Geometry in Islamic Art

Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Geometry in Islamic Art Carol Bier* Center for Islamic Studies, Graduate Theological Union, Berkeley, CA, USA The Textile Museum, George Washington University, Washington, DC, USA Geometric patterns in two and three dimensions comprise one of the key characteristics of arts and architecture of the Islamic world in many cultural traditions from the central Islamic lands of the Middle East to Spain, India, Indonesia, and sub-Saharan Africa (Bloom & Blair, 2009; Broug, 2013; Ettinghausen, Grabar, & Jenkins-Madina, 2001; Gerdes, 1999; Hillenbrand, 1994, 2009). Although geometry is present, either by conscious human choice in design or as an inherent feature of architectural production in all cultures, it seems to have assumed a much higher significance in Islamic centers of civilization (Grabar, 1992). Often attributed to a proscription against figural images, this interpretation is not borne out historically with reference to palace wall painting, ceramics, ivory, woodwork, and book illustration rich with pictorial narrative. There are, indeed, other more rational explanations for the emphasis on geometric figures (Allen, 1988; Freedberg, 1989; Belting, 2011) as well as many unanswered questions (Grabar, 1992). From the eleventh century throughout the central Islamic lands, the clustered and segmented vaults of muqarnas were used to effect spatial transitions for domes, vaults, and arches, or as a corbel to support balconies and cornices (Bloom, 1988; Tabbaa, 1985), exhibiting elaborations in succeeding centuries (Al-Assad, 1995; Golombek & Wilber, 1988). Simultaneously, one may trace the development of patterns in the plane from the empirical juxtaposition of geometric shapes to more complex arrangements with networks of intersecting polygons, which suggest direct relationships to academic studies of geometry in the Euclidean tradition (cf. Allen, 2004, who argues against such an interpretation). Yet precise intersections between the histories of architecture and mathematics have not been fully elucidated (Berggren, 2008; Bier, 2012; see also, Necipoğlu, 2015). The text of Abū’l Wafāʾ al-Būzjānī (d. 998) suggests direct links including conversations among artisans and mathematicians, leaving much to speculation (Chorbachi & Loeb, 1992; Özdural, 1995, 2000; see also Sarhangi, 2008b). Geometry in the Service of Religion: Kaaba, Qibla, Hajj, Awqaf, Salat, and Khatt Science, mathematics, and material culture also intersect in the service of Islam – in particular, the construction of astrolabes requires carefully placed markings that register a projection of threedimensional space in two dimensions to find the direction and distance to Mecca through astronomical measurements and calculations from a specific place (King, 1999). In architecture, geometrical considerations impact determination of qibla (the direction toward the Kaaba in Mecca), which orients both prayer space and worshipper for prayer, salat. The qibla wall of a mosque or prayer hall is often marked with an ornamented mihrab, or niche, which frequently incorporates geometric patterns and forms. And the times of prayer, awqaf, are based upon the sun’s position. The Kaaba itself (Arabic “cube”), which antedates Islam, is conceptually cubic in form, associated with rituals of circumambulation that comprise part of hajj, the pilgrimage prescribed for Muslims in the Qurʾān, and rites outside the formal hajj season (‘umra). Similar practices that entail circumambulation are associated in some instances with local shrines and tombs throughout the Islamic world (Mawani, 2016). Because Muslims believe the Qurʾān to have been revealed in Arabic, both language and script (khatt) have assumed elevated status within Islamic *Email: bier.carol@gmail.com Page 1 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 cultures, a status that contributed to the development of elaborate calligraphic forms often based upon systems of geometric proportion. Patterns in the Plane: Symmetry and Periodicity The presence of geometry is evident in the earliest Islamic monuments, including the octagonal plan and prismatic structure of the late seventh-century Dome of the Rock in Jerusalem (Grabar, 