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Islamic geometric patterns are one of the major forms of Islamic ornament, which tends to avoid using figurative images, as it is forbidden to create a representation of an important Islamic figure according to many holy scriptures. The geometric designs in
Islamic art Islamic art is a part of Islamic culture and encompasses the visual arts produced since the 7th century CE by people who lived within territories inhabited or ruled by Muslims, Muslim populations. Referring to characteristic traditions across ...
are often built on combinations of repeated squares and circles, which may be overlapped and
interlaced Interlaced video (also known as interlaced scan) is a technique for doubling the perceived frame rate of a video display without consuming extra bandwidth. The interlaced signal contains two fields of a video frame captured consecutively. Th ...
, as can arabesques (with which they are often combined), to form intricate and complex patterns, including a wide variety of
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
s. These may constitute the entire decoration, may form a framework for floral or
calligraphic Calligraphy () is a visual art related to writing. It is the design and execution of lettering with a pen, ink brush, or other writing instruments. Contemporary calligraphic practice can be defined as "the art of giving form to signs in an exp ...
embellishments, or may retreat into the background around other motifs. The complexity and variety of patterns used evolved from simple stars and lozenges in the ninth century, through a variety of 6- to 13-point patterns by the 13th century, and finally to include also 14- and 16-point stars in the sixteenth century. Geometric patterns occur in a variety of forms in
Islamic art Islamic art is a part of Islamic culture and encompasses the visual arts produced since the 7th century CE by people who lived within territories inhabited or ruled by Muslims, Muslim populations. Referring to characteristic traditions across ...
and
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and construction, constructi ...
. These include
kilim A kilim ( ; ; ) is a flat tapestry-weaving, woven carpet or rug traditionally produced in countries of the former Persian Empire, including Iran and Turkey, but also in the Balkans and the Turkic countries. Kilims can be purely decorative ...
carpets,
Persia Iran, officially the Islamic Republic of Iran (IRI) and also known as Persia, is a country in West Asia. It borders Iraq to the west, Turkey, Azerbaijan, and Armenia to the northwest, the Caspian Sea to the north, Turkmenistan to the nort ...
n girih and Moroccan
zellij Zellij (), also spelled zillij or zellige, is a style of mosaic tilework made from individually hand-chiseled tile pieces. The pieces were typically of different colours and fitted together to form various patterns on the basis of tessellations, ...
tilework,
muqarnas Muqarnas (), also known in Iberian architecture as Mocárabe (from ), is a form of three-dimensional decoration in Islamic architecture in which rows or tiers of niche-like elements are projected over others below. It is an archetypal form of I ...
decorative vaulting, jali pierced stone screens,
ceramics A ceramic is any of the various hard, brittle, heat-resistant, and corrosion-resistant materials made by shaping and then firing an inorganic, nonmetallic material, such as clay, at a high temperature. Common examples are earthenware, porce ...
,
leather Leather is a strong, flexible and durable material obtained from the tanning (leather), tanning, or chemical treatment, of animal skins and hides to prevent decay. The most common leathers come from cattle, sheep, goats, equine animals, buffal ...
,
stained glass Stained glass refers to coloured glass as a material or art and architectural works created from it. Although it is traditionally made in flat panels and used as windows, the creations of modern stained glass artists also include three-dimensio ...
, woodwork, and metalwork. Interest in Islamic geometric patterns is increasing in the West, both among craftsmen and
artist An artist is a person engaged in an activity related to creating art, practicing the arts, or demonstrating the work of art. The most common usage (in both everyday speech and academic discourse) refers to a practitioner in the visual arts o ...
s like
M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...
in the twentieth century, and among
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s and
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
s such as Peter J. Lu and
Paul Steinhardt Paul Joseph Steinhardt (born December 25, 1952) is an American theoretical physicist whose principal research is in cosmology and condensed matter physics. He is currently the Albert Einstein Professorship in Science, Albert Einstein Professor in ...
.


