:Islamic geometric patterns

Islamic geometric patterns are derived from simpler designs used in earlier cultures: Greek, Roman, and Sasanian. They are one of three forms of Islamic decoration, the others being the arabesque based on curving and branching plant forms, and Islamic calligraphy; all three are frequently used together.{{cite web |title=Geometric Patterns in Islamic Art |url=http://www.metmuseum.org/toah/hd/geom/hd_geom.htm |website=Heilbrunn Timeline of Art History |publisher=Metropolitan Museum of Art |access-date=1 December 2015}}{{cite book |last1=Hankin |first1=Ernest Hanbury |author-link=Ernest Hanbury Hankin |title=The Drawing of Geometric Patterns in Saracenic Art. Memoirs of the Archaeological Survey of India No. 15 |date=1925 |publisher=Government of India Central Publication Branch |url=http://www.islamic-art.org/ArticlesOnline/BAViewPDF.asp?ArticleID=108}} From the 9th century onward, a range of sophisticated geometric patterns based on polygonal tessellation began to appear in Islamic art, eventually becoming dominant.{{Cite book |last=Bonner |first=Jay |url=https://books.google.com/books?id=o9IxDwAAQBAJ&pg=PA6 |title=Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction |date=2017-08-17 |publisher=Springer |isbn=978-1-4419-0217-7 |page=6}}

Islamic art mostly avoids figurative images to avoid becoming objects of worship.{{cite web |last1=Bouaissa |first1=Malikka |title=The crucial role of geometry in Islamic art |url=http://www.alartemag.be/en/en-art/the-crucial-role-of-geometry-in-islamic-art/ |publisher=Al Arte Magazine |access-date=1 December 2015 |date=27 July 2013 |archive-date=27 March 2019 |archive-url=https://web.archive.org/web/20190327135234/http://www.alartemag.be/en/en-art/the-crucial-role-of-geometry-in-islamic-art/ |url-status=dead }}{{cite book | last=Bonner | first=Jay | title=Islamic geometric patterns : their historical development and traditional methods of construction | publisher=Springer | location=New York | year=2017 | isbn=978-1-4419-0216-0 | oclc=1001744138 | page=1}} This aniconism in Islamic culture caused artists to explore non-figural art, and created a general aesthetic shift toward mathematically based decoration.{{cite journal |last=Bier |first=Carol |date=Sep 2008 |title=Art and Mithãl: Reading Geometry as Visual Commentary |journal=Iranian Studies |volume=41 |issue=4 |pages=491–509 |doi=10.1080/00210860802246176 |jstor=25597484|s2cid=171003353 }}

Authors such as Keith Critchlow{{efn|Critchlow is a professor of architecture, and the author of a book on Islamic patterns.}} argue that Islamic patterns are created to lead the viewer to an understanding of the underlying reality, rather than being mere decoration, as writers interested only in pattern sometimes imply.{{cite book |author-link=Keith Critchlow |author=Critchlow, Keith |title=Islamic Patterns : an analytical and cosmological approach |publisher=Thames and Hudson |year=1976 |isbn=0-500-27071-6}}{{cite book |author=Field, Robert |title=Geometric Patterns from Islamic Art & Architecture |publisher=Tarquin Publications |year=1998 |isbn=978-1-899618-22-4}} In Islamic culture, the patterns are believed to be the bridge to the spiritual realm, the instrument to purify the mind and the soul.{{cite journal |last1=Ahuja |first1=Mangho |last2=Loeb |first2=A. L. |date=1995 |title=Tessellations in Islamic Calligraphy |jstor=1576154 |journal=Leonardo |volume=28 |issue=1 |pages=41–45|doi=10.2307/1576154 |s2cid=191368443 }} David Wade{{efn|Wade is the author of a series of books on pattern in various artforms.}} states that "Much of the art of Islam, whether in architecture, ceramics, textiles or books, is the art of decoration – which is to say, of transformation." Wade argues that the aim is to transfigure, turning mosques "into lightness and pattern", while "the decorated pages of a Qur’an can become windows onto the infinite."{{cite web |last1=Wade |first1=David |title=The Evolution of Style |url=http://patterninislamicart.com/background-notes/the-evolution-of-style |website=Pattern in Islamic Art |access-date=12 April 2016 |quote=Much of the art of Islam, whether in architecture, ceramics, textiles or books, is the art of decoration – which is to say, of transformation. The aim, however, is never merely to ornament, but rather to transfigure. ... The vast edifices of mosques are transformed into lightness and pattern; the decorated pages of a Qur’an can become windows onto the infinite. Perhaps most importantly, the Word, expressed in endless calligraphic variations, always conveys the impression that it is more enduring than the objects on which it is inscribed. |archive-date=19 May 2022 |archive-url=https://web.archive.org/web/20220519142349/https://patterninislamicart.com/background-notes/the-evolution-of-style |url-status=dead }} Against this, Doris Behrens-Abouseif{{efn|Behrens-Abouseif is a professor of the history of art and architecture at SOAS.}} states in her book Beauty in Arabic Culture that a "major difference" between the philosophical thinking of Medieval Europe and the Islamic world is exactly that the concepts of the good and the beautiful are separated in Arabic culture. She argues that beauty, whether in poetry or in the visual arts, was enjoyed "for its own sake, without commitment to religious or moral criteria".{{cite book |last1=Behrens-Abouseif |first1=Doris |title=Beauty in Arabic Culture |date=1999 |publisher=Markus Wiener |isbn=978-1-558-76199-5 |pages=7–8}}

File:Tiles inside the Jame Mosque of Yazd 01.JPG|Tiles inside the Jame Mosque of Yazd, Persia, with geometric and vegetal patterns

File:Bou inania DSCF2978.jpg|Bou Inania Madrasa, Fes, Morocco, originally c. 1350, with geometric patterns in zellij tilework

File:162-MoroccoOpenDoors.jpg|A variety of vernacular decorative Islamic styles in Morocco: wooden panels, zellij tilework, stucco calligraphy, and floral door panels

