The complex of disciplines composed of mathematics, architecture and art in Islamic civilisation has been an important field of recent research. The scholars showed the interaction between mathematical reflexion and procedures and their implementation in designing concrete and symbolic forms in buildings, decoration and design. Furthermore, recent scholarship pointed out the amazing progress that this marriage brought about in prefiguring outstanding mathematical results that scientists proved only in late 20th century. In the following survey, Professor Salim Al-Hassani explores the various facets of this exciting subject that is still full of discoveries to come. By drawing attention to the ongoing debates in scholarly circles among physicicts, mathematicians and historians of science, art and architecture, he shows how the connection between theoretical and applied mathematics was fruitful and creative in the Islamic tradition

**Note of the editor**

This essay is a revised and expanded version of a lecture presented at the 28th Annual Conference on the History of Arabic Sciences organised by the Institute for the History of Arabic Sciences, Aleppo University, Aleppo, Syria, in 25-27 April 2007. It was submitted for publication in the proceedings of the conference.

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**Table of contents**

1. Introduction

2. Geometry at the basis of Islamic architectural decoration

3. The 'Muqarnas' Project

4. Conflicting Views on Mathematical Islamic Art

4.1. Dr Zohor Idrisi's View

4.2. Professor George Saliba's View

5. The Work of Alpay Özdural

6. Combination of Mathematics, Astronomy, Art and Architecture

7. Conclusion

8. Bibliography and Resources

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"Sophisticated geometry in Islamic architecture", "Geometry meets artistry in medieval tile work", "Geometry meets Arts in Islamic tiles". These were some of the headlines we saw in February 2007 in the main news agencies and science dispatches giving coverage to an exciting discovery published by two American scholars, Peter J. Lu and Paul J. Steinhardt (respectively from the Department of Physics at Princeton and Harvard universities).^{[1]} The discovery is "that medieval Islamic artists produced intricate decorative patterns using geometrical techniques that were not understood by Western mathematics until the second half of the 20th century". The combinations of ornate stars and polygons that have adorned mosques and palaces since the 15th century were created using a set of just five template tiles, which could generate patterns with a kind of symmetry that eluded formal mathematical description for another 500 years. The authors suggest that the Islamic artisans who created these typical girih^{[2]} designs had an intuitive understanding of highly complex mathematical concepts. They also suggest that these could be proof of a major role of mathematics in medieval Islamic art or it could have been just a way for artisans to construct their art more easily.

Girih designs feature arrays of tessellating polygons of multiple shapes, and are often overlaid with a zigzag network of lines. It had been assumed that straightedge rulers and compasses were used to create them — an exceptionally difficult process as each shape must be precisely drawn. From the 15th century, however, some of these designs are symmetrical in a way known today as "quasi-crystalline". Such forms have either fivefold or tenfold rotational symmetry — meaning they can be rotated to either five or ten positions that look the same — and their patterns can be infinitely extended without repetition. The principles behind quasi-crystalline symmetry were calculated by the Oxford mathematician Roger Penrose in the 1970s, but it is now clear that Islamic artists were creating them more than five centuries earlier.

The present paper reviews this discovery and discusses related literature on the subject of mathematics and arts in Muslim heritage. In particular, it accounts the related works of:

1. Alpay Özdural who showed how such geometrical patterns were used to solve cubic algebraic equations and also used the manuscript of Abu'l-Wafa and other mathematical Islamic mathematical treatises as evidence that mathematicians instructed artisans,

2. Gülru Necipoglu who discussed geometry, muqarnas and the contribution of the mathematical sciences and

3. George Saliba who presented critical arguments against some of the derived conclusions.

4. Zohor Idrisi who belives that ongoing work on Islamic tiles lacks the essential historical context that is required to inform the reader of how and when these mathematical techniques developed.

It is hoped that this review paper will bring to life the debate on the subject of Mathematics and Islamic Art and Architecture.