Background


Islamic decoration

Islamic geometric patterns are derived from simpler designs used in earlier cultures:
Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...
, Roman, and
Sasanian The Sasanian Empire (), officially Eranshahr ( , "Empire of the Iranians"), was an Iranian empire that was founded and ruled by the House of Sasan from 224 to 651. Enduring for over four centuries, the length of the Sasanian dynasty's reign ...
. They are one of three forms of Islamic decoration, the others being the arabesque based on curving and branching plant forms, and
Islamic calligraphy Islamic calligraphy is the artistic practice of penmanship and calligraphy, in the languages which use Arabic alphabet or the Arabic script#Additional letters used in other languages, alphabets derived from it. It is a highly stylized and struc ...
; all three are frequently used together. From the 9th century onward, a range of sophisticated geometric patterns based on polygonal tessellation began to appear in Islamic art, eventually becoming dominant.
Islamic art Islamic art is a part of Islamic culture and encompasses the visual arts produced since the 7th century CE by people who lived within territories inhabited or ruled by Muslims, Muslim populations. Referring to characteristic traditions across ...
mostly avoids figurative images to avoid becoming objects of worship. This aniconism in Islamic culture caused artists to explore non-figural art, and created a general aesthetic shift toward mathematically based decoration.


Purpose

Authors such as Keith Critchlow 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. In Islamic culture, the patterns are believed to be the bridge to the spiritual realm, the instrument to purify the mind and the soul. David Wade 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." Against this, Doris Behrens-Abouseif 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". 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 Zellij (), also spelled zillij or zellige, is a style of mosaic tilework made from individually hand-chiseled tile pieces. The pieces were typically of different colours and fitted together to form various patterns on the basis of tessellations, ...
tilework File:162-MoroccoOpenDoors.jpg, A variety of vernacular decorative Islamic styles in
Morocco Morocco, officially the Kingdom of Morocco, is a country in the Maghreb region of North Africa. It has coastlines on the Mediterranean Sea to the north and the Atlantic Ocean to the west, and has land borders with Algeria to Algeria–Morocc ...
: wooden panels, zellij tilework, stucco calligraphy, and floral door panels File:Alhambra Detail 17.JPG, Arch in the
Alhambra The Alhambra (, ; ) is a palace and fortress complex located in Granada, Spain. It is one of the most famous monuments of Islamic architecture and one of the best-preserved palaces of the historic Muslim world, Islamic world. Additionally, the ...
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 Bursa () is a city in northwestern Turkey and the administrative center of Bursa Province. The fourth-most populous city in Turkey and second-most populous in the Marmara Region, Bursa is one of the industrial centers of the country. Most of ...
, Turkey (1424), with girih 10-point stars and pentagons


Pattern formation

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 In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
, including
pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
s and
octagon In geometry, an octagon () is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a ...
s. 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
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
s, which can extend indefinitely and thus suggest
infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
. They are constructed on grids that require only
ruler A ruler, sometimes called a rule, scale, line gauge, or metre/meter stick, is an instrument used to make length measurements, whereby a length is read from a series of markings called "rules" along an edge of the device. Usually, the instr ...
and
compass A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with No ...
to draw. Artist and educator Roman Verostko argues that such constructions are in effect
algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
s, making Islamic geometric patterns forerunners of modern algorithmic art. The circle symbolizes unity and diversity in nature, and many Islamic patterns are drawn starting with a circle. 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. A similar design forms the logo of the Mohammed Ali Research Center. 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 The Timurid dynasty, self-designated as Gurkani (), was the ruling dynasty of the Timurid Empire (1370–1507). It was a Sunni Islam, Sunni Muslim dynasty or Barlās clan of Turco-Mongol originB.F. Manz, ''"Tīmūr Lang"'', in Encyclopaedia of I ...
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. The mathematical properties of the decorative tile and
stucco Stucco or render is a construction material made of aggregates, a binder, and water. Stucco is applied wet and hardens to a very dense solid. It is used as a decorative coating for walls and ceilings, exterior walls, and as a sculptural and ...
patterns of the
Alhambra The Alhambra (, ; ) is a palace and fortress complex located in Granada, Spain. It is one of the most famous monuments of Islamic architecture and one of the best-preserved palaces of the historic Muslim world, Islamic world. Additionally, the ...
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. Moroccan geometric woodwork from the 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. File:Topkapi Scroll P290.JPG, Two-dimensional designs for two quarter-dome
muqarnas Muqarnas (), also known in Iberian architecture as Mocárabe (from ), is a form of three-dimensional decoration in Islamic architecture in which rows or tiers of niche-like elements are projected over others below. It is an archetypal form of I ...
– as a
seashell A seashell or sea shell, also known simply as a shell, is a hard, protective outer layer usually created by an animal or organism that lives in the sea. Most seashells are made by Mollusca, mollusks, such as snails, clams, and oysters ...
(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 A spandrel is a roughly triangular space, usually found in pairs, between the top of an arch and a rectangular frame, between the tops of two adjacent arches, or one of the four spaces between a circle within a square. They are frequently fil ...
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 A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
. File:OctagonalTesselations2.png, Analysis of
octagon In geometry, an octagon () is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a ...
al patterns in
Mughal architecture Mughal architecture is the style of architecture developed in the Mughal Empire in the 16th, 17th and 18th centuries throughout the ever-changing extent of their empire in the Indian subcontinent. It developed from the architectural styles of ea ...
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 Agra ( ) is a city on the banks of the Yamuna river in the Indian state of Uttar Pradesh, about south-east of the national capital Delhi and 330 km west of the state capital Lucknow. With a population of roughly 1.6 million, Agra is the ...
, 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


