Persia, a Museum of Materials · Hall 10 · Glazed Tile

Build a girih from five tiles

Girih means knot. The star-and-strap patterns on the walls of Isfahan look impossibly complex, but the craftsmen who made them could work with a kit of five shapes, each with the same edge length and a fixed set of lines inside. Fit the tiles edge to edge and the lines join into one endless knot. Tap a glowing edge to add a tile.

Choose a tile

Angle of the lines54° Show the five tiles

Material: Glazed tile, cut and painted
Place: Isfahan, capital of Safavid Iran from 1598
Time: 15th to 18th century; the oldest girih is more than 900 years old
The kit: Five tiles, one edge length, angles of 36° and its multiples
The knot: Lines that cross every edge at its middle, at 54°

Try this

Choose the pentagon and tap the glowing edges around the decagon. Ten of them fit exactly: 144° + 108° + 108° = 360°.
Add a bowtie and press Turn. The same tile placed a different way makes a different knot.
Drag the angle of the lines away from 54°. The pattern comes apart into odd shapes; at 54° the stars return.

The five tiles, their angles and the rule for the lines follow the analysis of Peter Lu and Paul Steinhardt (Science, 2007), who identified the kit from medieval buildings across Iran, Iraq and Anatolia, from the Maragheh tomb tower of 1197 to the Darb-e Imam in Isfahan, and from a pattern scroll of around 1500 in the Topkapı palace. The angle slider is a modern toy, not a historical tool. The colours are ours; real girih was carved in brick, cut from tile mosaic or painted on seven-colour tiles.

The five tiles

Every edge has the same length and every angle is a multiple of 36°, so the tiles fit together in countless ways. The lines inside each tile cross its edges at the midpoints, always at the same angle, so whatever you build, the lines join. The names are the traditional Persian ones.

Decagon

tabl

10 × 144°

Pentagon

pange

5 × 108°

Elongated hexagon

shesh band

72°, 144°, 144°, 72°, 144°, 144°

Bowtie

sormeh dan

72°, 72°, 216°, 72°, 72°, 216°

Rhombus

torange

72°, 108°, 72°, 108°

Patterns that never repeat

In 2007 two physicists, Peter Lu and Paul Steinhardt, argued that a pattern on the Darb-e Imam shrine in Isfahan, made in 1453, is built from three of these tiles, decagons, bowties and hexagons, at two scales at once: large decagons and bowties subdivided into small ones. The result never exactly repeats. It has the order of the tilings the mathematician Roger Penrose found in 1974, and of the quasicrystals physicists discovered in 1982, five centuries later.

Seven colours

The finest tilework was mosaic: each colour cut from a separate glazed slab and fitted like a jigsaw. It was slow. Under the Safavids, who covered Isfahan in tile after making it their capital in 1598, many walls were clad in haft-rang, “seven-colour” tiles: whole squares painted with several glazes and fired once, much faster and with a wider range of colours, if a little less crisp than mosaic.

The dome of Sheikh Lotfollah

On the east side of Isfahan’s great square stands a small mosque with no minaret, built between 1602 and 1619 for the private prayers of the court. Inside, the tile pattern of the dome shrinks toward the centre, where a peacock is drawn; the sunlight that enters seems to become its tail. The square itself has been a UNESCO World Heritage Site since 1979.

Episode 10 of 12

Still alive today

The tile in your kitchen is still a kashi in Persian, a name widely thought to come from Kashan, whose potters were making lustre tiles by around 1200.

Questions people ask

What does “girih” mean?

It is the Persian word for knot. The patterns are strapwork: bands that seem to pass over and under each other like a knotted cord, joining stars and polygons into one design.

Why are the angles all 36°, 72° and so on?

Because the patterns are built on tenfold symmetry: ten-pointed stars and pentagons. 360° divided by ten is 36°, and every angle in the five tiles is a multiple of it.

What is a quasicrystal?

An arrangement that is perfectly ordered but never repeats, like the Penrose tilings of 1974. Real quasicrystals were found in metal alloys in 1982, a discovery that won Dan Shechtman the Nobel Prize in Chemistry in 2011. Lu and Steinhardt argue that the Darb-e Imam pattern has the same kind of order, though a wall can only show a finite piece of it.

Where is the word “kashi” from?

In Persian a glazed tile is a kashi, a name widely thought to come from Kashan, the city between Isfahan and Tehran whose potters made lustre tiles from around 1200 to the 14th century.

Where can I see girih?

Almost everywhere in Iran, but the best-known walls are in Isfahan: the Friday Mosque, where ten-pointed girih appears as early as 1086 and which has been a World Heritage Site since 2012; the Darb-e Imam shrine; and the mosques on Naqsh-e Jahan Square. Outside Isfahan, see the Blue Mosque of Tabriz (1465) and the Gonbad-e Kabud tomb tower at Maragheh (1196–97), one of the earliest walls on which the five-tile kit appears.

Who worked out the geometry?

Mathematicians and craftsmen together. In the tenth century Abu al-Wafa Buzjani wrote a book of the geometric constructions a craftsman needs, many of them done with a ruler and a compass of fixed opening. An anonymous Persian treatise of the 13th to 15th century, “Interlocks of Similar or Complementary Figures”, shows a girih being drawn step by step, and the Topkapı scroll, 29.5 m long with 114 patterns, is a pattern book from around 1500.

Continue your visit

In the next episode…

About 26 million knots of wool become a single carpet, signed “a servant of the court, Maqsud of Kashan”: how many days would your own rug take?
Hall 11 · Wool · The Carpet →

Next hallHall 11 · Wool · A picture made of knotsEnter the hall →EarlierHall 9 · Paper & Ink · Measure the EarthEnter the hall →The museumThe entrance · all twelve halls, oldest firstBack to the entrance →

Sources: Peter J. Lu and Paul J. Steinhardt, “Decagonal and quasi-crystalline tilings in medieval Islamic architecture”, Science 315 (2007); Wikipedia, “Girih tiles”, “Girih”, “Sheikh Lotfollah Mosque”, “Shah Mosque (Isfahan)”, “Cuerda seca”, “Kashi Kari” and “Islamic geometric patterns”; MacTutor, “Abu’l-Wafa”; UNESCO World Heritage List, “Meidan Emam, Esfahan” (1979) and “Masjed-e Jāmé of Isfahan” (2012).

The drawing board and the tiles are coded by Bridge Labs after the five girih tiles of Lu and Steinhardt.