File:Alhambra Detail 17.JPG|Arch in the Alhambra with Mocárabe stalactite work

File:Jar Met 56.185.15.jpg|Ayyubid Raqqa ware stoneware glazed jar with overlapping circles grid pattern. Syria, 12th/13th century

File:Green mosque archway.JPG|An archway in the Ottoman Green Mosque, Bursa, Turkey (1424), with girih 10-point stars and pentagons

File:Shah Nematollah Vali Shrine 08.jpg, Mahan, Iran, 1431. The blue girih-tiled dome contains stars with, from the top, 5, 7, 9, 12, 11, 9 and 10 points in turn. 11-point stars are rare in Islamic art.{{cite book |author=Broug, Eric |title=Islamic Geometric Patterns |publisher=Thames and Hudson |year=2008 |isbn=978-0-500-28721-7 |pages=183–185, 193}}|alt=Dome of shrine decorated with many different shapes of star]]

{{further|Mathematics and art|Girih}}

Many Islamic designs are built on squares and circles, typically repeated, overlapped and interlaced to form intricate and complex patterns. A recurring motif is the 8-pointed star, often seen in Islamic tilework; it is made of two squares, one rotated 45 degrees with respect to the other. The fourth basic shape is the polygon, including pentagons and octagons. All of these can be combined and reworked to form complicated patterns with a variety of symmetries including reflections and rotations. Such patterns can be seen as mathematical tessellations, which can extend indefinitely and thus suggest infinity.{{cite web |last1=Hussain |first1=Zarah |title=Introduction to Islamic art |url=http://www.bbc.co.uk/religion/religions/islam/art/art_1.shtml |publisher=BBC |access-date=1 December 2015 |date=30 June 2009}} They are constructed on grids that require only ruler and compass to draw.{{cite web |last1=Bellos |first1=Alex |last2=Broug (Illustrator) |first2=Eric |title=Muslim rule and compass: the magic of Islamic geometric design |url=https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/feb/10/muslim-rule-and-compass-the-magic-of-islamic-geometric-design |work=The Guardian |access-date=1 December 2015 |date=10 February 2015}} Artist and educator Roman Verostko argues that such constructions are in effect algorithms, making Islamic geometric patterns forerunners of modern algorithmic art.{{cite web |last1=Verostko |first1=Roman |title=Algorithmic Art |url=http://www.verostko.com/algorithm.html |date=1999 |orig-year=1994}}

The circle symbolizes unity and diversity in nature, and many Islamic patterns are drawn starting with a circle.{{cite web |last1=Henry |first1=Richard |title=Geometry – The Language of Symmetry in Islamic Art |url=http://artofislamicpattern.com/resources/educational-posters/ |website=Art of Islamic Pattern |access-date=1 December 2015}} For example, the decoration of the 15th-century mosque in Yazd, Persia is based on a circle, divided into six by six circles drawn around it, all touching at its centre and each touching its two neighbours' centres to form a regular hexagon. On this basis is constructed a six-pointed star surrounded by six smaller irregular hexagons to form a tessellating star pattern. This forms the basic design which is outlined in white on the wall of the mosque. That design, however, is overlaid with an intersecting tracery in blue around tiles of other colours, forming an elaborate pattern that partially conceals the original and underlying design.{{cite web |last1=Lockerbie |first1=John |title=Islamic Design: Arabic / Islamic geometry 01 |url=http://www.catnaps.org/islamic/geometry.html#ribbon |website=Catnaps.org |access-date=2 December 2015 |archive-url=https://web.archive.org/web/20190413160232/http://www.catnaps.org/islamic/geometry.html |archive-date=13 April 2019 |url-status=live}} A similar design forms the logo of the Mohammed Ali Research Center.{{cite web |title=Islamic Art and Geometric Design |url=http://www.moha.center/index.php/news-moha-en/209-moha-logo-design-en# |publisher=MOHA |access-date=3 December 2015 |date=2014 |archive-url=https://web.archive.org/web/20151203133727/http://www.moha.center/index.php/news-moha-en/209-moha-logo-design-en |archive-date=3 December 2015}} The logo's construction is demonstrated in an animation on the MOHA website.

One of the early Western students of Islamic patterns, Ernest Hanbury Hankin, defined a "geometrical arabesque" as a pattern formed "with the help of construction lines consisting of polygons in contact." He observed that many different combinations of polygons can be used as long as the residual spaces between the polygons are reasonably symmetrical. For example, a grid of octagons in contact has squares (of the same side as the octagons) as the residual spaces. Every octagon is the basis for an 8-point star, as seen at Akbar's tomb, Sikandra (1605–1613). Hankin considered the "skill of the Arabian artists in discovering suitable combinations of polygons .. almost astounding." He further records that if a star occurs in a corner, exactly one quarter of it should be shown; if along an edge, exactly one half of it.

The Topkapı Scroll, made in Timurid dynasty Iran in the late-15th century or beginning of the 16th century, contains 114 patterns including coloured designs for girih tilings and muqarnas quarter or semidomes.{{cite book |title=Geometric Design in Timurid/Turkmen Architectural Practice: Thoughts on a Recently Discovered Scroll and Its Late Gothic Parallels |author=Gülru Necipoğlu |work=Timurid Art and Culture – Iran and Central Asia in the Fifteenth Century (eds (Golombek, L. and Subtelny, M.) |publisher=E.J. Brill |year=1992 |url=http://isites.harvard.edu/fs/docs/icb.topic570061.files/articles/Geometric_Design.pdf |access-date=2015-12-02 |archive-date=2015-12-22 |archive-url=https://web.archive.org/web/20151222005112/http://isites.harvard.edu/fs/docs/icb.topic570061.files/articles/Geometric_Design.pdf |url-status=dead }}{{cite journal |title=Artisans and Mathematicians in Medieval Islam. The Topkapi Scroll: Geometry and Ornament in Islamic Architecture by Gülru Necipoğlu (Review) |author=Saliba, George |year=1999 |journal=Journal of the American Oriental Society |volume=119 |issue=4 |pages=637–645 |jstor=604839 |doi=10.2307/604839}} (subscription required){{cite web |author=van den Hoeven, Saskia, van der Veen, Maartje |title=Muqarnas-Mathematics in Islamic Arts |url=http://www.wiskuu.nl/muqarnas/Muqarnas_english8.pdf |access-date=15 January 2016 |archive-url=https://web.archive.org/web/20130927070005/http://www.wiskuu.nl/muqarnas/Muqarnas_english8.pdf |archive-date=27 September 2013}}

