Girih tiles are a series of five tiles that were used in the creation of Islamic geometric patterns for the decoration of buildings in Islamic architecture. Girih meaning “knot” in Persian, are lines (strapwork) that decorate these tiles. In most cases, only the girih (and other small decorations such as flowers) are visible rather than the borders of the tiles themselves. Most tiles have a unique pattern of girih inside the tile that are continuous and follow the symmetry of the tile. However, the decagon has two possible girih patterns, one of which has only fivefold rather than tenfold rotational symmetry.

In 2007, physicists Peter J. Lu and Paul J. Steinhardt suggested that girih tessellations possessed properties consistent with self-similar fractal quasi-crystalline tessellations, such as Penrose tessellations.
A concept that was only discovered by western mathematicians and physicists in the 1970’s and 1980’s. If so, the medieval Islamic application of this geometry would be at least half a millennium older than its Western mastery. Scholars thought the girih were made by drawing a zigzag mesh of lines with a straight edge and a compass. But when Lu looked at it, he recognised the regular but non-repetitive patterns of Penrose tiles. This finding was supported both by analysis of patterns on surviving structures and by examination of 15th-century Persian scrolls prepared by master architects to document their techniques. However, there is no indication of how many architects may have known about the mathematics involved.

Templates found on scrolls, such as the Topkapi scroll, may have been consulted. The scroll shows a sequence of two- and three-dimensional geometric patterns. There is no text, there is a grid pattern and colour coding used to mark symmetries and distinguish three-dimensional projections. Drawings like the one on this scroll would have served as pattern books for the craftsmen who made the tiles, and the shapes of the girih tiles dictated how they could be combined to form large patterns. In this way, craftsmen could create highly complex designs without resorting to mathematics and without necessarily understanding their underlying principles.

Breathtakingly elaborate geometric tiles are a distinctive feature of medieval Islamic architecture in the Middle East and Central Asia. Art historians have long assumed that simpler elements of the patterns were created with elementary tools such as rulers and compasses. But there is no explanation as to how artists and architects could have created the distinctly complex tile patterns that adorn many medieval Islamic buildings.

While it is possible to create these patterns individually with basic tools, they are incredibly difficult to replicate on a larger scale without generating extensive geometric distortions. The most complex medieval Islamic tiles barely show any distortion.
Directives and compasses work fine for the recurring symmetries of the simplest (periodic) patterns, but much more powerful tools were probably needed to fully explain the elaborate tiling with decagonal symmetry.

‘TABL’ or Decagon
A regular decagon with ten interior angles of 144°

‘PANGE’ or Pentagon
A regular pentagon with five interior angles of 108°

‘SORMEH DAN’ or Bow-tie
A bow-tie (non-convex hexagon) with inner angles of 72°, 72°, 216°, 72°, 72°, 216°

‘SHESH BAND’ or Bobbin
An elongated (irregular convex) hexagon with interior angles of 72°, 144°, 144°, 72°, 144°, 144°

‘TORANGE’ or Rhombus
A diamond with interior angles of 72°, 108°, 72°, 108°

All the sides of these figures have the same length; and all their angles are multiples of 36 ° (π / 5 radians ). The girih tiles also incorporate decorative lines and each of these decorative lines intersects the centre of each edge at 72 and 108 degrees. These decorative lines allow for a continuous pattern across an entire tile.

All of them, except the pentagon, have bilateral (reflection) symmetry through two perpendicular lines. Some have additional symmetries. In particular, the hexagon has a tenfold rotational symmetry (rotation over 36°); and the pentagon has a fivefold rotational symmetry (rotation over 72°).

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

1628 AD. (Mogul period)
Tomb of I’timad-ud-Daula, Agra, India
(Photo © SmugMug, Inc./ Kim Carpenter)

Left: Tile pattern on the central plane (Tomb of I’timad-ud-Daula)
Right: Reconstruction of the pattern with the girih tiles

1424 AD. (Pre-Ottoman period)
Fragment arch in the Green Mosque in Bursa, Turkey

Reconstruction of the pattern with the girih tiles.

1453 AD. (“Black sheep” Turkman period)
Portal of the Darb-i Imam Shrine in Isfahan, Iran

Reconstruction of the pattern with the girih tiles.

1197 AD. (Great Seljuk period)
Gunbad-i Kabud in Maragha, Iran

Reconstruction of the pattern with the girih tiles.

15-16th c. AD.
Panel 28 of the Topkapi roll showing the five Girih tiles

The five Girih tiles (Persian name)
‘TABL’ or Decagon
“PANGE” or Pentagon
‘SORMEH DAN’ or Bow-tie
‘SHESH BAND’ or Bobbin
‘TORANGE’ or Rhombus

In ‘normal’ crystals, atoms are arranged regularly and periodically. The latter means that they have a certain geometric structure and that structure repeats itself with a certain symmetry. For example, there are crystals with a two-, three-, four- or sixfold symmetry. This means that the position of the crystals when you turn them 180 degrees around their axis is indistinguishable from the position the crystals were in before turning.

In 1984, researcher Dan Shechtman created a quasicrystal in a laboratory. Quasicrystals are crystals that consist of atoms with an apparently regular, but in reality aperiodic structure. The latter means that their organisation changes as they grow.

So quasicrystals are regular patterns that do not repeat themselves. They have everything to do with the well-known ‘Fibonacci sequence’, in which each number is the sum of the two preceding numbers. The ratio between two of these numbers comes closer and closer to the Golden Ratio, which occurs frequently in nature.

Quasi-crystal patterns are similar to Penrose mosaics, the so-called ‘aperiodic mosaics’ made of two different tiles. Such a pattern is regular, but never repeats itself.

In the thirteenth century, Islamic artists were already making tile patterns from five different tiles.

In 2009, a quasicrystal was found for the first time in nature in the east of Russia. The new mineral was discovered more specifically in the Khatyrka meteorite. It consists of aluminium, copper and iron and under the electron microscope it displays a neat tenfold symmetrical pattern. So it is not terrestrial.

Khatyrka-Meteorite