Islamic Geometric Patterns in Architecture: How They Are Drawn

Stand under a Mughal jaali at Fatehpur Sikri, look at the tile panels of the Alhambra in Granada, or study a carved wooden ceiling in Kashmir, and the same quality appears: an interlacing pattern of stars and polygons that seems to continue forever, never quite repeating the way the eye expects, and yet perfectly ordered. Islamic geometric patterns are among the great achievements of design anywhere in the world, and India holds some of their finest examples — in stone, marble, wood, brick and tile.

They are also widely copied and often copied badly. Today a geometric jaali for a pooja door, a living-room partition or a hotel facade is usually downloaded as an image, traced into a cutting file and sent to a CNC router. The result frequently has stars cut in half at the panel edge, lines that do not quite meet, bridges too thin to survive, and a module that is too small to read from across the room. Understanding how the patterns were actually drawn — with a compass, a straightedge and a grid — solves most of these problems.

This article explains, respectfully and step by step, how the traditional patterns are constructed: the circle divisions that generate stars, the grids that repeat them, the polygons-in-contact method recorded by historians of the craft, and the girih tiles studied by modern scientists. It then looks at Indian examples, and turns the geometry into practical rules, sizes and indicative costs for jaalis, tiles and panels in modern interiors.

Why geometry: the origins of the tradition

Geometric ornament existed long before Islam — in Roman mosaics, Byzantine floors and Sasanian stucco — but in the Islamic world from about the ninth and tenth centuries it developed into a sophisticated, mathematically rich art. Several things came together. In mosques and many other religious buildings, figurative images of people and animals were generally avoided, so geometry, the flowing vegetal ornament known as arabesque, and calligraphy became the main decorative languages. At the same time, scholars in Baghdad and elsewhere translated and extended Greek mathematics, including Euclid's geometry, and some of that knowledge reached builders and craftsmen. A tenth-century treatise attributed to the mathematician Abu'l-Wafa al-Buzjani dealt specifically with geometric constructions useful to craftsmen, showing how closely mathematicians and artisans could work.

Meanings are often attached to these patterns — unity, order, the infinite — and many viewers and makers have found such readings valuable. It is fair to say that the patterns express order and continuity and that they repeat without a natural end; specific symbolic claims about individual patterns are harder to document and are best treated as interpretation. What is certain is the method: the patterns were constructed geometrically, using a small set of tools and rules, and passed from master to apprentice, in pattern books and on scrolls.

The tools: compass, straightedge and a grid

A traditional geometric pattern needs no ruler marked in units. The craftsman starts with a line and a circle; every other point is found by intersecting circles and lines. This is why the patterns scale so easily — the same construction can produce a 50 mm star on a box lid or a 3 m star on a dome — and why they fit together without gaps: every proportion comes from the circle itself, not from a measurement that might be rounded.

The second idea is the repeat unit. Almost every pattern is built inside a small cell — a square, a rectangle, an equilateral triangle or a hexagon — that is then repeated in every direction, sometimes mirrored or rotated. Mathematicians classify such repeats into seventeen wallpaper symmetry groups, and researchers have found many of them in Islamic ornament. For a designer, the practical question is simpler: which family is the pattern in? That family decides which stars can appear, which grid to draw and where a panel can be cut without breaking the pattern.

Method 1 — Dividing the circle: how an 8-pointed star is drawn

The 8-pointed star — sometimes called khatam in Persian-speaking traditions — is the most widely used star in Islamic design and the one most Indian jaalis are based on. Its construction shows the whole logic of the craft in four steps, shown in Fig. 2 for a module of 300 mm, a common size for a jaali or tile panel.

Step one: draw a circle whose diameter equals the module — here 300 mm — and draw two diameters at right angles, horizontal and vertical. Step two: bisect each right angle with the compass to find the diagonals, which divides the circle into eight equal parts of 45°. Step three: join every second point to draw a square, then join the remaining four points to draw a second square rotated by 45°. Step four: keep only the outer outline of the two squares; the result is the 8-pointed star. Its inner corners lie on a smaller circle at about 0.77 times the radius, so for a 300 mm module the inner circle is about 230 mm across.

The same idea gives every other common star. Dividing the circle into six with the compass (the radius steps around the circumference exactly six times) gives the 6-pointed star from two triangles, and the 12-pointed star from joining points further around. Dividing into five or ten needs a more careful construction, which is one reason 5- and 10-fold patterns are associated with the most skilled workshops. Joining every second point of a 10-division gives a {10/2} pair of pentagons; joining every third gives the sharper {10/3} star common in Iranian tilework.

From a star to a field

A single star is a motif; a pattern needs a field. The star is placed in its repeat cell — for the 8-pointed star, a square of the same size as the module — and the lines of the star are extended until they meet the lines of the neighbouring stars. Where they meet, new shapes appear between the stars: crosses, kite shapes, small octagons. The way the lines are extended is what separates one classical pattern from another, and craftsmen developed several reliable methods for doing it. The most widely discussed is the polygons-in-contact method.