The mathematical properties of the decorative tile and stucco patterns of the Alhambra palace in Granada, Spain have been extensively studied. Some authors have claimed on dubious grounds to have found most or all of the 17 wallpaper groups there.{{cite journal |last1=Perez-Gomez |first1=R. |title=The Four Regular Mosaics Missing in the Alhambra |journal=Comput. Math. Applic. |date=1987|volume=14 |issue=2 |pages=133–137 |url=http://www.york.ac.uk/depts/maths/histstat/symmetry/alhambra.pdf |doi=10.1016/0898-1221(87)90143-x|doi-access=free }}{{cite journal |last1=Grünbaum |first1=Branko |title=What Symmetry Groups Are Present in the Alhambra? |journal=Notices of the AMS |date=June 2006 |volume=53 |issue=6 |pages=670–673 |url=https://www.ams.org/notices/200606/comm-grunbaum.pdf}} Moroccan geometric woodwork from the {{nowrap|14th to 19th centuries}} makes use of only 5 wallpaper groups, mainly p4mm and c2mm, with p6mm and p2mm occasionally and p4gm rarely; it is claimed that the "Hasba" (measure) method of construction, which starts with n-fold rosettes, can however generate all 17 groups.{{cite journal |last1=Aboufadil |first1=Y. |last2=Thalal |first2=A. |last3=Raghni |first3=M. A. E. I.|title=Symmetry groups of Moroccan geometric woodwork patterns |journal=Journal of Applied Crystallography |date=2013 |volume=46 |issue=6 |pages=1834–1841 |doi=10.1107/S0021889813027726 |url=http://scripts.iucr.org/cgi-bin/paper?S0021889813027726|url-access=subscription }}

File:Topkapi Scroll P290.JPG|Two-dimensional designs for two quarter-dome muqarnas – as a seashell (top), as a fan (bottom). Topkapı Scroll, 15th century

File:Darb-i Imam shrine spandrel.JPG|Girih tiling in the decagonal pattern on a spandrel from the Darb-e Imam shrine

File:Spandrel-large scale pattern.svg|Construction of girih pattern in Darb-e Imam spandrel (yellow line). Construction decagons blue, bowties red. The strapwork cuts across the construction tessellation.

File:OctagonalTesselations2.png|Analysis of octagonal patterns in Mughal architecture by Ernest Hanbury Hankin, 1925. 8-pointed stars emerge (lower right) where heavy black lines cross.

File:Decoration on the wall of the masoleoum of Itmad-ud-Daulah's tomb 1.jpg|Decoration in Tomb of I'timād-ud-Daulah, Agra, showing correct treatment of sides and corners. A quarter of each 6-point star is shown in each corner; half stars along the sides.

File:Architectural Drawing for brick vaulting, Iran, 1800-70.JPG|Architectural drawing for brick vaulting, Iran, probably Tehran, 1800–70

File:Grande Mosquée de Kairouan, carreaux lustrés - Kairouan's Great Mosque, luster tiles.jpg tiles in the Great Mosque of Kairouan, Tunisia. 836 onwards|alt=Simple early Islamic geometric tilework]]

The earliest geometrical forms in Islamic art were occasional isolated geometric shapes such as 8-pointed stars and lozenges containing squares. These date from 836 in the Great Mosque of Kairouan, Tunisia, and since then have spread all across the Islamic world.{{cite journal |author1=Abdullahi, Yahya |author2=Bin Embi, Mohamed Rashid|title=Evolution of Islamic geometric patterns |journal=Frontiers of Architectural Research |year=2013 |volume=2 |issue=2 |pages=243–251 |doi=10.1016/j.foar.2013.03.002|doi-access=free }}

File:Turkey, Konya - Alaeddin Mosque 03.jpg in the Alâeddin Mosque, Konya, Turkey. 1220 onwards|alt=Middle stage Islamic patterns]]

The next development, marking the middle stage of Islamic geometric pattern usage, was of 6- and 8-point stars, which appear in 879 at the Ibn Tulun Mosque, Cairo, and then became widespread.

A wider variety of patterns were used from the 11th century. Abstract 6- and 8-point shapes appear in the Tower of Kharaqan at Qazvin, Persia in 1067, and the Al-Juyushi Mosque, Egypt in 1085, again becoming widespread from there, though 6-point patterns are rare in Turkey.

In 1086, 7- and 10-point girih patterns (with heptagons, 5- and 6-pointed stars, triangles and irregular hexagons) appear in the Jameh Mosque of Isfahan. 10-point girih became widespread in the Islamic world, except in the Spanish Al-Andalus. Soon afterwards, sweeping 9-, 11-, and 13-point girih patterns were used in the Barsian Mosque, also in Persia, in 1098; these, like 7-point geometrical patterns, are rarely used outside Persia and central Asia.

Finally, marking the end of the middle stage, 8- and 12-point girih rosette patterns appear in the Alâeddin Mosque at Konya, Turkey in 1220, and in the Abbasid palace in Baghdad in 1230, going on to become widespread across the Islamic world.