Evolution


Early stage

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.


Middle stage

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 Qazvin (; ; ) is a city in the Central District (Qazvin County), Central District of Qazvin County, Qazvin province, Qazvin province, Iran, serving as capital of the province, the county, and the district. It is the largest city in the provi ...
, 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 Al-Andalus () was the Muslim-ruled area of the Iberian Peninsula. The name refers to the different Muslim states that controlled these territories at various times between 711 and 1492. At its greatest geographical extent, it occupied most o ...
. 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 Konya is a major city in central Turkey, on the southwestern edge of the Central Anatolian Plateau, and is the capital of Konya Province. During antiquity and into Seljuk times it was known as Iconium. In 19th-century accounts of the city in En ...
, Turkey in 1220, and in the
Abbasid The Abbasid Caliphate or Abbasid Empire (; ) was the third caliphate to succeed the prophets and messengers in Islam, Islamic prophet Muhammad. It was founded by a dynasty descended from Muhammad's uncle, Abbas ibn Abd al-Muttalib (566–653 C ...
palace in
Baghdad Baghdad ( or ; , ) is the capital and List of largest cities of Iraq, largest city of Iraq, located along the Tigris in the central part of the country. With a population exceeding 7 million, it ranks among the List of largest cities in the A ...
in 1230, going on to become widespread across the Islamic world.


Late stage

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 Fatehpur Sikri () is a town in the Agra District of Uttar Pradesh, India. Situated from the district headquarters of Agra, Fatehpur Sikri itself was founded as the capital of the Mughal Empire in 1571 by Mughal emperors, Emperor Akbar, servin ...
in
India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...
in 1571–1596, but in few other places.


Artforms

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.


Ceramics

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. Thus an unglazed earthenware water flask from
Aleppo Aleppo is a city in Syria, which serves as the capital of the Aleppo Governorate, the most populous Governorates of Syria, governorate of Syria. With an estimated population of 2,098,000 residents it is Syria's largest city by urban area, and ...
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.


Girih tilings and woodwork

Girih are elaborate interlacing patterns formed of five standardized shapes. The style is used in Persian Islamic architecture and also in decorative woodwork. Girih designs are traditionally made in different media including cut brickwork,
stucco Stucco or render is a construction material made of aggregates, a binder, and water. Stucco is applied wet and hardens to a very dense solid. It is used as a decorative coating for walls and ceilings, exterior walls, and as a sculptural and ...
, and mosaic
faience Faience or faïence (; ) is the general English language term for fine tin-glazed pottery. The invention of a white Ceramic glaze, pottery glaze suitable for painted decoration, by the addition of an stannous oxide, oxide of tin to the Slip (c ...
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. 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.


