among these of the was time. the Theuse widthof of a decimalthe arcadesnumeral was system additionally with zero determined as a digit, the by equilateralfoundations triangles of algebra whose by al- baseḪw correspondedārizmī (Latinized to the base Algoritmi,of the ca. triangle 780–846) and and whose the use tip wasof the located trigonom in theetric center functions of the oppositesine and backcosine. side These of the advances hall (Figure 7). ◦ quicklyThe reached angle ofthe an farthest equilateral ends triangle—60 of the Islamic—corresponds world, including roughly Córdoba. to the angle2 Theof Arabic-Indian the human field of ◦ numeralview. system While was the introduced human eye to is Córd able tooba perceive already a by much the widerinventor angle—almost Ibn Firnās (810–887/888). 180 —only a partAt the of that is court of the caliph, the mathematician Maslama al-Maǧrītī (ca. 950–1007/1008) edited the astronomic tables of al-Ḫwārizmī, translated Ptolemy’s star chart and improved the translation of his Almagest 19 ( Ewert and Wisshak 1981 , pp. 72–78, Figure 36). 20 (Ruggles 2000; Arnold 2017, p. 20). 1 For 21an overview(Arnold et al.see 2015 (Rashed; Arnold 1984). 2017 , pp. 103–10). 2 A detailed study of the mathematical knowledge in al-Andalus is still outstanding.

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These changes not only reduced significantly the size of the column shafts—a concern for builders trying to secure suitable materials of hard stone. They reduced the thickness of the columns needed to support an arcade. The ratio between bay width and column width was increased from 3:1 in the Corinthian order up to 8:1 in the 10th century, and even to 14:1 in the 11th century (see Table 2). The change becomes apparent also when the ratio between the thickness of the columns and the total height of the opening of the arcade is compared. In the Corinthian order, this ratio had been 1:10. In the 10th century, the ratio was increased to 1:11–15, in the 11th century up to 1:16–25. In other words, the thickness of the supports needed to carry an arcade was eventually cut in half compared Artsto the2018 Roman, 7, 35 period. 10 of 15 The aim of the architects at Córdoba appears to have been to reduce the physical impact of the arcade on the ground. The reduction of the column thickness reduced how much they impeded ◦ generallyvisibility across regarded a space as the and field increased of view. the Ordinary potential camera use of lensesthe floor thus surface. have aThis field was of 57a majorwhile concern artists ◦ 22 frameespecially their in picture mosque based architecture, on a field ofin viewwhich of 50–60the surface. A personarea available sitting below for prayer the southern needed arcade to be ofmaximized. the hall and Though looking the across faithful the direct hall had their his prayers view of thetoward water Mecca basin ( beyondqibla), it framedwould byhave the been outline an ofadded the northernvalue to arcade.see the Inprayer the same leader way (im aām person). Particularly sitting below during the the opposite, sermon, northern the columned arcade prayer of the hall of and the looking mosque across was viewed the hall by had the his congregation view of the no gardent only in framed the direction by the outlineof the naves, of the but southern also in arcade.a concentric At al-Rumm manner,an¯ in¯ıya, the visiondirection became of the a subjectpreacher of on mathematics—an his pulpit (minbar integral). In the concept extension not foundof the inmosque any known of Córdoba building executed before by and al- notḤakam applied II in as961–971, a common diagonal principle views until across the the Italian prayer Renaissance. hall were Thehighlighted design of by the the hall distribution of al-Rumm of angray¯ ¯ıya and became pink thecolumn prototype shafts of along several diagonal later palatial lines, directed halls, including toward onethe central in the Alc axisázar of the of C mosqueórdoba, (Figure two on the6).19 Alcazaba The same of attention Almería to and concentric one in the views Aljafer wasía ofalso Zaragoza, paid to 23 allthe dating throne to halls the of 11th Mad century.īnat al-Zahrāʾ, in which a large number of people gathered to greet the caliph.

Figure 6. Reconstructed ground plan of the extension of the mosque of Córdoba executed by al-Ḥakam Figure 6. Reconstructed ground plan of the extension of the mosque of Córdoba executed by al-H. akam II in 961–971, with distribution of columncolumn shafts in reddish-pink breccia (indicated in red) and in dark gray limestone (indicated in black). Drawing by FelixFelix Arnold, based on the plan ofof ChristianChristian Ewert.Ewert.