File:Mihrab at Jama Masjid, Fatehpur Sikri.jpg. 1571-5|alt=Elaborate late stage Islamic woodwork]]

The beginning of the late stage is marked by the use of simple 16-point patterns at the Hasan Sadaqah mausoleum in Cairo in 1321, and in the Alhambra in Spain in 1338–1390. These patterns are rarely found outside these two regions. More elaborate combined 16-point geometrical patterns are found in the Sultan Hassan complex in Cairo in 1363, but rarely elsewhere. Finally, 14-point patterns appear in the Jama Masjid at Fatehpur Sikri in India in 1571–1596, but in few other places.{{efn|One such place is the Mustansiriyya Madrasa in Baghdad, as illustrated by Broug.{{cite book |author=Broug, Eric |title=Islamic Geometric Design |publisher=Thames and Hudson |year=2013 |isbn=978-0-500-51695-9 |page=173}}}}

Several artforms in different parts of the Islamic world make use of geometric patterns. These include ceramics, girih strapwork, jali pierced stone screens, kilim rugs, leather, metalwork, muqarnas vaulting, shakaba stained glass, woodwork, and zellij tiling.

{{further information|Islamic pottery}}

Ceramics lend themselves to circular motifs, whether radial or tangential. Bowls or plates can be decorated inside or out with radial stripes; these may be partly figurative, representing stylised leaves or flower petals, while circular bands can run around a bowl or jug. Patterns of these types were employed on Islamic ceramics from the Ayyubid period, 13th century. Radially symmetric flowers with, say, 6 petals lend themselves to increasingly stylised geometric designs which can combine geometric simplicity with recognisably naturalistic motifs, brightly coloured glazes, and a radial composition that ideally suits circular crockery. Potters often chose patterns suited to the shape of the vessel they were making.{{cite web |title=Geometric Decoration and the Art of the Book. Ceramics |url=http://www.discoverislamicart.org/exhibitions/ISL/geometric/exhibition.php?theme=1 |publisher=Museum with no Frontiers |access-date=7 December 2015}} Thus an unglazed earthenware water flask{{efn|Leaving the flask porous allowed evaporation, keeping the water cool.}} from Aleppo in the shape of a vertical circle (with handles and neck above) is decorated with a ring of moulded braiding around an Arabic inscription with a small 8-petalled flower at the centre.{{cite web |title=Flask |url=https://collections.vam.ac.uk/item/O82540/flask/ |publisher=Victoria and Albert Museum |access-date=7 December 2015}}

{{Main article|Girih}}

Girih are elaborate interlacing patterns formed of five standardized shapes. The style is used in Persian Islamic architecture and also in decorative woodwork.{{cite encyclopedia |title=Gereh-Sāzī |url=http://www.iranicaonline.org/articles/gereh-sazi |publisher=Encyclopaedia Iranica |access-date=2 December 2015}} Girih designs are traditionally made in different media including cut brickwork, stucco, and mosaic faience tilework. In woodwork, especially in the Safavid period, it could be applied either as lattice frames, left plain or inset with panels such as of coloured glass; or as mosaic panels used to decorate walls and ceilings, whether sacred or secular. In architecture, girih forms decorative interlaced strapwork surfaces from the 15th century to the 20th century. Most designs are based on a partially hidden geometric grid which provides a regular array of points; this is made into a pattern using 2-, 3-, 4-, and 6-fold rotational symmetries which can fill the plane. The visible pattern superimposed on the grid is also geometric, with 6-, 8-, 10- and 12-pointed stars and a variety of convex polygons, joined by straps which typically seem to weave over and under each other.{{cite web |title=Gereh-Sazi |url=http://english.tebyan.net/newindex.aspx?pid=177143 |publisher=Tebyan |access-date=4 December 2015 |date=20 August 2011}} The visible pattern does not coincide with the underlying construction lines of the tiling. The visible patterns and the underlying tiling represent a bridge linking the invisible to the visible, analogous to the "epistemological quest" in Islamic culture, the search for the nature of knowledge.{{Cite journal |last=Hooman |first=Koliji |date=April 2016 |title=Gazing Geometries: Modes of Design Thinking in Pre-Modern Central Asia and Persian Architecture |journal=Nexus Network Journal |volume=18 |pages=105–132 |doi=10.1007/s00004-016-0288-6|doi-access=free }}

File:Kairo Ibn Tulun Moschee BW 6.jpg: window with girih-style 10-point stars (at rear), with floral roundels in octagons forming a frieze at front]]

{{Main article|Jali}}

Jali are pierced stone screens with regularly repeating patterns. They are characteristic of Indo-Islamic architecture, for example in the Mughal dynasty buildings at Fatehpur Sikri and the Taj Mahal. The geometric designs combine polygons such as octagons and pentagons with other shapes such as 5- and 8-pointed stars. The patterns emphasized symmetries and suggested infinity by repetition. Jali functioned as windows or room dividers, providing privacy but allowing in air and light.{{cite web |title=For Educators: Geometric Design in Islamic Art: Image 15 |url=http://www.metmuseum.org/learn/for-educators/publications-for-educators/art-of-the-islamic-world/unit-three/featured-works-of-art/image-15 |publisher=Metropolitan Museum of Art |access-date=2 December 2015}} Jali forms a prominent element of the architecture of India.{{cite web |title=intypes. perforate |url=https://intypes.cornell.edu/expanded.cfm?erID=107 |publisher=Cornell University |access-date=18 January 2016}} The use of perforated walls has declined with modern building standards and the need for security. Modern, simplified jali walls, for example made with pre-moulded clay or cement blocks, have been popularised by the architect Laurie Baker.{{cite news |last1=Varanashi |first1=Satyaprakash |title=The multi-functional jaali|url=http://www.thehindu.com/todays-paper/tp-features/tp-propertyplus/the-multifunctional-jaali/article1138372.ece |access-date=18 January 2016 |work=The Hindu|date=30 January 2011}} Pierced windows in girih style are sometimes found elsewhere in the Islamic world, such as in windows of the Mosque of Ibn Tulun in Cairo.{{cite book |last1=Mozzati |first1=Luca |title=Islamic Art |date=2010 |publisher=Prestel |isbn=978-3-7913-4455-3 |page=27}}

{{Main article|Kilim}}

File:Wolf's Mouth Kilim Motif.jpg such as the Wolf's Mouth (Kurt Aǧzi), to protect the flocks against wolves, are often woven into tribal kilims.]]