2006). Geometric patterns in the plane or radial symmetry appear extensively in floor mosaics, carved stucco, and wall ornaments at Khirbat al-Mafjar near Jericho (Hamilton, 1959), drawing upon the Late Antique artistic heritage of the late Roman, Byzantine, and Sasanian empires. In the ninth century, periodic patterns in the plane exhibit symmetry in the palaces at Samarra and in other more distant Abbasid centers such as Balkh, with similarities of style noted in architectural stucco, woodwork, ceramics, as well as in objects of other materials, such as rock crystal, inlaid wood, and ivory (Ettinghausen et al., 2001). All such periodic patterns have visible or implied square, rectangular, triangular, rhombic, or hexagonal grids and utilize symmetries of reflection, translation, glide reflection, and rotations of orders 2, 3, 4, and 6. At Nishapur in northeastern Iran, a wall panel in carved stucco (Fig. 1) exhibits a plane pattern with overlapping, interlacing hexagons, each spatial unit of the tiling filled with vegetal ornament that is also a characteristic of an Abbasid style. In these early examples of periodic patterns in Islamic art, there is a notable absence of pentagons and decagons, which offer geometric challenges in the spatial dimension. Unlike other regular polygons (equilateral triangles, squares, and hexagons), their interior angles are not factors of 360 . An early complex interlaced pattern with local tenfold rotations set within an implied rhombic grid, as reconstructed by Bonner (Fig. 2a), appears on the stellated octagonal tower of Mas’ud III in Ghazna (Fig. 2b), dating to the late eleventh or early twelfth century (Bonner, 2015). And a pattern with concentric Fig. 1 Nishapur, Iran. Carved stucco wall panel. Tenth century. Excavated by The Metropolitan Museum of Art (Acc. No. 40.170.442, Rogers Fund, 1940). www.metmuseum.org Page 2 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 2 Ghazna, Afghanistan. Tower of Mas’ud III, cut brick.1099–1115. (a). Interlaced pattern with tenfold rotations set within an implied rhombic grid, pattern reconstruction by Jay Bonner (2015). (b). Detail, registers of brick patterns (note Bonner’s analysis in Fig. 2a is of pattern that is third from top on left flange). (# David Collection, Copenhagen. Photo: Andre Leth. http://www.davidmus.dk/en/collections/islamic/dynasties/ghaznavids-and-ghurids/architecture/masud-iii) decagrams (ten-pointed stars), also set within an implied rhombic grid, may be discerned in the tympanum of the Gonbad-e Alaviyyan in Hamedan in western Iran (Fig. 3). The use of decagons and decagrams in periodic patterns is highly unusual and may indicate a major achievement in the utilization of geometry for architectural ornament, overcoming the intransigence of pentagons and decagons to tessellate the plane (see also Nava’i (2012) and Sarhangi (2013a)). Page 3 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 3 Hamadan, Iran. Gonbad-e Alaviyyan. Cut brick pattern of decagrams set within an implied rhombic grid. Late twelfth century. Analysis by C. Bier Fig. 4 Isfahan, Iran. Masjid-e Jomeh. North Dome (Gonbad-e Khaq), with central pentagon. 1088–89 (Josephine Powell photograph, Courtesy of Special Collections, Fine Arts Library, Harvard University) Forms and Patterns Based upon Divisions of the Circle In Spain, three domed chambers in front of the mihrab of the Great Mosque of Cordoba, dating from an extension of the building in the mid-tenth century, exhibit the earliest evidence of ribbed domes, constructed using four pairs of parallel ribs, which interlace to form an octagon and eight-pointed star (Fuentes González & Huerta, 2010). The earliest appearance of a pentagon seems to be in the north dome of the Masjid-e Jomeh, Isfahan, dated to 1088–1089 (Fig. 4), possibly to be associated with the mathematician, Omar Khayyam (Özdural, 1995, 1998). In addition, there are two patterns having local fivefold symmetries (convex and concave polygons) that also appear in the north dome chamber in recessed tympana in the transition zone below the dome. Page 4 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 5 Bukhara, Uzbekistan. Tomb of the Samanids, exterior view of articulated brickwork. Tenth century. (Courtesy: # Frank Harold, 