Visibility became a major concern to architects of Islamic Córdoba. Literary descriptions of A precondition for measuring the human view was an appreciation for the optic characteristics of garden mansions (munya) in the suburbs of Córdoba show that the view (nāẓir) onto the landscape the human eye. Ground breaking studies in this regard were being made at this time by Ibn Haitam was appreciated already in the 8th century.20 By the late 10th century, the framing of the view from¯ (Latinized as “Alhacen”, 965–1040) at the Fatimid court in Cairo24. His treatise Kitab¯ al-Manazir¯ “Book palatial halls became an important motive in palace design. The earliest example is the garden hall of of Optics“, written in 1028 and translated into Latin under the title De Aspectibus or Perspectiva, would al-Rummānīya, a palace built by the finance minister of al-Ḥakam II in 965/966 CE.21 The hall was become a reference work for Renaissance artists, including Filippo Brunelleschi and Leon Battista Alberti. It is rather unlikely that the architects at Córdoba were already aware of the scientific advanced made19 (Ewert by IbnandHai Wisshaktam. 1981, Both pp. the 72–78, works Figure of Ibn 36). Hai tam and the architects at Córdoba are an expression 20 ¯ ¯ for (Rugglesa new attitude 2000; Arnold toward 2017, man p. and20). space, however, an attitude based on a new mathematical concept 21 of space. (Arnold et al. 2015; Arnold 2017, pp. 103–10).

22 (Arnold 2017, pp. 39–40, Figure 2.2–3). 23 (Arnold 2017, pp. 156–72, Figure 3.27). 24 (Lindberg 1976; Moreno Castillo 2007; Belting 2008). The book was known in 11th-century Zaragoza. (Hogendijk 1995). Arts 2018, 7, x FOR PEER REVIEW 10 of 15 built on a dam that separated a large water reservoir from an extensive garden. Arcades opened in both directions, thus giving the hall a transparency not found in other buildings of the time. The width of the arcades was additionally determined by equilateral triangles whose base corresponded to the base of the triangle and whose tip was located in the center of the opposite back side of the hall (Figure 7). The angle of an equilateral triangle—60°—corresponds roughly to the angle of the human field of view. While the human eye is able to perceive a much wider angle—almost 180°—only a part of that is generally regarded as the field of view. Ordinary camera lenses thus have a field of 57° while artists frame their picture based on a field of view of 50–60°.22 A person sitting below the southern arcade of the hall and looking across the hall had his view of the water basin beyond framed by the outline of the northern arcade. In the same way a person sitting below the opposite, northern arcade of the hall and looking across the hall had his view of the garden framed by the outline of the southern arcade. At al-Rummānīya, vision became a subject of mathematics—an integral concept not found in any known building before and not applied as a common principle until the Italian Renaissance. The design of the hall of al-Rummānīya became the prototype of several later palatial halls, including one in the Alcázar of Córdoba, two on the Alcazaba of Almería and one in the Arts 2018 7 Aljafería, ,of 35 Zaragoza, all dating to the 11th century.23 11 of 15

Article

Mathematics and the Islamic Architecture of Córdoba Figure 7. ApplicationApplication of of equilateral equilateral triangles triangles to to the the de designsign of ofarcade arcade openings: openings: western western hall hall of al- of Felix Arnold Ḫ ʾ al-RummRummānīan¯ya¯ıya in Córdoba in Córdoba (top (top); northern); northern hall hall of the of Aljafería the Aljafer iní aZaragoza in Zaragoza (bottom (bottom). Drawings). Drawings by Felix by FelixArnold. Arnold. German Archaeological Institute, Calle Serrano 159, Madrid 28002, Spain; [email protected]