Method 2 — Polygons in contact

In the early twentieth century, E. H. Hankin, a scientist based in Agra who studied the monuments around him, published an analysis for the Archaeological Survey of India of how geometric patterns in Indian and other Islamic buildings could be drawn. He observed that many patterns can be produced by first covering the surface with a network of simple polygons that touch edge to edge, and then drawing a pair of lines from the midpoint of every edge, at a chosen angle, until they meet lines from the neighbouring edges. The polygon network is then erased, leaving the pattern. Later researchers, notably the scholar and designer Jay Bonner, have explored this approach in great depth and shown how widely it explains historic designs.

The method is powerful because two decisions generate an entire family of patterns. The first is the polygon network: octagons and squares give 8-pointed stars; hexagons and triangles give 6- and 12-pointed stars; decagons, pentagons and rhombuses give 10-fold designs. The second is the contact angle — the angle at which the lines leave each midpoint. Small changes in the angle turn a calm, open pattern into a sharp, dense one, which is exactly the control a designer needs when adapting a historic pattern to a new size or material.

Method 3 — Girih tiles and the strapwork of Iran and Central Asia

The most mathematically celebrated patterns are the 5- and 10-fold designs common in Iranian and Central Asian architecture. Their strapwork lines — called girih, a Persian word for knot — are difficult to draw directly over a large surface. In 2007, the physicists Peter Lu and Paul Steinhardt published a study in the journal Science arguing that by around 1200 craftsmen had begun to use a set of five equal-edged tile shapes, each decorated with lines, to lay out such patterns: a decagon, a pentagon, an elongated hexagon, a bowtie and a rhombus. The decoration lines cross every tile edge at its midpoint at the same angle, so the tiles can be fitted together in many ways and the lines always continue smoothly.

The same study pointed to the Darb-i Imam shrine in Isfahan, dated 1453, as an example of a girih pattern with a very complex, nearly quasi-crystalline order — a kind of ordered but non-repeating structure that Western mathematics described only in the 1970s. Historians have also studied the Topkapı Scroll, a long pattern scroll associated with Timurid and later workshops, in which geometric designs are drawn with their construction lines, confirming that patterns were designed on paper with careful geometry before being transferred to buildings. Scholars continue to debate how widely girih tiles were used as a working method, but the shapes themselves are a clear way to understand 10-fold patterns.

Interlace, strapwork and the materials that carried the patterns

A geometric line is only a drawing. In a building it becomes a band with width, and often an interlace: bands that pass alternately over and under one another, like woven straps. Interlace adds depth and rhythm, and it is one of the clearest signs of a carefully made pattern — in good work, every crossing alternates correctly across the whole field. The band width matters too: in carved stone and stucco it is usually a modest fraction of the star size, wide enough to carve and read from a distance but narrow enough to keep the pattern light.

Each material carried the geometry in its own way. Glazed tile mosaic, cut piece by piece from fired tiles and assembled face down, is the technique of Morocco's zellige and Spain's alicatado. Carved and moulded stucco covers walls and arches in the Alhambra and in Iranian buildings. Brick laid in patterned bonds created geometry with no added ornament in Iran and Central Asia. Pierced stone produced the jaalis of India. Wood carried geometry in carved doors, in the lattice screens known in the Arab world as mashrabiya, and in the Kashmiri crafts of pinjrakari (lattice work) and khatamband — ceilings assembled from small wooden pieces fitted into geometric patterns. Painted, inlaid and enamelled versions appear on everything from ceilings to metalwork.

Islamic geometry in India: where to look

India's Islamic architecture absorbed geometric design from Iran and Central Asia and combined it with local stone-carving traditions of great skill. The Alai Darwaza in the Qutb complex in Delhi, built in 1311, has finely carved stone screens and is often cited as an early, accomplished example of Indo-Islamic ornament. In Gujarat, the Sultanate period produced mosques and tombs whose stone screens mix geometric and floral patterns; the Sidi Saiyyed Mosque in Ahmedabad is famous for its tree-of-life screens, which show that the jaali tradition was never limited to geometry, while other Gujarat Sultanate buildings, such as those at Sarkhej Roza and Champaner, carry geometric lattices.

Under the Mughals the geometric jaali reached a remarkable refinement. Humayun's Tomb in Delhi, completed around 1570, uses geometric stone screens and inlaid geometric borders; the marble tomb of Salim Chishti at Fatehpur Sikri is wrapped in screens of interlocking stars; the tomb of Itmad-ud-Daula in Agra (1622–28) combines geometric inlay and pierced marble; and the Taj Mahal encloses its cenotaphs within an octagonal marble jaali. In Kashmir, khatamband ceilings and pinjrakari lattices continue the tradition in wood. Readers who want the full story of the jaali itself can find it in this site's article on the Indian jaali.

