The basic tools for creating Islamic geometric designs were a compass and a ruler. The circle became the basis of Islamic pattern. The circle plays an important role in calligraphy, described by the Arabs as ‘the geometry of the line’, and gives structure to all complex Islamic patterns containing geometrical shapes. These patterns have three basic characteristics:

1. Repetition of geometric elements.
The simple forms of the circle, square and straight line are the basis of the patterns. Most patterns are based on one or two grid types – one consisting of equilateral triangles, the other of squares. A third type of grid, consisting of hexagons, is a variant of the triangle theme. 
The mathematical term for these grids is ‘regular tile pattern’ (from the Latin tesserae, meaning pieces of mosaic), in which a regular polygon is repeated to cover the plane.

2. A background and foreground pattern.
Plant patterns occur against a contrasting background in which the plant-like forms are linked and interwoven in a way that emphasises the decoration of the foreground. In other cases, the background is replaced by a contrast of light and shadow.

Sometimes it is impossible to distinguish the foreground from the background. Some geometric designs are made by bringing all the polygonal shapes together like the pieces of a jigsaw puzzle, with no empty spaces, and therefore no need for a spatial interplay between the foreground and background.
The concept of space in Islamic art is completely different from Western models, which usually use a linear perspective and divide the space of the picture into foreground, middle zone and background.
Most artists from the Islamic world used a kind of three-dimensional space, in which figures overlapped each other. This space offered multiple points of view and used a bird and frog perspective at the same time.

3. They were not designed to fit within a framework.
Geometric decorations in Islamic art suggest a remarkable freedom. The complex arrangements and combinations of elements can be extended ad infinitum, the framework around the pattern seems arbitrary and the basic composition is sometimes a unit from which the rest of the pattern repeats itself.

1436 AD. (Bahmanid period)
Tomb of Nematollah Vali, Kerman, Iran
The blue girih tiled dome contains stars with 5, 7, 9, 12, 11, 9 and 10 points alternating from the top. 11-point stars are rare in the geometric patterns of Islamic art.

1876 AD. (Qajar period)
Nasir al Molk Mosque, Shiraz, Iran

13th c. AD. (Anatolian beylik period)
Eşrefoğlu Mosque, Beyşehir, Turkey
One of the few surviving and best preserved mosques with wooden muqarnas. Mosques with wooden columns were once much more common, but over the centuries their numbers have been greatly reduced.

11th c. AD. (Kara-Khanid period)
Bukhara Mosque, Uzbekistan
(Photo © Freepik)

17th c. AD.
Sher-Dor Madrasa (Islamic School), Samarkand, Uzbekistan
Entrance decorated with lions, deer, Mongolian faces and Zoroastrian inspired suns. This was controversial for the Islamic traditions of the time, as the depiction of living animals or people was not accepted.
(Photo © Freepik)

17th c. AD.
Interior dome in the Tilya Kori Madrasa, Samarkand, Uzbekistan
(Photo © Freepik)

15th c. AD.
Ulugh Beg Madrasa, Samarkand, Oezbekistan
Statue of Ulugh Beg, notable for his work in astronomy-related mathematics, such as spherical geometry. He build the great Ulugh Beg Observatory in 1429 in Samarkand.
(Photo © Freepik)

17th c. AD. (Safavid period)
Masjid-i Shah Mosque (Jame Abbasi), Isfahan, Iran
(Photo © Freepik)

18th c. AD.
Interior of the Vakil Mosque, Shiraz, Iran
(Photo © Freepik)

1324 AD. (pre-Mugol period)
Screen in the Mausoleum of Shah Rukn-e-Alam, Multan, Pakistan 

13-14th c. AD. (Nazari period or Nasrid dynasty)
Geometric plant pattern on a wall of Alhambra palace, Andalusia, Spain
(Photo © Steve Miller)

13-14th c. AD. (Nazari period or Nasrid dynasty)
Geometric patterns from the Mexuar hall of Alhambra palace, Andalusia, Spain
(Photo © Sir Cam)

1573 AD. (Sultanate Gujarat period)
Jali, Sidi Saiyyed Mosque of Ahmedabad, Gujarat, India
(Photo © Vrajesh jani)

1605-1627 AD. (Mogul period)
Jali, India
Louvre-Lens Museum, France
(Photo © Algoet Stefaan)

Introduction by Eric Broug to complex Islamic geometry.

Another three-dimensional geometric art form is the muqarnas, originally a kind of stalactite vault. In the beginning, the function of muqarnas was to create a beautiful transition from the vertical walls to the dome in a square building with a round dome. The muqarnas soon took on a life of their own, and sometimes the entire interior of the dome is covered with a stalactite vault. Also known as honeycomb or stalactite vaults.

Above: 3D muqarnas Mosque of the Shah
Below: 2D projection drawing of these muqarnas, by Shiro Takahashi

15-16th c. AD.
Design of a muqarnas quarter vault from the Topkapi roll

13-14th c. AD. (Nazari period or Nasrid dynasty)
Hall of the murqanas, lion palace, Alhambra, Andalusia, Spain

19th c. AD.(Qajar period)
Nasir al-MulkMosque, Shiraz, Iran
(Photo © Freepik)

19th c. AD. (Qajar period)
Nasir al-Mulk Mosque, Shiraz, Iran
(Photo © Freepik)

13-14th c. AD. (Nazari period or Nasrids)
Hall of the Murqanas, Lion Palace, Alhambra, Andalusia, Spain
(Photo © Freepik)

In medieval Islamic times, simple mathematics was used in surveying and for administrative purposes. Higher-level mathematics was also practised, namely algebra (quadratic and cubic equations), number theory, geometry according to the Greeks, and trigonometry. Geometry and trigonometry were used by astronomers.

According to the Iranian geometrician and astronomer Abu’l-Wafa al-Buzjani (940-998 AD.), there were two groups: the mathematically trained geometrists who could give theoretical proofs but had little experience in practical drawing, and the creators of ornaments who could draw practically but had no knowledge of proofs.
This is also why, in most Arabic texts on Euclidean geometry, there is no mention of geometrical ornaments. These texts were written by theoretically trained mathematicians, and not by the makers of geometrical ornaments.

The geometrical ornaments are clearly of a higher mathematical content than what is necessary for surveying and administration. The question now is whether the designers and makers of such ornaments were the same mathematicians who also practised astronomy, or whether they were a completely different group.

Which mathematics did these medieval designers use? The designers were not trained in the Elements of Euclid. They had a different kind of mathematical knowledge, which was largely not recorded in writing. Such scriptless traditions also exist in other cultures and are sometimes called ‘Ethnomathematics’. 

This knowledge, and the accompanying drawings of patterns, may have been passed down from father to son in families of craftsmen, and perhaps the methods of construction were kept secret through generations.

Excerpts from
– ‘Middeleeuwse islamitische geometrische ornamentiek’, by Jan P. Hogendijk
– ‘Islamic Art and Geometric Patterns’, by The Metropolitan Museum of Art