1999, https://depts.washington.edu/silkroad/cities/uz/bukhara/bukhara.html) With a circular or polygonal plan, numerous prismatic tomb towers built of fired brick were constructed in Iran and neighboring lands during the ninth through twelfth centuries. The tradition continues with great architectural variation (Hillenbrand, 1994), including the extensive use of glazed bricks and ceramic mosaic, which led to the introduction of tiles. The design of these mausolea begins with a center point and circle; the circle may then be divided into equal segments. Early examples include the monuments at Radkan, Demavend, Lajim and Kharraqan in Iran, and Bust in Afghanistan. The most common form is an octagonal prism with a variety of conical or polyhedral roofing arrangements (Ashkan & Ahmad, 2012). Several different architectural structures also suggest experimentation with geometry and form; examples include the unique tapering stellated decagonal tower at Gonbad-e Qabus (dated 1006–1007) with its smooth conical roof and the cubic “tomb of the Samanids” in Bukhara (Fig. 5), with its elaborately reticulated brickwork arranged to display decorative patterns. Broug (2013, Fig. 1, pp. 21–24) clearly explains pattern families derivative of divisions of the circle into four or eight segments, three or six segments, and five segments in Islamic geometric patterns. In analyzing plane patterns in Islamic art, Bonner (2015) amplifies these categories based upon an elaboration of obtuse, median, or acute angles at points of intersections of bisected lines; he also adds a category for more complex patterns with two-point sharing, introduced in the late twelfth century. Typically many different geometric patterns appear on a single monument. From the early Islamic period until the mid-eleventh century, such two-dimensional patterns seem to be composed by design algorithms, whether executed in stone or glass mosaic (as in the Dome of the Rock and at Khirbat al-Mafjar) or in brick (as in Bukhara, Demavend, Isfahan, and elsewhere). A unit is repeated according to an organizing principle (today identified as a symmetry group), which results in a pattern. The geometry is emergent. For example, ornamental brickwork relies upon bricks being set in different orientations Page 5 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 6 Demavend, Iran. Tomb tower, showing vertical and horizontal orientation of bricks. Eleventh century. (Robert Byron, 1933–34, # Conway Library, Courtauld Institute of Art, http://archnet.org/media_contents/40177 [Harvard: administered by British Library]) (vertical and horizontal) or bricks used in proportions of 1:2:3, as at Demavend (Fig. 6), in the earlier tower at Kharraqan (dated 1067), and in blind niches at Gonbad-e Surkh at Maragha. Sometimes the bricks are set with mortar that appears to be intentionally set back, thereby more clearly articulating algorithmic patterns through the play of sunlight and shadow, projection, and recess, as in Bukhara, and on the facades of the two tomb towers at Kharraqan in western Iran. By the late eleventh century, there seems to be a change, first evident on the entry facades at Kharraqan (Fig. 7) and then on the façade of the Gonbad-e Surkh (Fig. 8), indicated by bricks having been cut to size to create different patterns. Despite its out-of-the-way location off the Hamadan-Qazvin road, the later tomb tower at Kharraqan (dated 1093) provides the earliest evidence for the introduction of glazed ceramic elements (Pickett, 1997; Stronach & Young, 1966). In the succeeding 100 years, the use of glazed architectural ceramics increased dramatically (Wilber, 1939). Blue-glazed bricks, using a copper colorant for the glaze, were used extensively to cover the long narrow triangular facets of tall pyramidal roofs with polygonal bases, which reflected sunlight from every direction – as noted by the historian, Yaqut (d. 1229), who describes their visibility from miles away serving as a beacon for travelers (Wilber, 1939, p. 23, n.25). A noteworthy early example of the use of glazing to highlight a particular pattern is the mid-twelfth-century tympanum of the Gonbad-e Sorkh at Maragha (Fig. 8). Its arcuate shape encloses three networks of intersecting hexagons, dodecagons, and nonagons