Received: 29 May 2018; Accepted: 25 July 2018; Published: 7 August 2018 5. GeometricA precondition Pattern for as measuring Structure the human view was an appreciation for the optic characteristics of the human eye. Ground breaking studies in this regard were being made at this time by Ibn Haiṯam At Córdoba, the applicationAbstract: of mathematics In 10th-century to architecture Córdoba, is most mathematics—and visible in the use ofparticularly geometric geometry—was applied to (Latinized as “Alhacen”, 965–1040) at the Fatimid court in Cairo24. His treatise Kitāb al-Manāzir “Book motives and patterns for decorativearchitectural purposes. design Geometric in new motives ways, constituting are already found a “mathematical in the Late Antique turn” of Islamic architecture. In the of Optics“, written in 1028 and translated into Latin under the title De Aspectibus or Perspectiva, would and Visigothic period but becomemosque especially of Córdoba prevalent and in thein the 10th palaces century. of At Madīnat al-Zahrāʾ frames , geometry was employed in the become a reference work for Renaissance artists, including Filippo Brunelleschi and Leon Battista of doors and arcades are frequentlydesign decoratedof ground byplans, patterns elevations, of squares, decorative hexagons, patterns, octagons, and even 6- and to measure the human view. Alberti. It is rather unlikely that the architects at Córdoba were already aware of the scientific 8-pointed stars, swastikas, meanders,While Roman and other architects geometric like Vitruvius motives.25 hadSometimes used mathematics geometric to motives place each element of a building in are interwoven with plant decoration, its appropriate including relation circles, to squares all other and elements triangles. of a building, the architects at Córdoba employed 22 (ArnoldUsually 2017, decoration pp. 39–40, wasFiguregeometry applied 2.2–3). toto thecreate architectural a spatial web structure in which as aall separate parts are layer, equal with to each no other and part of a single, correlation23 (Arnold 2017, between pp. 156–72, decorative Figureunified pattern 3.27). space. and architectural The architects form. of InCórdoba Islamic thus architecture, pointed thisthe distinctionway to new possibilities of designing between24 (Lindberg architecture 1976; Moreno and decorationCastilloarchitecture, 2007; was Belting increasinglypossibilities 2008). The which blurred. book were was Early toknown be examples tested in 11th-century further are window by Zaragoza.architects grills, of Gothic and Renaissance which(Hogendijk are regularly 1995). decoratedarchitecture, with geometric though patterns to different from the ends. Visigoth period onward.26 The grills are perforated according to a geometric pattern, decoration, and structure, thus becoming a unity. In the 10th century, the geometryKeywords: of window Mezquita grill patterns de becomesCórdoba; ever Mad moreīnat elaborate, al-Zahrā incorporatingʾ; architectural proportions; geometry; octagons, stars, and other geometricVitruvius; motives. interlocking arches; ribbed vault; muqarnaṣ This equation of geometric pattern and architectural structure was eventually applied to the building as a whole. A first step was taken in the first phase of the mosque of Córdoba, where the arcades were composed of two levels of arches. In the Roman and Late Antique Period arches were always understood as semicircular1. Introduction perforations of walls. Arcades could be superimposed one above the other—for example in the aqueducts of Segovia and Mérida—but each arcade always received a The proclamation of the caliphate at Córdoba in 929 CE marked a turning point not only in horizontal top. In the mosque of Córdoba, the lower level of arcades received free standing arches—a political and ideological terms but also in the history of art and architecture. With the foundation of first step toward the isolation of the pure geometric shape from the wall plane. a new capital city—Madīnat al-Zahrāʾ—in 936 CE, the architects of the caliphate initiated the creation of a novel way of designing, constructing, and decorating buildings. Among the signature elements to come out of this new “style” of designing architecture are the interlocking arch and the ribbed 25 (Kubisch 1997). vault. An innovation of even greater consequence was the way mathematics was applied to the 26 (Brisch 1966; Kubisch 1999). Fragments of the Visigoth period from Córdoba are published in (Sánchez Velasco 2006, pp. 64–68). design of architecture. Mathematics had been an integral element of architectural design since ancient times. The architects of the caliphate applied mathematics—and geometry in particular—to architecture in new ways, however, which signified a mathematical turn in architectural design. The following paper focuses on how geometry was applied in different ways to the design of architecture in 10th-century Córdoba and what was achieved by doing so. Taken together, these innovative ways of applying mathematics to the design of architecture came to influence not only the further development of Islamic architecture in the region but also European architecture in general. Since the 8th century, crucial advances had been made in the field of mathematics at the court of the Abbasid caliphs in Bagdad by combining the knowledge of Classical Greek mathematicians like Euclid with innovations made in India in the 6th and 7th century.1 Among these was the use of a decimal numeral system with zero as a digit, the foundations of algebra by al-Ḫwārizmī (Latinized Algoritmi, ca. 780–846) and the use of the trigonometric functions sine and cosine. These advances quickly reached the farthest ends of the Islamic world, including Córdoba. 2 The Arabic-Indian numeral system was introduced to Córdoba already by the inventor Ibn Firnās (810–887/888). At the court of the caliph, the mathematician Maslama al-Maǧrītī (ca. 950–1007/1008) edited the astronomic tables of al-Ḫwārizmī, translated Ptolemy’s star chart and improved the translation of his Almagest

1 For an overview see (Rashed 1984). 2 A detailed study of the mathematical knowledge in al-Andalus is still outstanding.

