The math behind Islamic geometric patterns

Most Islamic geometric patterns are mathematical tessellations of the plane that conform to one of 17 possible wallpaper symmetry groups, built using rosette structures with base angles in multiples of 22.5°, constructible with ruler and compass by craftsmen who often did not formally know they were doing mathematics. The depth of mathematical sophistication in these patterns is real, was discovered late by Western scholarship, and continues to surface new structure even today. This post walks through the actual mathematics — what's going on under the hood when you look at a zellij wall or a girih panel.

The substance draws on Wichmann and Wade's Islamic Design: A Mathematical Approach (especially Chapters 7 and 8 and Appendix A), the Lu and Steinhardt 2007 paper on quasi-crystalline tilings, Conway, Burgiel, and Goodman-Strauss's The Symmetries of Things, and the International Tables for Crystallography.

Tessellation: the foundation

A tessellation, or tiling, is a covering of the plane by shapes that fit together with no gaps and no overlaps. Islamic geometric patterns are almost always tessellations in this strict mathematical sense. The shapes (called tiles in this context, regardless of the actual material used) fit edge-to-edge: where two tiles meet, they share exactly one complete edge.

This constraint is much stronger than it sounds. It rules out almost all freeform compositions. A rectangle of random shapes scattered across a plane is not a tessellation. An Islamic geometric pattern is a precise mathematical structure.

The 17 wallpaper groups

If you classify all the possible ways to tessellate the plane with a repeating pattern, you end up with exactly seventeen distinct categories. These are the wallpaper symmetry groups, formalized in the late 19th and early 20th centuries (the proof that there are exactly seventeen, no more and no fewer, was completed by 1924).

Each group is defined by the symmetries that preserve the pattern — translations, rotations, reflections, and glide reflections — and the relationships between them. The 17 groups have systematic notation: p1, p2, p3, p4, p6, pm, pg, cm, pmm, pmg, pgg, cmm, p3m1, p31m, p4m, p4g, p6m. Most of these are unfamiliar to anyone who hasn't studied crystallography. They cover everything from minimal symmetry (p1, which has only translation) to maximum symmetry (p6m, which has rotation, reflection, and glide reflection in multiple axes).

The remarkable fact: Islamic craftsmen produced patterns in all 17 groups over the course of centuries, without knowing the mathematical classification existed. The Alhambra alone contains examples of patterns in most or all 17 groups, depending on how strictly you classify the examples. This was first systematically documented in the 20th century by Edith Müller, who completed a doctoral thesis at the University of Zürich in 1944 cataloging the symmetry groups present at the Alhambra. The craftsmen were doing crystallography by hand, six centuries early.

For the formal mathematical details, see the International Tables for Crystallography Volume A on space-group symmetry. For a more accessible treatment, Conway, Burgiel, and Goodman-Strauss's The Symmetries of Things is the standard modern reference.

The four allowable rotations

A subtle but important fact about wallpaper groups: only certain rotational symmetries are possible. A repeating pattern in the plane can have 2-fold, 3-fold, 4-fold, or 6-fold rotational symmetry, and nothing else. No 5-fold. No 7-fold. No 8-fold. This is called the crystallographic restriction and is provable in a few lines of geometric argument.

So how do Islamic patterns end up with 8-pointed and 10-pointed and 16-pointed stars? Because rotational symmetry of the star is different from rotational symmetry of the overall pattern. A pattern can contain 8-pointed stars while the overall pattern only has 4-fold rotational symmetry (the star looks the same after a 45° rotation, but rotating the whole tessellation by 45° doesn't preserve it; rotating by 90° does). The motifs are more symmetric locally than the global tiling.

The exception is the famous Penrose tilings and the related Islamic girih patterns discovered by Lu and Steinhardt: these are non-periodic patterns that don't repeat, and therefore aren't constrained by the crystallographic restriction. They can have 5-fold and 10-fold symmetry in ways no truly periodic pattern can. The 15th-century Persian craftsmen who produced these patterns at the Darb-e Imam shrine in Isfahan had figured out non-periodic tessellation centuries before Roger Penrose described it formally in the 1970s.

The rosette: the iconic motif

The most recognizable Islamic pattern element is the rosette — a central star surrounded by polygons. The structure is highly conventionalized.

A rosette consists of:

• A central polygon, almost always a regular star polygon with 6, 8, 10, 12, 16, or more points
• A ring of "petals" around the star — typically six-sided polygons whose shape depends on the star's geometry
• "Kites" — four-sided convex polygons filling the gaps
• Sometimes "darts" — concave four-sided polygons

The 8-pointed khatem star anchors a family of more than 140 documented patterns in the Maghreb tradition (zellij). The 10-pointed star anchors the Persian tradition (girih). The 12-pointed star is more common in Mamluk Egyptian and Mughal Indian work.

The rosette structure exists at multiple scales. A typical Islamic pattern might have rosettes at two different scales — large rosettes filling the major positions, smaller rosettes at the intersections — creating the "two-level patterns" discussed in Wichmann Ch 15. The Topkapı Scroll documents many examples of this multi-scale construction.

For more on the dominant motif of the Maghreb tradition specifically, see the khatem — the 8-pointed star.

The sine rule and base angles

The classical construction technique uses base angles in multiples of 22.5° (which is 360° divided by 16). The reason: most rosette geometry resolves cleanly at these angles. An 8-pointed star has internal angles related by 22.5° increments. A 16-pointed rosette uses 22.5° as its fundamental unit. Working in these increments lets the craftsman use simple ruler-and-compass construction.