sharing the same center point, with nonagons highlighted in turquoise glaze (Bier, 2012). The composition clearly reflects extraordinary mathematical understanding of how polygons and polygonal networks relate to one another geometrically. Surrounding the geometric composition, an elegant historical inscription gives the building’s date of construction as 4 March 1148 CE and the name of the patron, Fakhr al-Din Abu ʾl-ʿIzz ʿAbd al-ʿAziz ibn Mahmud, a local ruler also known as Qawam-e Azarbayjan, who 15 years later was host to a famous Jewish mathematician, Abu Nasr Samawʾal b. Yahya al-Magh̲ribi, born in Baghdad (his father was from Page 6 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 7 Kharraqan, Iran. Tomb tower, cut brick façade. Dated by historical inscription to 486H/1092–3 (Photo by Ann Gunter) Fig. 8 Maragha, Iran. Gonbad-e Sorkh, tympanum of cut brick and turquoise glaze. Dated by historical inscription to 542H/ 1148. Color transparency (C162) by Hans C. Seherr-Thoss, detail (Hans C. and Sonia P. Seherr-Thoss Photographs of Islamic Architecture, Freer Gallery of Art and Arthur M. Sackler Gallery Archives, Smithsonian Institution, Washington, D.C. Gift of Sonia P. Seherr-Thoss) Spain). Samawʾal retired to Maragha and converted to Islam on Friday, 8 November 1163, as he reports in his autobiography, which was completed in 1167 (Schmidtke, 2013). Page 7 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 9 Nakhchivan, Azerbaijan. Tomb tower of Mo’mine Khatun, detail of cut brick façade with turquoise glaze with 11-gons and 13-gons. Dated by historical inscription to 1186. Analysis by Nathan Voirol From the late twelfth century until the Mongol conquests of the thirteenth century, brick monuments in Iran, Iraq, and Azerbaijan demonstrate an aesthetic predilection for patterns with intersecting polygons, including stars (Pope & Ackerman, 1938/1939). The earlier use of intersecting squares, octagons, hexagons, and dodecagons, as at Kharraqan in Iran (Stronach & Young, 1966) in the late eleventh century and at Chist-e Sharif in Afghanistan (Broug, 2013, Fig. 1.1), of the late twelfth century, is superseded by patterns in which the arms of star polygons are extended to create “petals,” and lines are further extended to meet edge extensions of other star polygons by means of point joining. In several instances, such techniques are carried to an extreme: the decagonal prismatic mausoleum commissioned by Mo’mine Khatun at Nakhchivan in cut brick (Fig. 9) shows evidence of 11- and 13-gons (dated 1186), and a pattern executed in cut brick on the Choli Minaret at Erbil dating from the late twelfth century (Fig. 10) shows a pattern with 11- and 14-gons as reconstructed (Ajlouni & Justa, 2011). In the citadel at Aleppo, a pattern carved in wood from an Ayyubid panel in the early thirteenth century shows a progression of rayed polygons forming a sequence from 12-, 11-, 10-, 11-, and 12-gons, as analyzed by Herzfeld (1943), which is noted and discussed more recently by several authors (Allen, 1988, p. 53 and Fig. 57; Tabbaa, 2001, p. 96, Fig. 45). Bonner (2015) notes the presence of a more elegant rendition of the same pattern on the portal of a Seljuk monument in Anatolia, the Erkilet Kiosk in Kayseri (dated 1241). The design appears in Schneider (1980, no. 417). Such complex patterns, with or without illusionary interlacing, suggest an extensive array of experimentation with geometric shapes in polygonal networks and an ongoing effort to achieve more elegant refinement. Page 8 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 10 Erbil, Iraq. Minaret Choli, panel of cut brick and turquoise glaze with 11-gons and 14-gons. Pattern reconstruction, as a sequential process as developed by Ajlouni and Justa (2011, Fig. 4) Girih: Grid-Based Patterns In the scholarly literature, the term girih (Persian, “knot”) has been applied to this ornamental tradition although its first documented appearance is attributed to architects working in nineteenth-century Central Asia (Baklanov, 1947). Necipoğlu (1995) carefully recounts the historical use of this term and consciously applies it to describe an artistic mode that is expressed through diagrams in the Topkapi scroll, which is an undated, anonymous scroll without text. By