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A high point in the equation of geometric pattern and architectural structure was reached in the 27 reign of al-H. akam II with the extension of the mosque in 961–971. The arcades of the maqs.ura¯ area were designed as a complex superposition of different arches, including regular half-circle arches, Arts 2018, 7, x FOR PEER REVIEW 12 of 15 some of horseshoe shape and some multi-foil—a shape first introduced at the Salón Rico a decade decorationearlier. The are arches still span being either debated, a single some bay favoring or several an baysorigin at in once, the thuseast, crossing others in each the otherwest29 (Figure. Among8). Athe common earliest examples motive is in the the interlocking west are the arch, ceiling in which of the archespalace spanof Mu twoʿtaṣ baysim in atAlmería once.In and this the way, ceiling the ofrole the of palaces the arch of as Qal an elementʿat Banī spanningḤammād fromin Algeria, one support both dating to the nextto the was 11th abandoned century.30 in Whatever a strict sense, the actualthe arch origin now of playing the idea, more muqarna a decorativeṣ decoration role thanis the serving most definitive a structural expression purpose. of Atthe theapplication same time, of mathematicsdecoration and to structurearchitectural now design. became identical, much like in the window grills before.

Figure 8. The Capilla de Villaviciosa in the mosque of Córdoba, part of the extension executed by al- Figure 8. The Capilla de Villaviciosa in the mosque of Córdoba, part of the extension executed by Ḥakam II. Photograph by R. Friedrich, German Archaeological Institute. al-H. akam II. Photograph by R. Friedrich, German Archaeological Institute. Interlocking arches, ribbed vaulting, and muqarnaṣ decoration all formed an integral part of the subsequentIn the extensiondevelopment of al- ofH. Islamicakam II, architecture this play between in the western decoration Mediterranean patterns and region, structural including elements the 28 wasNasrid applied architecture also to theof Granada design of and vaults. the MarinidIn a structural architecture sense, of domesMorocco. were Both deconstructed the interlocking into arches,arches andeach the spanning ribbed fromvaulting one sideas first of thedeveloped vaulted spacein the toextension another of (Figure the mosque8). The domeof Córdoba could bealso composed played a crucialof four orrole more in the such evolution arches, arranged of Gothic according architectu to differentre, though geometric the route patterns. from Córdoba The end resultto the wasÎle-de- the Franceevolution cannot of the be ribbed traced vault, with in whichcertainty. ribs highlightEarly cases the for structural the use composition of interlocking of the arches vault. Inin contrastchurch architectureto later examples are the the transept ribs of of the Ste.-Honorine vaults of the in mosque Graville of and Có rdobaa church serve tower a decorative in Allemagne purpose near moreCaen (boththan a built structural in Normandy one and arearound the result 1070), of early the application examples for of geometrythe use of to ribbed decoration, vaults and the inCathedral this case of to Durhamarchitectural (England, design. begun in 1093), all constructed by Normans, who may well have been acquainted with EventuallyIslamic architecture, the equation either between on the decorationIberian Peninsula and architectural or in Sicily. structure31 led to the evolution of the muqarnaThe Islamics. decoration, architecture a form of the of three-dimensional,Iberian Peninsula and geometric the Gothic ornamentation architecture inof whichthe Île-de-France a vaulting both employed pointed arches and ribbed domes. The way these elements were used is quite different, however. Gothic architecture employs pointed arches and ribbed vaults as a way of clarifying27 (Ewert 1968 the). hierarchy of spatial elements. Ribbed vaults allow the architects of a Gothic church to 28 (Giese-Vögeli 2007; Marfil Ruiz 2004).

29 For a recent study see (Carrillo 2014). 30 (Golvin 1974; Al-ʿUdrī 1965, p. 85; Arnold 2017, pp. 133–35, 161). 31 Cf. Arnold (2017), p. 176 deals with the question whether the transmission of architectural forms from Islamic to Late Romanesque architecture went hand in hand with the ideas and concepts behind these forms, specifically whether the association between Islamic philosophy and Scholasticism on the one hand and between Gothic architecture and Islamic architecture on the other was related in some way.

Arts 2018, 7, x FOR PEER REVIEW 13 of 15

divide the roof of the central nave into segments that are equal in width to the naves of the side aisles. Pointed arches offer the possibility of creating a hierarchy of arches of different widths, independent of their height. Pointed arches and ribbed vaults become part of a logical system. In Islamic architecture, the same elements are employed to a very different end. The architects of Córdoba did not attempt to create a hierarchy but a web in which all parts would be equal to each other. The interlocking arch as employed in Islamic architecture does not create order but rather ambiguity, weaving together elements in a way that is beyond structural logic or even comprehension (cf. Figure 8 again). The ideal was the suspension of differences in space, the ultimate aim the blurring of the distinction between interior and exterior space.

