Girih

Girih patterns have been used to decorate varied materials including stone screens, as at Fatehpur Sikri; plasterwork, as at mosques and madrasas such as the Hunat Hatun Complex in Kayseri; metal, as at Mosque-Madrassa of Sultan Hassan in Cairo; and in wood, as at the Mosque–Cathedral of Córdoba.

The girih style of ornamentation is thought to have been inspired by 2nd century AD Syrian Roman knotwork patterns.

The Umayyad Mosque (709–715) in Damascus, Syria has window screens made of interlacing undulating strapwork in the form of six-pointed stars.

[3] Early examples of Islamic geometric patterns made of straight strap lines can be seen in the architecture of the surviving gateway of the Ribat-i Malik caravanserai, Uzbekistan, built in 1078.

Architecture was classified in the field of practical geometry in the early Islamic period, and building projects always involve a muhandis (geometer).

[5] In addition, no clear border was established between science and craft;[5] thus, the craftsmen usually followed the mathematicians’ principles and guidelines directly.

[7] In woodwork, one of the earliest surviving examples of Islamic geometric art is the 13th-century minbar (pulpit) of the Mosque of Ibn Tulun, Cairo.

[2] In 10th century a systematic investigation of geometric patterns was conducted by Persian mathematician and astronomer Abu al-Wafa' Buzjani in the House of Wisdom.

It became a dominant design element in the 11th and 12th centuries, as in the carved stucco panels with interlaced girih of the Kharraqan towers (1067) near Qazvin, Iran.

Today, artisans using traditional techniques use a pair of dividers to leave an incision mark on a paper sheet that has been left in the sun to make it brittle.

[a][10] Girih patterns made this way are based on tessellations, tiling the plane with a unit cell and leaving no gaps.

Although there is evidence that some ancient girih tilings used a subdivision rule to draw a two-level pattern, there are no known historic examples that can be repeated an infinite level of times.

For example, the pattern used in the spandrel of the Darb-i Imam shrine (see figure) consists only of decagons and bowties, while the subdivision rule uses an elongated hexagon tile alongside these two shapes.

It is therefore impossible to tile the plane periodically with a figure that has five-fold rotational symmetry, such as a five-pointed star or a decagon.

Patterns with infinite perfect quasi-periodic translational order can have crystallographically forbidden rotational symmetries such as pentagonal or decagonal shapes.

The use of such a subdivision rule would serve as evidence that Islamic artisans of the 15th century were aware that girih tiles can produce complex patterns that never exactly repeat themselves.

It appears that medieval Islamic artisans were using a tool that had the potential of creating highly complex patterns, but they never realized it.

As E. Makovicky argues,[24] The artisans were satisfied by creating a large fundamental domain without being concerned with the mathematical notion of indefinitely expandable quasiperiodic patterns.

[23] Ibn al-Razzaz al-Jazari's Compendium of Science and Useful Practice in the Mechanical Arts contains explicit templates for special purposes such as cast bronze doors.

This method refers to that the number of star polygons applied to the pattern are highly dependent on the change of the dome curvature.

In addition, girih patterns vary a lot on the surface, with different geometric shapes including decagons, hexagons, bowties and rhombuses.