means of comparative stylistic and contextual analysis, Necipoğlu (1995, p. ix) ascribes this seminal historical document to the Timurid-Turkmen realm of Iran in the late fifteenth-early sixteenth century, but accepts that it is indicative of a much longer developmental evolution of designs used for architectural ornamentation. Other authors use the term girih with slightly different inflections (Denny, 2002; Rogers, 1973; Tabbaa, 2001). The Topkapi scroll, published in a 2:1 facsimile edition (Necipoğlu, 1995), serves as a primary source, the data from which has been utilized in numerous contemporary studies in recent years to understand mathematical aspects of pattern construction in two and three dimensions, knowledge of which had been lost. Lu and Steinhardt (2007a) introduced the term “girih tiles” for what Bonner (2003) had termed sub-grids that may have allowed for the possibility of modular constructions (see also Sarhangi, Jablan, & Sazdanovic, 2005; Sarhangi, 2008a). The Topkapi scroll illustrates several design templates that demonstrate such sub-grids (see Cromwell, 2010c), in which one-point or two-point matching rules apply. Bonner (2003) associates these patterns with the design methodology he calls the “polygonal technique,” which is similar to the “polygons in contact” identified by Hankin (1925), taken up by Kaplan (2000a, 2000b, 2004, 2005) in relation to the use of computer technologies to generate Islamic star patterns. In Encyclopedia Iranica, the entry for gereh-sāzī defines the term as “making knot” and treats it in relation to woodwork (Blair, 2012) and architecture (Milwright, 2012). The section on woodwork focuses on the early fourteenth century in Iran, but omits reference to its important Zengid and Ayyubid antecedents, which reflect the acknowledged skill of three generations of woodworkers from Aleppo Page 9 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 11 Jerusalem. Al-Aqsa mosque, minbar (destroyed by arson in 1969). Detail. Carved and inlaid wood, made in Aleppo, late twelfth century. (Photo by K. A. C. Creswell. # Creswell Archive, Ashmolean Museum, neg. EA.CA.5009. Image courtesy of Fine Arts Library, Harvard College Library, http://archnet.org/media_contents/35325 [Harvard: # administered by Ashmolean]). named al-Ma’ali, whose names are inscribed in their works (Mayer, 1958, pp. 48, 63, 65; Tabbaa, 2001, pp. 91, 94, 96). The value accorded such works is affirmed by historical circumstances: the task of installing the famous minbar (Fig. 11) commissioned by Nur al-Din al-Zanji (1146–1174) in the Aqsa Mosque in Jerusalem when the city was restored to Muslim rule after the Crusades fell to his successor, Saladin (Singer, 2008). And the Damascus-born thirteenth century biographer, Ibn Abi Usaibiʾa, mentions a carpenter given the title muhandis (Arabic: architect or engineer), who had studied geometry with the greatest mathematician of his age, Sharaf al-Dīn al-Ṭūsī (c.1135–1213 CE) (Tabbaa, 2001, p. 88; Mayer, 1958, p.53). Blair (2012) describes the product of the gereh-sāzī technique as geometric interlaced strapwork ornament within a lattice framework and notes that it became “very popular in the Safavid period.” From the Safavid period onward, colored glass windows, orosi, follow similar layouts of geometric patterning (Koliji, 2012b). And the design scrolls of the Qajar state architect, Mirza Akbar, now in the Victoria and Albert Museum, London, attest to a continuation of the tradition into the late nineteenth century (Wade, 2015 http://patterninislamicart.com/drawings-diagrams-analyses/7/mirza-akbararchitectural-scrolls). Periodic and Quasiperiodic Patterns Contemporary crystallographic interest in quasicrystals and the mathematical study of quasiperiodic patterns have drawn recent attention to historic two-dimensional designs present in Islamic architecture Page 10 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 12 Maragha, Iran. Gonbad-e Kabud, decagonal tomb tower. Dated by historical inscription to 563H/1196-97. Pattern analysis and generation by Rima Ajlouni (2012, Fig. 7), showing long-range quasi-periodic pattern with local five-fold and global ten-fold radial symmetries from late twelfth century Iran (Bonner, 2003; Castéra, 2011; Lu & Steinhardt, 