6. Conclusions In 10th-century Córdoba, mathematics—and geometry in particular—was applied to the design of architecture in different ways and at different levels—to organize ground plans and elevations, as a way to measure the human field of view, and as a means of decoration. In all cases, the use of geometry was based on a specific, novel concept of space. Architectural space was conceived in an abstract way as a holistic unity. The difference to how Roman architects had employed mathematics in architecture highlights how revolutionary this new attitude toward architectural design truly was. In Roman architecture mathematics—in the form of arithmetic and more rarely geometric ratios— was employed to place each element in its appropriate relation to all other elements of the building and thus to achieve what Vitruvius called simmetria, or aesthetic harmony. The architects of Islamic Córdoba instead employed geometry to create a spatial web, in which all parts are equal to each other, and part of a single unified space. Both the advances made in the field of mathematics since the 9th century and the application of mathematics to architecture in the 10th century can be viewed as the result of what might be termed a “mathematical turn” of Islamic culture. The universe was considered in a new, abstract way, as governed by abstract rules and, in the case of art and architecture, by a web of regulating lines. This new world view pointed the way toward new possibilities of designing architecture, possibilities that were to be tested further by the architects of Gothic architecture and eventually the Renaissance, though in different ways.

Arts 2018, 7, 35 Funding: This research received no external funding.13 of 15 Conflicts of Interest: The author declares no conflict of interest. is fractionized into a large number of miniature squinches or corbles. The origins of the muqarnas Reference . decoration are still being debated, some favoring an origin in the east, others in the west29. Among the earliest examples in the west are the ceiling of(Al- theʿUdr palaceī 1965) of MuAl-ʿtaUdrs.imī, A inḥmad Almer ibnía ʿ andUmar. the 1965. ceiling Fragmentos Geográfico-Histórico de al-Masālik ilā Ğamīʿ al- 30 of the palaces of Qal at Ban¯ı H. ammad¯ in Algeria, bothMam datingālik. Edited to the by 11th ʿAbd century. al-ʿAziz̄ al-AhwaWhatevernī .̄ Madrid: the Instituto de Estudios Islámicos. actual origin of the idea, muqarnas. decoration is the(Arias most Páramo definitive 2008) Arias expression Páramo, of Lorenzo. the application 2008. Geometría of y Proporción en la Arquitectura Prerománica Asturiana. mathematics to architectural design. Anejos de Archivo Español de Arqueología 49. Madrid: Consejo Superior de Investigaciones Científicas. Interlocking arches, ribbed vaulting, and muqarna(Arnolds. decoration 2012) Arnold, all Felix. formed 2012. an Architek integraltur part als kulturgeschichtliche of the Quelle. In Politische Räume in Vormodernen subsequent development of Islamic architecture in theGesellschaften: western Mediterranean Gestaltung, Wahrnehmung, region, including Funktion the. Edited by Ortwin Dally, Friedericke Fless and Rudolf Nasrid architecture of Granada and the Marinid architectureHaensch. of Morocco.Rahden: Leidorf, Both the pp. interlocking 291–300. arches and the ribbed vaulting as first developed in the(Arnold extension 2017) of theArnold, mosque Felix.of 2017. Có rdobaIslamic alsoPalace played Architecture a in the Western Mediterranean: A History. New York: crucial role in the evolution of Gothic architecture, thoughOxford the routeUniversity from Press. Córdoba to the Île-de-France cannot be traced with certainty. Early cases for the(Arnold use of et interlocking al. 2015) Arnold, arches Felix, in Alberto church Ca architecturento García, and Antonio Vallejo Triano. 2015. Munyat ar-Rummaniya: Ein Islamischer Landsitz bei Córdoba. 1. Palastanlagen. Madrider Beitrage 34. Mainz: Reichert. are the transept of Ste.-Honorine in Graville and a church tower in Allemagne near Caen (both built (Arnold, forthcoming) Arnold, Felix. Forthcoming. Félix Hernández y el codo islámico. Cuadernos de Madinat in Normandy around 1070), early examples for the use of ribbed vaults the Cathedral of Durham al-Zahra 8. (England, begun in 1093), all constructed by Normans, who may well have been acquainted with (Belting 2008) Belting, Hans. 2008. Florenz und Bagdad: Eine Westöstliche Geschichte des Blicks. Munich: C. H. Beck. Islamic architecture, either on the Iberian Peninsula or in Sicily.31 (Binding 1993) Binding, Günther. 1993. Baubetrieb im Mittelalter. Darmstadt: Primus. The Islamic architecture of the Iberian Peninsula and the Gothic architecture of the Île-de-France (Bischoff 1981) Bischoff, Bernhard. 1981. Mittelalterliche Studien. Ausgewählte Aufsätze zur Schriftkunde und both employed pointed arches and ribbed domes. The way these elements were used is quite different, Literaturgeschichte 3. Stuttgart: Hiersemann. however. Gothic architecture employs pointed arches and ribbed vaults as a way of clarifying the hierarchy of spatial elements. Ribbed vaults allow the architects of a Gothic church to divide the roof of the central nave into segments that are equal in width to the naves of the side aisles. Pointed arches offer the possibility of creating a hierarchy of arches of different widths, independent of their height. Pointed arches and ribbed vaults become part of a logical system. In Islamic architecture, the same elements are employed to a very different end. The architects of Córdoba did not attempt to create a hierarchy but a web in which all parts would be equal to each other. The interlocking arch as employed in Islamic architecture does not create order but rather ambiguity, weaving together elements in a way that is beyond structural logic or even comprehension (cf. Figure8 again). The ideal was the suspension of differences in space, the ultimate aim the blurring of the distinction between interior and exterior space.