Jali

Jali are pierced stone screens with regularly repeating patterns. They are characteristic of Indo-Islamic architecture, for example in the
Mughal dynasty The Mughal dynasty () or the House of Babur (), was a Central Asian dynasty of Turco-Mongol tradition, Turco-Mongol origin that ruled large parts of the Indian subcontinent from the early 16th to the 19th century. The dynasty was a cadet branch ...
buildings at
Fatehpur Sikri Fatehpur Sikri () is a town in the Agra District of Uttar Pradesh, India. Situated from the district headquarters of Agra, Fatehpur Sikri itself was founded as the capital of the Mughal Empire in 1571 by Mughal emperors, Emperor Akbar, servin ...
and the
Taj Mahal The Taj Mahal ( ; ; ) is an ivory-white marble mausoleum on the right bank of the river Yamuna in Agra, Uttar Pradesh, India. It was commissioned in 1631 by the fifth Mughal Empire, Mughal emperor, Shah Jahan () to house the tomb of his belo ...
. 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. Jali forms a prominent element of the architecture of India. 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. 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.


Kilim

A kilim is an Islamic flatwoven carpet (without a pile), whether for household use or a prayer mat. The pattern is made by winding the
weft In the manufacture of cloth, warp and weft are the two basic components in weaving to transform thread (yarn), thread and yarn into textile fabrics. The vertical ''warp'' yarns are held stationary in tension on a loom (frame) while the horizo ...
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. Kilim patterns are often characteristic of specific regions. 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 The wolf (''Canis lupus''; : wolves), also known as the grey wolf or gray wolf, is a canine native to Eurasia and North America. More than thirty subspecies of ''Canis lupus'' have been recognized, including the dog and dingo, though gr ...
.


Leather

Islamic leather is often embossed with patterns similar to those already described. Leather book covers, starting with the
Quran The Quran, also Romanization, romanized Qur'an or Koran, is the central religious text of Islam, believed by Muslims to be a Waḥy, revelation directly from God in Islam, God (''Allah, Allāh''). It is organized in 114 chapters (, ) which ...
where figurative artwork was excluded, were decorated with a combination of kufic script, medallions and geometric patterns, typically bordered by geometric
braid A braid (also referred to as a plait; ) is a complex structure or pattern formed by interlacing three or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strand ...
ing.


Metalwork

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. 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 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.


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.


Stained glass

Geometrically patterned
stained glass Stained glass refers to coloured glass as a material or art and architectural works created from it. Although it is traditionally made in flat panels and used as windows, the creations of modern stained glass artists also include three-dimensio ...
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 Azerbaijan, officially the Republic of Azerbaijan, is a Boundaries between the continents, transcontinental and landlocked country at the boundary of West Asia and Eastern Europe. It is a part of the South Caucasus region and is bounded by ...
, 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. Traditions of stained glass set in wooden frames (not lead as in Europe) survive in workshops in Iran as well as Azerbaijan. 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. The old city of
Sana'a Sanaa, officially the Sanaa Municipality, is the ''de jure'' capital and largest city of Yemen. The city is the capital of the Sanaa Governorate, but is not part of the governorate, as it forms a separate administrative unit. At an elevation ...
in Yemen has stained glass windows in its tall buildings.


''Zellij''

''
Zellij Zellij (), also spelled zillij or zellige, is a style of mosaic tilework made from individually hand-chiseled tile pieces. The pieces were typically of different colours and fitted together to form various patterns on the basis of tessellations, ...
'' () is geometric tilework with glazed terracotta tiles set into plaster, forming colourful mosaic patterns including regular and semiregular
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
s. The tradition is characteristic of
Morocco Morocco, officially the Kingdom of Morocco, is a country in the Maghreb region of North Africa. It has coastlines on the Mediterranean Sea to the north and the Atlantic Ocean to the west, and has land borders with Algeria to Algeria–Morocc ...
, but is also found in Moorish Spain. ''Zellij'' is used to decorate mosques, public buildings and wealthy private houses.