Wichmann Chapter 8 works through a complete construction of an Alhambra pattern. Starting from a single base length, every other dimension in the pattern can be derived using the sine rule (sin A / a = sin B / b = sin C / c). The pattern is fully specified by one number; the rest is geometry.

This is one of the most striking things about Islamic geometric design from a mathematical perspective. Patterns that look impossibly complex turn out to be exact derivations from a single starting measurement. The craftsmen who built them, often without the formal sine rule, used techniques in sand boxes that achieved the same result by physical construction.

The mismatch problem

Most Islamic patterns work cleanly at common rotational symmetries (8-fold, 12-fold). Some don't.

A 24-pointed star, for instance, doesn't fit cleanly inside an octagonal pattern. The angles don't resolve. The mathematically exact pattern produces gaps or overlaps. Wichmann Chapter 10 documents how Mamluk craftsmen solved this: by adjusting the dimensions slightly, accepting tiny imperfections invisible to the eye, and producing the pattern anyway. The sand-box method allowed manual fine-tuning that the strict mathematics didn't.

Modern parametric tools handle this problem differently: by computing the exact mathematical solution and either accepting the imperfection numerically or solving with adjusted parameters. The tool I built for my own work, SVG Stack Studio, makes these calculations explicit and visible to the artist. See the about page for the technical details.

The Penrose connection

In 1974, Roger Penrose described aperiodic tilings — coverings of the plane that don't repeat but use only two distinct tile shapes. The discovery was treated as one of the major mathematical surprises of the 20th century, and it laid the groundwork for the discovery of quasi-crystals in physical materials (which won Daniel Shechtman the 2011 Nobel Prize in Chemistry).

In 2007, Peter Lu and Paul Steinhardt published a paper in Science demonstrating that 15th-century Persian craftsmen producing the girih patterns at the Darb-e Imam shrine in Isfahan had effectively figured out aperiodic tiling 500 years before Penrose. Their analysis showed the patterns could be decomposed into a small set of "girih tiles" — five specific shapes — combined in ways that produce quasi-crystalline order without periodic repetition.

This is the single most striking mathematical fact about Islamic geometric art, and it's worth pausing on. The craftsmen weren't working from a formal mathematical theory. They were working from accumulated craft knowledge, design templates passed through generations, and aesthetic intuition. But what they produced was mathematically equivalent to a theory the West wouldn't develop for half a millennium.

From craft to code

The mathematics of Islamic geometric patterns is now actively taught and applied in design schools, mathematics departments, and parametric design tools. Eric Broug's books (Islamic Geometric Patterns and Islamic Geometric Design) teach the construction techniques to general audiences. Adam Williamson and others run workshops teaching the traditional methods. Parametric tools — Grasshopper for Rhino, custom plugins for Illustrator, dedicated apps like my own SVG Stack Studio — automate the geometry while preserving the underlying mathematical relationships.

The continuity matters: the same mathematics that governed a 14th-century zellij panel governs a 21st-century laser-cut layered piece. The material changes; the geometry doesn't.

For the broader tradition, see Islamic geometric art: a complete guide. For the most iconic specific motif, see the khatem.

FAQ

What's special about the math in Islamic geometric patterns?

Two things. First, Islamic craftsmen produced patterns in all 17 wallpaper symmetry groups (the complete classification of repeating planar patterns) centuries before the mathematical classification existed. Second, 15th-century Persian craftsmen produced aperiodic tessellations equivalent to Penrose tilings 500 years before Penrose, as shown by the 2007 Lu and Steinhardt paper in Science.

How many possible Islamic patterns are there?

Effectively unlimited within the constraints of the tradition. The 17 wallpaper symmetry groups provide the structural categories; within each group, any combination of motifs, scales, and colorings produces a distinct pattern. The Maghreb octagonal family alone has more than 140 documented patterns. Counting all regions and periods, the total documented corpus exceeds several thousand distinct designs.

Did the original craftsmen know the math?

In a practical sense, yes. In a formal sense, mostly no. Craftsmen learned construction techniques (the sand box, ruler-and-compass methods, design templates) from masters and applied them by feel and accumulated knowledge. Some, especially in court workshops, would have been familiar with Euclidean geometry through Arabic translations of Greek texts. The formal mathematical theory of symmetry groups didn't exist until the 19th century, and the formal theory of aperiodic tessellation didn't exist until 1974.

Can you make Islamic patterns with a computer?

Yes, and increasingly people do. Parametric design tools — Grasshopper, custom Illustrator scripts, dedicated apps — can generate Islamic patterns from mathematical rules and produce output files suitable for laser cutting, CNC milling, or digital printing. The mathematics is identical; the workflow is different. My own work uses a tool I built called SVG Stack Studio specifically for this purpose.

Sources

• Wichmann, Brian, and David Wade. Islamic Design: A Mathematical Approach. Springer, 2017. Chapter 7, Chapter 8, Appendix A.
• Lu, Peter J., and Paul J. Steinhardt. "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture." Science 315 (2007).
• Conway, J. H., H. Burgiel, and C. Goodman-Strauss. The Symmetries of Things. A K Peters, 2008.
• International Tables for Crystallography Volume A — formal classification of the 17 wallpaper groups.
• Necipoğlu, Gülru. The Topkapı Scroll. Getty Center, 1995.
• Edith Müller's 1944 doctoral thesis on Alhambra symmetry groups, University of Zürich.
• Broug, Eric. Islamic Geometric Design. Thames & Hudson, 2013.