A kilim is an Islamic{{cite web |last1=Thompson |first1=Muhammad |last2=Begum|first2=Nasima |title=Islamic Textile Art and how it is Misunderstood in the West – Our Personal Views |url=http://www.turkotek.com/salon_00101/salon.html |publisher=Salon du Tapis d'Orient |access-date=3 December 2015}} flatwoven carpet (without a pile), whether for household use or a prayer mat. The pattern is made by winding the weft threads back over the warp threads when a colour boundary is reached. This technique leaves a gap or vertical slit, so kilims are sometimes called slit-woven textiles. Kilims are often decorated with geometric patterns with 2- or 4-fold mirror or rotational symmetries. Because weaving uses vertical and horizontal threads, curves are difficult to generate, and patterns are accordingly formed mainly with straight edges.{{cite web |title=CARPETS v. Flat-woven carpets: Techniques and structures |url=http://www.iranicaonline.org/articles/carpets-v |publisher=Encyclopaedia Iranica |access-date=3 December 2015}} Kilim patterns are often characteristic of specific regions.{{cite web |title=Turkish Kilim Rug |url=http://www.rugs-oriental.net/turkish-kilim-rug.html |website=Through the Collector's Eye |access-date=3 December 2015}} Kilim motifs are often symbolic as well as decorative. For example, the wolf's mouth or wolf's foot motif (Turkish: Kurt Aǧzi, Kurt İzi) expresses the tribal weavers' desires for protection of their families' flocks from wolves.{{cite book |author=Erbek, Güran |title=Kilim Catalogue No. 1 |publisher =May Selçuk A. S. Edition=1st |date=1998}}

Islamic leather is often embossed with patterns similar to those already described. Leather book covers, starting with the Quran where figurative artwork was excluded, were decorated with a combination of kufic script, medallions and geometric patterns, typically bordered by geometric braiding.{{cite web |title=Geometric Decoration and the Art of the Book. Leather |url=http://www.discoverislamicart.org/exhibitions/ISL/geometric/exhibition.php?theme=4 |publisher=Museum with no Frontiers |access-date=7 December 2015}}

Metal artefacts share the same geometric designs that are used in other forms of Islamic art. However, in the view of Hamilton Gibb, the emphasis differs: geometric patterns tend to be used for borders, and if they are in the main decorative area they are most often used in combination with other motifs such as floral designs, arabesques, animal motifs, or calligraphic script. Geometric designs in Islamic metalwork can form a grid decorated with these other motifs, or they can form the background pattern.{{cite book |last=Gibb |first=Sir Hamilton Alexander Rosskeen |author-link=Hamilton Alexander Rosskeen Gibb |title=The Encyclopaedia of Islam |url=https://books.google.com/books?id=_JY3AAAAIAAJ&pg=PA991 |year=1954 |publisher=Brill Archive|pages=990–992 |id=GGKEY:N71HHP1UY5E}}

Even where metal objects such as bowls and dishes do not seem to have geometric decoration, still the designs, such as arabesques, are often set in octagonal compartments or arranged in concentric bands around the object. Both closed designs (which do not repeat) and open or repetitive patterns are used. Patterns such as interlaced six-pointed stars were especially popular from the 12th century. Eva Baer{{efn|Baer is Emeritus Professor of Islamic Studies at Tel Aviv University.{{cite book |last1=Baer |first1=Eva |title=Ayyubid Metalwork With Christian Images |year=1989 |publisher=BRILL |isbn=9004089624 |url=https://books.google.com/books?id=IZ7ZIpb-TAgC |access-date=15 January 2016}}}} notes that while this design was essentially simple, it was elaborated by metalworkers into intricate patterns interlaced with arabesques, sometimes organised around further basic Islamic patterns, such as the hexagonal pattern of six overlapping circles.{{cite book |last=Baer |first=Eva |title=Metalwork in Medieval Islamic Art |url=https://books.google.com/books?id=s__yi4pD-VEC&pg=PA122 |year=1983 |publisher=SUNY Press |isbn=978-0-87395-602-4 |pages=122–132}}

{{Main article|Muqarnas}}

Muqarnas are elaborately carved ceilings to semi-domes, often used in mosques. They are typically made of stucco (and thus do not have a structural function), but can also be of wood, brick, and stone. They are characteristic of Islamic architecture of the Middle Ages from Spain and Morocco in the west to Persia in the east. Architecturally they form multiple tiers of squinches, diminishing in size as they rise. They are often elaborately decorated.{{cite web |last1=Tabbaa |first1=Yasser |title=The Muqarnas Dome: Its Origin and Meaning |url=http://archnet.org/system/publications/contents/4158/original/DPT0869.pdf?1384781045 |website=Archnet |pages=61–74 |access-date=2 December 2015 |archive-url=https://web.archive.org/web/20151208040619/http://archnet.org/system/publications/contents/4158/original/DPT0869.pdf?1384781045 |archive-date=8 December 2015 |url-status=dead }}