Illustrations

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 Kashan (; ) is a city in the Central District (Kashan County), Central District of Kashan County, in the northern part of Isfahan province, Isfahan province, Iran, serving as capital of both the county and the district. History Earlies ...
, 1262, in the shapes of the
Sufi Sufism ( or ) is a mysticism, mystic body of religious practice found within Islam which is characterized by a focus on Islamic Tazkiyah, purification, spirituality, ritualism, and Asceticism#Islam, asceticism. Practitioners of Sufism are r ...
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 Samarkand ( ; Uzbek language, Uzbek and Tajik language, Tajik: Самарқанд / Samarqand, ) is a city in southeastern Uzbekistan and among the List of oldest continuously inhabited cities, oldest continuously inhabited cities in Central As ...
,
Uzbekistan , image_flag = Flag of Uzbekistan.svg , image_coat = Emblem of Uzbekistan.svg , symbol_type = Emblem of Uzbekistan, Emblem , national_anthem = "State Anthem of Uzbekistan, State Anthem of the Republ ...
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 Fatehpur Sikri () is a town in the Agra District of Uttar Pradesh, India. Situated from the district headquarters of Agra, Fatehpur Sikri itself was founded as the capital of the Mughal Empire in 1571 by Mughal emperors, Emperor Akbar, servin ...
,
India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...
File:Vintage Turkish Kilim Geometric Patterned Rug.jpg, alt=Simple geometric patterns on a flatweave rug, Woven wool
Kilim A kilim ( ; ; ) is a flat tapestry-weaving, woven carpet or rug traditionally produced in countries of the former Persian Empire, including Iran and Turkey, but also in the Balkans and the Turkic countries. Kilims can be purely decorative ...
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 A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...
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 Bronze is an alloy consisting primarily of copper, commonly with about 12–12.5% tin and often with the addition of other metals (including aluminium, manganese, nickel, or zinc) and sometimes non-metals (such as phosphorus) or metalloid ...
door, Mosque-Madrassa of Sultan Hassan,
Cairo Cairo ( ; , ) is the Capital city, capital and largest city of Egypt and the Cairo Governorate, being home to more than 10 million people. It is also part of the List of urban agglomerations in Africa, largest urban agglomeration in Africa, L ...
, decorated with strapwork File:Isfahan Royal Mosque entrance.JPG, alt=Elaborate stepped vaulting in Iran,
Muqarnas Muqarnas (), also known in Iberian architecture as Mocárabe (from ), is a form of three-dimensional decoration in Islamic architecture in which rows or tiers of niche-like elements are projected over others below. It is an archetypal form of I ...
in Shah Mosque,
Isfahan Isfahan or Esfahan ( ) is a city in the Central District (Isfahan County), Central District of Isfahan County, Isfahan province, Iran. It is the capital of the province, the county, and the district. It is located south of Tehran. The city ...
,
Iran Iran, officially the Islamic Republic of Iran (IRI) and also known as Persia, is a country in West Asia. It borders Iraq to the west, Turkey, Azerbaijan, and Armenia to the northwest, the Caspian Sea to the north, Turkmenistan to the nort ...
File:Shaki khan palace 1.jpg, alt=Colourful stained glass patterns from Azerbaijan, Geometric
shabaka Neferkare Shabaka, or Shabako ( Meroitic: 𐦰𐦲𐦡𐦐𐦲 (sha-ba-ka), Egyptian: 𓆷𓃞𓂓 ''šꜣ bꜣ kꜣ'', Assyrian: ''Ša-ba-ku-u'', Šabakû ) was the third Kushite pharaoh of the Twenty-fifth Dynasty of Egypt, who reigned fr ...
stained glass in the 1797 Palace of Shaki Khans,
Azerbaijan Azerbaijan, officially the Republic of Azerbaijan, is a Boundaries between the continents, transcontinental and landlocked country at the boundary of West Asia and Eastern Europe. It is a part of the South Caucasus region and is bounded by ...
File:Traditional Window, Sana'a (11024852966).jpg, Traditional Window,
Sana'a Sanaa, officially the Sanaa Municipality, is the ''de jure'' capital and largest city of Yemen. The city is the capital of the Sanaa Governorate, but is not part of the governorate, as it forms a separate administrative unit. At an elevation ...
,
Yemen Yemen, officially the Republic of Yemen, is a country in West Asia. Located in South Arabia, southern Arabia, it borders Saudi Arabia to Saudi Arabia–Yemen border, the north, Oman to Oman–Yemen border, the northeast, the south-eastern part ...
File:Mekhnes Place El-Hedine Mosaique.jpg, alt=Decorative brightly coloured tiling in Morocco, Glazed tile
Zellij Zellij (), also spelled zillij or zellige, is a style of mosaic tilework made from individually hand-chiseled tile pieces. The pieces were typically of different colours and fitted together to form various patterns on the basis of tessellations, ...
at Place el-Hedim in Meknes,
Morocco Morocco, officially the Kingdom of Morocco, is a country in the Maghreb region of North Africa. It has coastlines on the Mediterranean Sea to the north and the Atlantic Ocean to the west, and has land borders with Algeria to Algeria–Morocc ...