6. Conclusions In 10th-century Córdoba, mathematics—and geometry in particular—was applied to the design of architecture in different ways and at different levels—to organize ground plans and elevations, as a way to measure the human field of view, and as a means of decoration. In all cases, the use of geometry was based on a specific, novel concept of space. Architectural space was conceived in an abstract way as a holistic unity. The difference to how Roman architects had employed mathematics in architecture highlights how revolutionary this new attitude toward architectural design truly was. In Roman architecture mathematics—in the form of arithmetic and more rarely geometric ratios—was employed to place each element in its appropriate relation to all other elements of the building and thus to achieve what Vitruvius called simmetria, or aesthetic harmony. The architects of Islamic Córdoba instead employed geometry to create a spatial web, in which all parts are equal to each other, and part of a single unified space.

29 For a recent study see (Carrillo 2014). 30 (Golvin 1974; Al- Udr¯ı 1965, p. 85; Arnold 2017, pp. 133–35, 161). 31 Cf. Arnold(2017), p. 176 deals with the question whether the transmission of architectural forms from Islamic to Late Romanesque architecture went hand in hand with the ideas and concepts behind these forms, specifically whether the association between Islamic philosophy and Scholasticism on the one hand and between Gothic architecture and Islamic architecture on the other was related in some way. Arts 2018, 7, x FOR PEER REVIEW 13 of 15 divide the roof of the central nave into segments that are equal in width to the naves of the side aisles. Pointed arches offer the possibility of creating a hierarchy of arches of different widths, independent of their height. Pointed arches and ribbed vaults become part of a logical system. In Islamic architecture, the same elements are employed to a very different end. The architects of Córdoba did not attempt to create a hierarchy but a web in which all parts would be equal to each other. The interlocking arch as employed in Islamic architecture does not create order but rather ambiguity, weaving together elements in a way that is beyond structural logic or even comprehension (cf. Figure 8 again). The ideal was the suspension of differences in space, the ultimate aim the blurring of the distinction between interior and exterior space.

6. Conclusions In 10th-century Córdoba, mathematics—and geometry in particular—was applied to the design of architecture in different ways and at different levels—to organize ground plans and elevations, as a way to measure the human field of view, and as a means of decoration. In all cases, the use of geometry was based on a specific, novel concept of space. Architectural space was conceived in an abstract way as a holistic unity. The difference to how Roman architects had employed mathematics in architecture highlights how revolutionary this new attitude toward architectural design truly was. In Roman architecture mathematics—in the form of arithmetic and more rarely geometric ratios— was employed to place each element in its appropriate relation to all other elements of the building and thus to achieve what Vitruvius called simmetria, or aesthetic harmony. The architects of Islamic Córdoba insteadArts employed2018, 7, 35 geometry to create a spatial web, in which all parts are equal to each 14 of 15 other, and part of a single unified space. Both the advancesBoth made the advancesin the field made of mathemat in the fieldics since of mathematics the 9th century since and the the 9th application century and of the application mathematics toof architecture mathematics in the to architecture10th century incan the be 10thviewed century as the can result be viewedof what might as the be result termed of what might be a “mathematicaltermed turn” a of “mathematical Islamic culture. turn” The of universe Islamic culture. was considered The universe in a wasnew, considered abstract way, in a new,as abstract way, governed by abstractas governed rules and, by abstract in the case rules of and,art and in thearchitecture, case of art by and a web architecture, of regulating by alines. web This of regulating lines. new world viewThis pointed new worldthe way view toward pointed new the possibilities way toward of designing new possibilities architecture, of designing possibilities architecture, that possibilities were to be testedthat further were to by be the tested architects further of by Go thethic architects architecture of Gothic and architectureeventually the and Renaissance, eventually the Renaissance, though in differentthough ways. in different ways.