{{Main article|Shabaka (window)}}

Geometrically patterned stained glass is used in a variety of settings in Islamic architecture. It is found in the surviving summer residence of the Palace of Shaki Khans, Azerbaijan, constructed in 1797. Patterns in the "shabaka" windows include 6-, 8-, and 12-point stars. These wood-framed decorative windows are distinctive features of the palace's architecture. Shabaka are still constructed the traditional way in Sheki in the 21st century.{{cite book |author=King, David C. King |title=Azerbaijan |publisher=Marshall Cavendish |year=2006 |page=[https://archive.org/details/azerbaijan00king/page/99 99] |isbn=978-0-7614-2011-8 |url-access=registration |url=https://archive.org/details/azerbaijan00king/page/99 }}{{cite journal|last1=Sharifov |first1=Azad |title=Shaki Paradise in the Caucasus Foothills |url=http://azer.com/aiweb/categories/magazine/62_folder/62_articles/62_shaki.html |journal=Azerbaijan International |date=1998 |volume=6 |issue=2 |pages=28–35}} Traditions of stained glass set in wooden frames (not lead as in Europe) survive in workshops in Iran as well as Azerbaijan.{{cite web |last1=Alin |first1=Marina |title=Wood, glass, geometry – stained glass in Iran and Azerbaijan |url=http://islamic-arts.org/2014/wood-glass-geometry-stained-glass-in-iran-and-azerbaijan/ |publisher=Islamic Arts & Architecture |access-date=18 January 2016 |date=21 January 2014 |archive-url=https://web.archive.org/web/20160126151556/http://islamic-arts.org/2014/wood-glass-geometry-stained-glass-in-iran-and-azerbaijan/ |archive-date=26 January 2016 |url-status=dead }} Glazed windows set in stucco arranged in girih-like patterns are found both in Turkey and the Arab lands; a late example, without the traditional balance of design elements, was made in Tunisia for the International Colonial Exhibition in Amsterdam in 1883.{{cite web |title=Carved stucco and stained glass window |url=http://islamic-arts.org/2011/carved-stucco-and-stained-glass-window/ |date=16 December 2011 |publisher=Islamic Arts & Architecture |access-date=18 January 2016 |archive-url=https://web.archive.org/web/20160126160758/http://islamic-arts.org/2011/carved-stucco-and-stained-glass-window/ |archive-date=26 January 2016 |url-status=dead }} The old city of Sana'a in Yemen has stained glass windows in its tall buildings.{{cite web |last1=Hansen |first1=Eri c |title=Sana'a Rising – "a Venice built on sand." |url=http://islamic-arts.org/2011/sanaa-rising/ |publisher=Islamic Arts & Architecture |access-date=18 January 2016 |date=21 December 2011 |archive-url=https://web.archive.org/web/20160126173718/http://islamic-arts.org/2011/sanaa-rising/ |archive-date=26 January 2016 |url-status=dead }}

{{Main article|Zellij}}

Zellij ({{Langx|ar|الزَّلِيْج}}) is geometric tilework with glazed terracotta tiles set into plaster, forming colourful mosaic patterns including regular and semiregular tessellations. The tradition is characteristic of Morocco, but is also found in Moorish Spain. Zellij is used to decorate mosques, public buildings and wealthy private houses.{{cite book |last=Njoku |first=Raphael Chijioke |title=Culture and Customs of Morocco |url=https://books.google.com/books?id=JRV1ZTDoSsUC&pg=PA58 |year=2006 |publisher=Greenwood Publishing Group |isbn=978-0-313-33289-0 |page=58}}

File:Persia p1070299.jpg|alt=Decoratively patterned ceramic bowl from Persia|Safavid bowl with radial and circular motifs, Persia, 17th century

File:Lustre tiles Iran Sufi divine breath shapes.JPG|Lustre tiles from Iran, probably Kashan, 1262, in the shapes of the Sufi symbols for the divine breath

File:Samarkand Shah-i Zinda Tuman Aqa complex cropped2.jpg|alt=Complicated strapwork tiling|Glazed tilework Girih at Shah-i-Zinda in Samarkand, Uzbekistan

File:Side of a Minbar.jpg|alt=Finely carved Islamic wooden pulpit|Side of a wooden Minbar (pulpit) with 12-point stars. 14th century. Turkish and Islamic Arts Museum

File:Salim Chishti Tomb-2.jpg|alt=Indian pierced stone screens|Jali pierced stone screens at the tomb of Salim Chishti, Fatehpur Sikri, India

File:Vintage Turkish Kilim Geometric Patterned Rug.jpg|alt=Simple geometric patterns on a flatweave rug|Woven wool Kilim from Turkey

File:'Ali ibn Abi Talib - Prayer Book - Walters W579 - Closed Top View A.jpg|alt=Finely worked leather bookbinding|Leather prayer book cover, Persia, 16th century

File:Kairo Rifai Moschee BW 5.jpg|alt=Iron gate from Egypt, forming a pattern of stars and kites|Iron gate with 10-point stars and kites at Al-Rifa'i Mosque, Cairo (1869–1912)

File:Flickr - Gaspa - Cairo, madrasa di Hasan (12).jpg|alt=Bronze doors from Egypt, decorated with strapwork|Detail of bronze door, Mosque-Madrassa of Sultan Hassan, Cairo, decorated with strapwork

File:Isfahan Royal Mosque entrance.JPG|alt=Elaborate stepped vaulting in Iran|Muqarnas in Shah Mosque, Isfahan, Iran

File:Shaki khan palace 1.jpg|alt=Colourful stained glass patterns from Azerbaijan|Geometric shabaka stained glass in the 1797 Palace of Shaki Khans, Azerbaijan

File:Traditional Window, Sana'a (11024852966).jpg|Traditional Window, Sana'a, Yemen

File:Mekhnes Place El-Hedine Mosaique.jpg|alt=Decorative brightly coloured tiling in Morocco|Glazed tile Zellij at Place el-Hedim in Meknes, Morocco

File:Alhambra-p3-closeup.jpg of glazed ceramic tiles forming colourful geometric patterns in the Alhambra, Spain, which inspired M. C. Escher{{cite book |author=Locher, J. L. |year=1971 |title=The World of M. C. Escher |publisher=Abrams |isbn=0-451-79961-5 |page=17}}|alt=Colourful geometric tiling in the Alhambra, Spain]]

{{further|Islamic influences on Western art}}

It is sometimes supposed in Western society that mistakes in repetitive Islamic patterns such as those on carpets were intentionally introduced as a show of humility by artists who believed only Allah can produce perfection, but this theory is denied.{{cite web |title=Islamic Textile Art: Anomalies in Kilims |author=Thompson, Muhammad |author2=Begum, Nasima |url=http://www.turkotek.com/salon_00101/salon.html |work=Salon du Tapis d'Orient |publisher=TurkoTek |access-date=25 August 2009}}{{cite book |title=The future of art in a digital age: from Hellenistic to Hebraic consciousness |url=https://archive.org/details/futureartdigital00alex |url-access=limited |last=Alexenberg |first=Melvin L. |year=2006 |publisher=Intellect |isbn=1-84150-136-0 |page=[https://archive.org/details/futureartdigital00alex/page/n55 55]}}{{cite web |title=Only God is Perfect |url=http://www.geometricdesign.co.uk/perfect.htm |work=Islamic and Geometric Art |author=Backhouse, Tim |access-date=25 August 2009}}