Outside Islamic art


In Western culture

It is sometimes supposed in Western society that mistakes in repetitive Islamic patterns such as those on
carpet A carpet is a textile floor covering typically consisting of an upper layer of Pile (textile), pile attached to a backing. The pile was traditionally made from wool, but since the 20th century synthetic fiber, synthetic fibres such as polyprop ...
s were intentionally introduced as a show of humility by artists who believed only Allah can produce perfection, but this theory is denied. Major Western collections hold many objects of widely varying materials with Islamic geometric patterns. The
Victoria and Albert Museum The Victoria and Albert Museum (abbreviated V&A) in London is the world's largest museum of applied arts, decorative arts and design, housing a permanent collection of over 2.8 million objects. It was founded in 1852 and named after Queen ...
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. The
Metropolitan Museum of Art The Metropolitan Museum of Art, colloquially referred to as the Met, is an Encyclopedic museum, encyclopedic art museum in New York City. By floor area, it is the List of largest museums, third-largest museum in the world and the List of larg ...
in New York has among other relevant holdings 124 mediaeval (1000–1400 A.D.) objects bearing Islamic geometric patterns, including a pair of Egyptian minbar (pulpit) doors almost 2 m. high in rosewood and mulberry inlaid with ivory and ebony; and an entire
mihrab ''Mihrab'' (, ', pl. ') is a niche in the wall of a mosque that indicates the ''qibla'', the direction of the Kaaba in Mecca towards which Muslims should face when praying. The wall in which a ''mihrab'' appears is thus the "''qibla'' wall". ...
(prayer niche) from
Isfahan Isfahan or Esfahan ( ) is a city in the Central District (Isfahan County), Central District of Isfahan County, Isfahan province, Iran. It is the capital of the province, the county, and the district. It is located south of Tehran. The city ...
, decorated with polychrome mosaic, and weighing over 2,000 kg. 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. The Dutch artist
M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...
was inspired by the
Alhambra The Alhambra (, ; ) is a palace and fortress complex located in Granada, Spain. It is one of the most famous monuments of Islamic architecture and one of the best-preserved palaces of the historic Muslim world, Islamic world. Additionally, the ...
's intricate decorative designs to study the mathematics of
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
, transforming his style and influencing the rest of his artistic career. In his own words it was "the richest source of inspiration I have ever tapped." which cites


Influence on the sciences

Cultural organisations such as the Mathematical Sciences Research Institute and the
Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
run events on geometric patterns and related aspects of Islamic art. 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, Jay Bonner, Eric Broug, Hacali Necefoğlu and Reza Sarhangi. 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. 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
algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
ically. Two physicists, Peter J. Lu and
Paul Steinhardt Paul Joseph Steinhardt (born December 25, 1952) is an American theoretical physicist whose principal research is in cosmology and condensed matter physics. He is currently the Albert Einstein Professorship in Science, Albert Einstein Professor in ...
, attracted controversy in 2007 by claiming that girih designs such as that used on the Darb-e Imam shrine in
Isfahan Isfahan or Esfahan ( ) is a city in the Central District (Isfahan County), Central District of Isfahan County, Isfahan province, Iran. It is the capital of the province, the county, and the district. It is located south of Tehran. The city ...
were able to create quasi-periodic tilings resembling those discovered by
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...
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). 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
stent In medicine, a stent is a tube usually constructed of a metallic alloy or a polymer. It is inserted into the Lumen (anatomy), lumen (hollow space) of an anatomic vessel or duct to keep the passageway open. Stenting refers to the placement of ...
s 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.


Notes


References


External links


Museum with no Frontiers: Geometric Decoration

Victoria and Albert Museum: Teachers' resource: Maths and Islamic art & design
{{DEFAULTSORT:Islamic Geometric Patterns Islamic architectural elements Islamic art Mathematics in the medieval Islamic world