Funding: This researchFunding: receivedThis research no external received funding. no external funding. Conflicts of Interest:Conflicts The author of Interest: declaresThe no author conflict declares of interest. no conflict of interest.

Reference References

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Carrillo, Alicia. 2014. Architectural Exchanges between North Africa and the Iberian Peninsula: Muqarnas in al-Andalus. The Journal of North African Studies 19: 68–82. [CrossRef] Ewert, Christian. 1968. Spanisch-Islamische Systeme Sich Kreuzender Bögen I. Die Senkrechten Ebenen Systeme Sich Kreuzender Bögen als Stützkonstruktion der Vier Rippenkuppeln in der Ehemaligen Hauptmoschee von Córdoba. Madrider Forschungen 2. Berlin: Walter de Gruyter. Ewert, Christian. 1978. Spanisch-Islamische Systeme Sich Kreuzender Bögen III. Die Aljafería in Zaragoza. 1. Madrider Forschungen 12.2. Berlin: Walter de Gruyter. Ewert, Christian, and Jens-Peter Wisshak. 1981. Forschungen zur Almohadischen Moschee. I. Vorstufen. Madrider Beiträge 9. Mainz: Philipp von Zabern. Fernández-Puertas, Antonio. 2000. Mezquita de Córdoba. Trazado proporcional de su planta general (siglos VIII-X). Archivo Espanol de Arte 73: 217–48. [CrossRef] Fernández-Puertas, Antonio. 2008. Mezquita de Córdoba. ‘Abd al-Rahman¯ I (169/785–786). El trazado proporcional de la planta y alzado de las arquerías del oratorio. La qibla y el mihrab¯ del siglo VIII. Archivo Espanol de Arte 81: 333–56. [CrossRef] Giese-Vögeli, Francine. 2007. Das islamische Rippengewölbe: Ursprung, Form, Verbreitung. Berlin: Gebrüder Mann. Golvin, Lucien. 1974. Les plafonds à Muqarnas de la Qal’a des Banû Hammâd et leur influence possible sur l’art de la Sicile à la période normande. Revue des Mondes Musulmans et de la Méditerranée 17: 63–69. [CrossRef] Hecht, Konrad. 1969. Maß und Zahl in der Gotischen Baukunst. Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft 21. Hildesheim: Georg Olms. Hernández Giménez, Félix. 1961. El codo en la Historiografía Árabe de la Mezquita Mayor de Córdoba: Contribución al Estudio del Monumento. Madrid: Maestre.

Article Mathematics and the Islamic Architecture of Córdoba

Felix Arnold Ḫ ʾ

German Archaeological Institute, Calle Serrano 159, Madrid 28002, Spain; [email protected] Arts 2018, 7, 35 15 of 15 Received: 29 May 2018; Accepted: 25 July 2018; Published: 7 August 2018