File:Arabesques alhambra.jpgs, arabesques and calligraphy on a wall of the Myrtle court, Alhambra, Granada, Spain|alt=Combination of geometric patterns with arabesque swirls and elegant calligraphy in the Alhambra, Spain]]

Major Western collections hold many objects of widely varying materials with Islamic geometric patterns. The Victoria and Albert Museum in London holds at least 283 such objects, of materials including wallpaper, carved wood, inlaid wood, tin- or lead-glazed earthenware, brass, stucco, glass, woven silk, ivory, and pen or pencil drawings.{{cite web |title=Search the Collections "Islamic geometric pattern" |url=https://collections.vam.ac.uk/search/?page=1&page_size=15&q=islamic+geometric+pattern|publisher=Victoria and Albert Museum |access-date=2 December 2015}} The Metropolitan Museum of Art in New York has among other relevant holdings 124 mediaeval (1000–1400 A.D.) objects bearing Islamic geometric patterns,{{cite web |title=Islamic geometric pattern A.D. 1000–1400 |url=http://www.metmuseum.org/collection/the-collection-online/search?&noqs=true&ao=on&ft=Islamic+geometric+pattern&when=A.D.+1000-1400 |publisher=Metropolitan Museum of Art |access-date=2 December 2015}} including a pair of Egyptian minbar (pulpit) doors almost 2 m. high in rosewood and mulberry inlaid with ivory and ebony;{{cite web |title=Pair of Minbar Doors |url=http://www.metmuseum.org/collection/the-collection-online/search/449537?rpp=30&pg=1&ao=on&ft=Islamic+geometric+pattern&when=A.D.+1000-1400&pos=2|publisher=Metropolitan Museum of Art|access-date=2 December 2015}} and an entire mihrab (prayer niche) from Isfahan, decorated with polychrome mosaic, and weighing over 2,000 kg.{{cite web |title=Mihrab (Prayer Niche) |url=http://www.metmuseum.org/collection/the-collection-online/search/449537?rpp=30&pg=1&ao=on&ft=Islamic+geometric+pattern&when=A.D.+1000-1400&pos=3 |publisher=Metropolitan Museum of Art |access-date=2 December 2015}}

File:Wooden box inlaid with ivory in floral and zellige-like geometrical motifs Italy 15th century.jpg

Islamic decoration and craftsmanship had a significant influence on Western art when Venetian merchants brought goods of many types back to Italy from the 14th century onwards.{{cite book |last=Mack |first=Rosamond E. |title=Bazaar to Piazza: Islamic Trade and Italian Art, 1300-1600 |publisher=University of California Press |year=2001 |isbn=0-520-22131-1 |pages=Chapter 1}}

The Dutch artist M. C. Escher was inspired by the Alhambra's intricate decorative designs to study the mathematics of tessellation, transforming his style and influencing the rest of his artistic career.{{cite book |last=Roza |first=Greg |title=An Optical Artist: Exploring Patterns and Symmetry |year=2005 |publisher=Rosen Classroom |isbn=978-1-4042-5117-5 |page=20}}{{cite book |last=Monroe |first=J. T. |title=Hispano-Arabic Poetry: A Student Anthology |url=https://books.google.com/books?id=3ZzXksIVQuYC&pg=PA65 |year=2004 |publisher=Gorgias Press LLC |isbn=978-1-59333-115-3 |page=65}} In his own words it was "the richest source of inspiration I have ever tapped."{{cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |title=Maurits Cornelius Escher |url=https://mathshistory.st-andrews.ac.uk/Biographies/Escher/ |website=Biographies |publisher=University of St Andrews |access-date=2 November 2015 |date=May 2000}} which cites {{cite news |author=Strauss, S. |title=M C Escher |work=The Globe and Mail |date=9 May 1996}}