2007a, 2007b; Makovicky, 1992, 2007, 2008), with quasiperiodic octagonal lattices and decagonal patterns also recognized in Spain and Morocco (Makovicky & Fenoll Hach-Alí, 1996; Makovicky, Pérez, & Fenoll Hach-Alí, 1998). Castera (2003) considers self-similarity in zellij work in Andalusia and Morocco, and muqarnas, offering information concerning both transformation rules and the recursive process in two and three dimensions; the examples he considers have fourfold symmetry. Debate continues concerning the presence of periodicity – in which translational symmetry is key – in contrast to the long-range global order of quasiperiodic patterns, in which there is a high degree of symmetry without translation (Al Ajlouni, 2012; Bier, 2013; Cromwell, 2009; Saltzman, 2008) (Figs. 12, 13 and 14). Lu and Steinhardt (2007a) focus on one early antecedent of the quasiperiodicity of patterns being studied today, which are at the forefront of mathematical thinking and identifying its underlying tiling. This pattern, which surrounds the shaft of the decagonal Gunbad-e Kabud in Maragha, had earlier been studied in detail by Makovicky (1992), a crystallographer (see also Makovicky, 2007, 2008 and Bier, 2011). As mentioned above, Bonner (2015) emphasizes that four varieties of pattern families (Fig. 13) may be produced based on the angles of intersecting lines (acute, median, obtuse, and two-point). He also notes that the underlying polygonal structures found in the fivefold system (especially the “ring of pentagons” that typically surround the decagons) very likely informed artists in their composition of more complex underlying generative tilings for patterns with multiple centers of localized symmetry. Based on this interpretation of design methodologies, Bonner (2015) argues that Lu and Steinhardt’s girih tiles (2007a) comprise a limited subset of the larger set of fivefold modules of the Islamic ornamental tradition, namely, only the median pattern family in which 72-degree angles are paramount. Page 11 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 13 A polygonal network (for example, with decagons and a ring of pentagons as illustrated) serves as a sub-grid to yield four pattern families based upon angles of intersection at mid-points (acute, obtuse, median) or two-points. Analysis developed by Jay Bonner (2003) Fig. 14 Istanbul, Turkey. Şehzade Mosque, circular medallion of minbar. 1543–1548 (Photo and analysis by Miroslaw Majewski (2011, Fig. 91)) Page 12 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Fig. 15 Istanbul, Turkey.Detail of mother-of-pearl work, Ottoman, 17th century. (Photography by Roger Von Oech. http:// blog.creativethink.com/images/2008/09/14/topkapi_inlay_450.gif) From the height of the Ottoman Empire in the sixteenth century, two carved and pierced medallions framing the marble minbar, or pulpit, of the Şehzade Mosque in Istanbul (Fig. 14) and the Selimiye Mosque in Edirne (both monuments designed by the great Ottoman imperial architect Sinan) also show quasiperiodic patterns (Bier, 2014). Other pierced openwork reticulated marble screens show periodic patterns of intersecting polygonal networks, demonstrating a systematic study of polygonal nets at least 50 years before publication of Kepler’s Harmonices Mundi in 1619 (Bier, 2014). From the Risale-ye Mi’mariyya, a seventeenth century treatise on Mehmed Agha, a chief imperial architect trained by Sinan, we know that architects were trained in mother-of-pearl work (Fig. 15) in the imperial gardens of Topkapi Palace, as a means of learning the “science of geometry,” which was considered the basis of architectural knowledge (Crane, 1987). Handasa/Muhandis The term handasa (Arabic: geometry) is a loan word from Middle Persian, its use dating from before the Arab conquests of the seventh century (Suter, 2014). ‘Ilm al-handasa is both “knowledge of geometry” and “the science of geometry,” a genitive-possessive construction. The title muhandis reflects an Arabic grammatical construction applied to the Persian loan word as if it were an Arabic verb with a quadriliteral root. Although it came to mean architect or engineer, its original primary meaning would have been associated with agency, as “geometrician” or “one who works with geometry.” In