Hogendijk, Jan P. 1995. Al-Mu’tamanAbstract: ibnIn H10th-centuryud.¯ 11th-century Córdoba, king of mathematics—and Saragossa and brilliant particularly mathematician. geometry—was applied to Historia Mathematica 22:architectural 1–18. [CrossRef design] in new ways, constituting a “mathematical turn” of Islamic architecture. In the Kubisch, Natascha. 1997. Dermosque geometrische of Córdoba Dekor and des in Reichen the palaces Saales vonof Madīnat al-Zahrāʾ . , Madridergeometry was employed in the Mitteilungen 38: 300–65.design of ground plans, elevations, decorative patterns, and even to measure the human view. Kubisch, Natascha. 1999. DieWhile Fenstergitter Roman von architects like Vitruvius. Madrider had used Mitteilungen mathematics40: 290–307. to place each element of a building in Lindberg, David C. 1976. Theoriesits ofappropriate Vision from al-Kindi relation to Keplerto all. Chicagoother elements and London: of Universitya building, of Chicagothe architects Press. at Córdoba employed Marfil Ruiz, Pedro. 2004. Estudiogeometry de las to linternas create a y spatial extradó sweb de las in cwhichúpulas deall laparts maqsura are equal de la Catedralto each deother and part of a single, Córdoba, antigua mezquitaunified aljama. space.Arqueolog The íarchitectsa de la Arquitectura of Córdoba3: 91–107. thus [CrossRef pointed] the way to new possibilities of designing Moreno Castillo, Ricardo. 2007.architecture,Alhacen. El ArquimedespossibilitiesÁrabe which. Madrid: were Nivola. to be tested further by architects of Gothic and Renaissance Rashed, Roshdi. 1984. Entre Arithmarchitecture,étique et Alg thoughèbre: Recherches to different sur l’Histoire ends. des Mathématiques Arabes. Paris: Société d’Edition Les Belles Lettres. Riobóo Camacho, Francisco.Keywords: 2015. Laqubba Mezquita no cuadrada: de Córdoba; Influencias Mad yī repercusionesnat al-Zahrā enʾ; laarchitectural arquitectura proportions; geometry; hispanomusulmana. Al-MulkVitruvius;13: 75–101. interlocking arches; ribbed vault; muqarnaṣ Riobóo Camacho, Francisco. 2016. La geometría en el Salón Rico de Madinat al-Zahra. Al-Mulk 14: 51–81.

Ruggles, D. Fairchild. 2000. Gardens, Landscape and Vision in the Palaces of Islamic Spain. University Park: Penn State Press. Sánchez Velasco, Jerónimo. 2006. Elementos Arquitectónicos de Época Visigoda en el Museo Arqueológico de Córdoba. Monografías del Museo Arqueológico de Córdoba 1. Córdoba: Junta de Andalucía. 1. Introduction Vallejo Triano, Antonio. 2010. La Ciudad Califal de : Arqueología de su Arquitectura. Jaén: Almuzara. Velázquez Bosco, Ricardo. 1923. ExcavacionesThe proclamation en Medina of Azahara the caliphate. Junta Superior at Córdoba de Excavaciones in 929 CE y Antigüedadesmarked a turning point not only in 54. Madrid: Junta Superiorpolitical del Tesoro and ideological Artístico. terms but also in the history of art and architecture. With the foundation of Vitruvius. ca.25 BCE. De architectura:a new capital Libri decem city—Mad. īnat al-Zahrāʾ—in 936 CE, the architects of the caliphate initiated the creation Wilson Jones, Mark. 2000. Principlesof a novel of Roman way of Architecture designing,. New constructing, Haven and London:and decorating Yale Univesrity buildings. Press. Among the signature elements Wu, Nancy Y. 2002. Ad Quadratum:to come The out Practical of this Application new “style” of Geometry of designing in Medieval arch Architectureitecture .are Avista the Studiesinterlocking in arch and the ribbed the History of Medievalvault. Science, An Technology innovation and of Art even 1. London: greater Routledge.consequence was the way mathematics was applied to the © 2018design by the of author. architecture. Licensee Mathematics MDPI, Basel, had Switzerland. been an integral This article element is an openof architectural access design since ancient articletimes. distributed The underarchitects the terms of the and conditionscaliphate ofapplied the Creative mathematics—and Commons Attribution geometry in particular—to (CCarchitecture BY) license (http://creativecommons.org/licenses/by/4.0/ in new ways, however, which signified a ).mathematical turn in architectural design. The following paper focuses on how geometry was applied in different ways to the design of architecture in 10th-century Córdoba and what was achieved by doing so. Taken together, these innovative ways of applying mathematics to the design of architecture came to influence not only the further development of Islamic architecture in the region but also European architecture in general. Since the 8th century, crucial advances had been made in the field of mathematics at the court of the Abbasid caliphs in Bagdad by combining the knowledge of Classical Greek mathematicians like Euclid with innovations made in India in the 6th and 7th century.1 Among these was the use of a decimal numeral system with zero as a digit, the foundations of algebra by al-Ḫwārizmī (Latinized Algoritmi, ca. 780–846) and the use of the trigonometric functions sine and cosine. These advances quickly reached the farthest ends of the Islamic world, including Córdoba. 2 The Arabic-Indian numeral system was introduced to Córdoba already by the inventor Ibn Firnās (810–887/888). At the court of the caliph, the mathematician Maslama al-Maǧrītī (ca. 950–1007/1008) edited the astronomic tables of al-Ḫwārizmī, translated Ptolemy’s star chart and improved the translation of his Almagest

1 For an overview see (Rashed 1984). 2 A detailed study of the mathematical knowledge in al-Andalus is still outstanding.

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