Cultural organisations such as the Mathematical Sciences Research Institute and the Institute for Advanced Study run events on geometric patterns and related aspects of Islamic art.{{cite web |title=Geometric Patterns in Islamic Art |url=http://www.mathfest.org/sessions/geometric-patterns-in-islamic-art-4/ |website=National Math Festival |access-date=3 December 2015 |url-status=dead |archive-url=https://web.archive.org/web/20151208051517/http://www.mathfest.org/sessions/geometric-patterns-in-islamic-art-4/ |archive-date=8 December 2015 }} In 2013 the Istanbul Center of Design and the Ensar Foundation ran what they claimed was the first ever symposium of Islamic Arts and Geometric Patterns, in Istanbul. The panel included the experts on Islamic geometric pattern Carol Bier,{{efn|Bier is a historian of Islamic art who studies pattern.{{cite web |title=Selected Works of Carol Bier |url=http://works.bepress.com/carol_bier/|website=SelectedWorks |access-date=3 December 2015}}}} Jay Bonner,{{efn |Bonner is an architect specialising in Islamic ornament.{{cite web |last1=Bonner |first1=Jay |title=About |url=http://bonner-design.com/index.php/about/ |publisher=Bonner Design |access-date=3 December 2015 |archive-date=8 December 2015 |archive-url=https://web.archive.org/web/20151208135026/http://bonner-design.com/index.php/about/ |url-status=dead }}}}{{cite book | last=Bonner | first=Jay | title=Islamic geometric patterns : their historical development and traditional methods of construction | publisher=Springer | location=New York | year=2017 | isbn=978-1-4419-0216-0 | oclc=1001744138}} Eric Broug,{{efn|Broug writes books and runs courses on Islamic geometric design.{{cite web |title=School of Islamic Geometric Design |url=http://www.sigd.org/ |publisher=Eric Broug |access-date=1 December 2015 |archive-date=8 April 2019 |archive-url=https://web.archive.org/web/20190408133823/http://www.sigd.org/ |url-status=dead }}}} Hacali Necefoğlu{{efn|Necefoğlu is a professor of chemistry at Kafkas University interested in pattern and crystallography.{{cite web |title=Prof.Dr. Hacali Necefoğlu (Fen Edebiyat Fakültesi) |url=http://abs.kafkas.edu.tr/arama/goster/172 |website=Akademik Bilgi Sistemi |access-date=3 December 2015 |language=tr |archive-date=5 May 2018 |archive-url=https://web.archive.org/web/20180505221422/http://abs.kafkas.edu.tr/arama/goster/172 |url-status=dead }}}} and Reza Sarhangi.{{efn|Sarhangi is the founder of The Bridges Organization. He studies the mathematics of Persian architecture and mosaic design.{{cite web |url=http://pages.towson.edu/gsarhang/ |title=Reza Sarhangi |publisher=Towson University |access-date=3 December 2015 |url-status=dead |archive-url=https://web.archive.org/web/20151127061124/http://pages.towson.edu/gsarhang/ |archive-date=27 November 2015 }}}}{{cite web |title=Istanbul hosts first ever Islamic geometric arts symposium |url=http://www.worldbulletin.net/haber/118929/istanbul-hosts-first-ever-islamic-geometric-arts-symposium |archive-url=https://web.archive.org/web/20151208133258/http://www.worldbulletin.net/haber/118929/istanbul-hosts-first-ever-islamic-geometric-arts-symposium |url-status=usurped |archive-date=December 8, 2015 |website=World Bulletin |access-date=3 December 2015 |date=25 September 2013}} In Britain, The Prince's School of Traditional Arts runs a range of courses in Islamic art including geometry, calligraphy, and arabesque (vegetal forms), tile-making, and plaster carving.{{cite web |title=Introduction to Islamic Art |url=http://www.psta.org.uk/openprogramme |publisher=The Prince's School of Traditional Arts |access-date=4 December 2015 |archive-date=3 December 2015 |archive-url=https://web.archive.org/web/20151203233824/http://www.psta.org.uk/openprogramme/ |url-status=dead }}

File:Kharaghan.jpg princes at Kharaghan, Qazvin province, Iran, covered with many different brick patterns like those that inspired Ahmad Rafsanjani to create auxetic materials]]

Computer graphics and computer-aided manufacturing make it possible to design and produce Islamic geometric patterns effectively and economically. Craig S. Kaplan explains and illustrates in his Ph.D. thesis how Islamic star patterns can be generated algorithmically.{{cite web |last1=Kaplan |first1=Craig S. |title=Computer Graphics and Geometric Ornamental Design: Chapter 3. Islamic Star Patterns |url=http://www.cgl.uwaterloo.ca/~csk/phd/ |publisher=University of Waterloo (PhD thesis) |access-date=4 December 2015 |date=2002 |archive-date=30 December 2015 |archive-url=https://web.archive.org/web/20151230155649/http://www.cgl.uwaterloo.ca/~csk/phd/ |url-status=dead }}

Two physicists, Peter J. Lu and Paul Steinhardt, attracted controversy in 2007 by claiming{{cite journal |last1=Lu |first1=P. J. |last2=Steinhardt |first2=P. J. |title=Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture |journal=Science |date=2007 |volume=315 |issue=5815 |pages=1106–1110 |url=http://www.peterlu.org/content/decagonal-and-quasi-crystalline-tilings-medieval-islamic-architecture |doi=10.1126/science.1135491 |pmid=17322056 |bibcode=2007Sci...315.1106L |s2cid=10374218 |access-date=2015-12-04 |archive-date=2015-12-08 |archive-url=https://web.archive.org/web/20151208152411/http://www.peterlu.org/content/decagonal-and-quasi-crystalline-tilings-medieval-islamic-architecture |url-status=dead |url-access=subscription }} that girih designs such as that used on the Darb-e Imam shrine{{efn|Illustrated above.}} in Isfahan were able to create quasi-periodic tilings resembling those discovered by Roger Penrose in 1973. They showed that rather than the traditional ruler and compass construction, it was possible to create girih designs using a set of five "girih tiles", all equilateral polygons, secondarily decorated with lines (for the strapwork).{{cite journal |last1=Ball |first1=Philip |author-link1=Philip Ball |title=Islamic tiles reveal sophisticated maths |url=http://www.nature.com/news/2007/070219/full/news070219-9.html |journal=Nature |access-date=4 December 2015 |date=22 February 2007|pages=news070219–9 |doi=10.1038/news070219-9 |s2cid=178905751 |doi-access=free |url-access=subscription }}

In 2016, Ahmad Rafsanjani described the use of Islamic geometric patterns from tomb towers in Iran to create auxetic materials from perforated rubber sheets. These are stable in either a contracted or an expanded state, and can switch between the two, which might be useful for surgical stents or for spacecraft components. When a conventional material is stretched along one axis, it contracts along other axes (at right angles to the stretch). But auxetic materials expand at right angles to the pull. The internal structure that enables this unusual behaviour is inspired by two of the 70 Islamic patterns that Rafsanjani noted on the tomb towers.{{cite web |last1=Webb |first1=Jonathan |title=Islamic art inspires stretchy, switchable materials |url=https://www.bbc.co.uk/news/science-environment-35818924 |publisher=British Broadcasting Corporation |access-date=16 March 2016 |date=16 March 2016}}

{{notelist}}

{{reflist|28em}}

{{Commons category}}

• [http://www.discoverislamicart.org/exhibitions/ISL/geometric/index.php Museum with no Frontiers: Geometric Decoration]
• [http://www.vam.ac.uk/content/articles/t/teachers-resource-maths-and-islamic-art-and-design/ Victoria and Albert Museum: Teachers' resource: Maths and Islamic art & design]

{{Islamic architecture}}

{{Islamic art}}

{{Islamic studies}}

{{Mathematics and art}}

{{DEFAULTSORT:Islamic Geometric Patterns}}

Category:Islamic architectural elements

Category:Islamic art

Category:Mathematics in the medieval Islamic world