Using the patterns today: rules for jaalis, tiles and panels

The same construction rules that made the historic patterns work solve most problems in modern work. The first is to choose the module size for the viewing distance. In the jaali guide on this site, interior screens read well with a module of about 75–150 mm and facades with about 200–450 mm; for a star pattern, the module is the star's repeat cell. The second is to cut the panel on lines of symmetry — through the centres of stars or along the mirror lines between them — so that every edge looks intentional. The third is to set the line width as a bridge that the material can survive.

Two more checks are worth making before approving any cutting file. Ask for the openness ratio — the percentage of the panel that is open — and match it to the job: about 25–40 per cent for a living–dining partition, lower for privacy, higher for a decorative frame. And check the drawing at full size, not on a phone screen: print one module at 1:1 or view it on a large monitor at true scale, and see whether the star points, bridges and crossings look the way they should.

Using sacred-origin patterns respectfully

Geometric patterns are shared heritage. They appear in mosques and tombs, but also in palaces, homes, gardens, markets and everyday objects, and they have been used by craftspeople of many faiths, including the Hindu, Jain and Muslim stone carvers of Gujarat and Rajasthan. Using a star pattern in a jaali, a tile or a ceiling is a long and widely accepted tradition. Calligraphy is different. Arabic inscriptions in historic buildings are often verses from the Qur'an, and many Muslims consider it inappropriate to place Qur'anic text on floors, in bathrooms, or anywhere it may be stepped on or treated casually. Designers who want calligraphy should commission it with knowledge of its content and place it with care; in most homes, geometry alone carries the character without that concern.

Respect also means accuracy. A pattern that claims a historic source should be drawn properly, with its symmetry intact and its crossings correct, rather than approximated from a low-resolution photograph. When a design is adapted — enlarged, simplified or cut in a new material — it is better to say it is inspired by a source than to present it as a copy.

The future: computation meets an old craft

The methods described here are, in effect, algorithms, which is why computers handle them so naturally. Parametric software can generate a polygons-in-contact pattern, vary the contact angle across a screen, and output a cutting file for a router, laser or waterjet in minutes. Architects have used geometric screens as climate devices on modern buildings: the Institut du Monde Arabe in Paris, completed in 1987, used a facade of mechanical apertures inspired by mashrabiya, and the Al Bahar Towers in Abu Dhabi use a responsive shading screen based on the same idea. In India, CNC-cut jaalis in MDF, metal and GFRC have made geometric screens affordable for ordinary homes. The risk is that ease of production replaces understanding. The patterns that will last are the ones drawn with the same care as the old ones — by people who know where every line comes from.

Rules for drawing and specifying a geometric pattern

• Identify the family first — 4-fold, 6-fold or 10-fold — and draw the matching grid before the motif.
• Build stars from circle divisions with compass and straightedge (or their software equivalents), never by eye.
• Choose the module from the viewing distance: about 75–150 mm for interior screens, 200–450 mm for facades.
• Make panel widths a whole number of modules and cut panels on lines of symmetry.
• Set bridges for the material: about the sheet thickness for metal, at least 1.5 × thickness for MDF, HDHMR and WPC.
• Check the openness ratio — about 25–40 per cent for most partitions.
• Keep interlace crossings alternating correctly if the pattern is woven.
• Review the cutting file at full scale before approving it.
• Use calligraphy only with knowledge of its content, and never on floors or in bathrooms.

Frequently asked questions

How are Islamic geometric patterns made?

Traditionally with only a compass and a straightedge. A circle is divided into equal parts, stars and polygons are drawn from the division points, and the unit is repeated on a square, triangular or hexagonal grid. Methods such as polygons in contact and girih tiles were used to extend the lines into large fields.

What is the 8-pointed star in Islamic art called?

It is often called khatam in Persian-speaking traditions, and it is also widely known as the octagram or the Rub el Hizb when drawn as two overlapping squares. It is the most common star in Islamic geometric design and in Indian geometric jaalis.

What are girih tiles?

A set of five equal-edged shapes — decagon, pentagon, elongated hexagon, bowtie and rhombus — decorated with lines that cross every edge at its midpoint at the same angle. Researchers have argued that craftsmen used them to lay out complex 10-fold strapwork patterns, such as those of the Darb-i Imam shrine in Isfahan.

Can I use Islamic geometric patterns in my home?

Yes. Geometric patterns are widely used in homes, palaces and public buildings as well as religious ones, and they suit jaalis, tiles, ceilings and furniture. Calligraphy, which may contain Qur'anic verses, should be used with care and never on floors or in bathrooms.

What size should a geometric jaali pattern be?

For interior screens viewed from 1–4 m, a module of about 75–150 mm reads well; for partitions in larger rooms, 150–250 mm; for facades seen from the street, about 200–450 mm. Panel widths should be a whole number of modules.