the thirteenth century, a historian of the Delhi Sultanate, Juzjani, extols the palatial architecture of the Ghurids in his hometown of Firuzkuh as being what “no geometrician has made manifest.” Discounting fulsome panegyric, it seems reasonable to assume that this reflects the actual practice of builders and craftsmen – making geometry manifest. Page 13 of 21 Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures DOI 10.1007/978-94-007-3934-5_10111-1 # Springer Science+Business Media Dordrecht 2015 Geometry as Cultural Expression (Geometry Made Manifest) That geometry itself was held in high cultural esteem is clear from the role it plays in various script styles, both related to calligraphy of the Qurʾān which Muslims accept as having been revealed in Arabic and in the chancery (Blair, 1992, 1998, p. 206; Schimmel, 1984), as well as calligraphic inscriptions on architectural monuments. A particular focus appears in the early calligraphic reforms of Ibn Muqla (d. 940) (Tabbaa, 1991, 1994, 2001), and the spread of Arabic scripts to other languages must also be noted. Geometry is also referenced in Arabic and Persian poetry (e.g., ‘Unsuri, Sana’i) and in the late twelfth century, Nezami (Soucek, 1975); it also figures in poetic descriptions of palace architecture (Meisami, 2001). A theological dimension may also be inferred; when Sana’i (d. 1131–41), for example, was poet laureate of the Ghaznavid court, he likened “the ability to behold the divine manifestation. . .to the intellectual way of perception of a geometrician” – “You only see with your imagination and your senses when you have not learned about lines, planes, and points” (De Bruijn, 1983, p. 238). Points, lines, and angles, yielding intersecting lines and patterns in the plane, occur with such frequency in the pre-Mongol monuments of Afghanistan, Iran, Azerbaijan, and Iraq, as to suggest that they concern a visual discourse related to mathematical ideas. The dynamic play with patterns in the plane in the eleventh and twelfth centuries parallels the design innovations in three-dimensional forms including varieties of prismatic structures for tomb monuments, conical and pyramidal roofing arrangements, and the early development of muqarnas (Ashkan & Ahmad, 2012; Bloom, 1988; Tabbaa, 1985). Scholarship on geometry in Islamic art and architecture in the twentieth century has often reflected Western prejudices, including notions that geometric patterns are abstract, repetitive, and nonrepresentational and therefore bear no meaning. Framing the interpretation of ornament as ornamental and decorative is a discourse based in the academic disciplines of the histories of art and architecture, which may have precluded cultural understanding of geometric patterns within their original cultural contexts. One renewed effort to define aesthetics in Islamic art relates optical illusion and aesthetic metaphor and explores decorative patterns of the Alhambra in terms of phenomenology and semiotics (Gonzalez, 2001). Another compares artistry to nature and identifies “reality [to be] less impressive than its artistic image,” as the “quintessence of the Islamic visual arts” (Behrens-Abouseif, 1999, p. 141), drawing upon the works of al-Farabi, Ibn al-Haytham, and Ibn Sīnā. Belting (2011) also draws upon the theories of vision of Ibn al-Haytham, linking Arabic science to the development of linear perspective and Renaissance art. If Islamic geometric ornament is reexamined within the particular cultural context of its creative formative period in the eleventh and twelfth centuries, it may well be found that it is an expressive rather than symbolic artistic form and may be reinterpreted in relation to other forms of cultural expression – including Qurʾānic inscriptions and tafsir (Arabic exegesis or commentary), poetry, and contemporary texts on mathematics. In approaching geometry in Islamic art in this relational manner, we may be able to chart a new course for the study of pattern as lines of mathematical thought expressed visually. References Abas, S. J., Salman, A. S., Moustafa, A., & Atiyah, M. (1995). Symmetries of Islamic geometrical patterns. Singapore: World Scientific. Ajlouni, R., & Justa, P. (2011). 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