**2. Geometry at the basis of Islamic architectural decoration**

A study of medieval Islamic art has shown that some of its geometric patterns use principles established only centuries later by modern mathematicians. In particular, recent research has provided the ground for the astonishing claim that 15th century Muslim architects and artists used techniques inspired by what mathematicians nowadays call "quasicrystalline geometry". This indicates intuitive understanding of complex mathematical formulae, even if the artisans had not worked out the underlying theory.

**Figure 1:** A computer reconstruction of the quasicrystalline patterns of the Darb-i Imam shrine (Isfahan, Iran), which was built in 1453 (Science Magazine, vol. 315, n° 1106, 2007).

The discovery was published in the journal Science in February 2007 by Paul J. Steinhardt and Peter J. Lu.^{[3]} The research shows that an important breakthrough had occurred in Islamic mathematics and design by 1200. The core of the discovery claims that Muslim architects of central Asia made tilings that reflected mathematics, they were so sophisticated that they were only figured out in the last decades of our age.

The similarity between ancient Islamic designs and contemporary quasicrystalline geometry lies in the fact that both use symmetrical polygonal shapes to create patterns that can be extended indefinitely. Until now, the conventional view was that the complicated star-and-polygon patterns of Islamic design were conceived as zigzagging lines drafted using straightedge rulers and compasses.

With this discovery, one can conclude that the combinations of ornate stars and polygons that have adorned mosques and palaces since the 15th century were created using a set of just five template tiles, which could generate patterns with a kind of symmetry that eluded formal mathematical description for another 500 years.

The discovery suggests that the Islamic artisans who created these typical girih designs had an intuitive understanding of highly complex mathematical concepts. "We can't say for sure what it means," says Lu, a graduate student in physics at Harvard's Graduate School of Arts and Sciences. "It could be proof of a major role of mathematics in medieval Islamic art or it could have been just a way for artisans to construct their art more easily. It would be incredible if it were all coincidence, though. At the very least, it shows us a culture that we often don't credit enough was far more advanced than we ever thought before." ^{[4]}

**Figure 2A & 2B:** Girih tile reconstruction of the strapwork pattern on an interior archway in the Sultan's Lodge in the Green Mosque in Bursa, Turkey. Adapted from *Science Magazine*, vol. 315, n° 1106, 2007) and Hamish Johnston, "Islamic quasicrystals' predate Penrose tiles", Physicsworld.com, Feb 22, 2007

Girih designs feature arrays of tessellating polygons of multiple shapes, and are often overlaid with a zigzag network of lines. It had been assumed that straightedge rulers and compasses were used to create them — an exceptionally difficult process as each shape must be precisely drawn. From the 15th century, however, some of these designs are symmetrical in a way known today as "quasi-crystalline". Such forms have either fivefold or tenfold rotational symmetry — meaning they can be rotated to either five or ten positions that look the same — and their patterns can be infinitely extended without repetition. The principles behind quasi-crystalline symmetry were calculated by the Oxford mathematician Roger Penrose in the 1970s, but it is now clear that Islamic artists were creating them more than 500 years earlier.

Peter Lu, one of the authors of this discovery, began wondering whether there were quasi-crystalline forms in Islamic art after seeing decagonal artworks in Uzbekistan, while he was there for professional reasons. On returning to Harvard, he started searching the university's vast library of Islamic art for quasi-crystalline designs. He found several, as well as architectural scrolls that contained the outlines of five polygon templates — a ten-sided decagon, a hexagon, a pentagon, a rhombus and a bow-tie shape — that can be combined and overlaid to create such patterns.

In keeping with the Islamic tradition of not depicting images of people or animals, many religious buildings were decorated with geometric star-and-polygon patterns, often overlaid with a zigzag network of lines. Lu and Steinhardt show in their study published in the journal Science that by the 13th century Islamic artisans had begun producing patterns using a small set of decorated, polygonal tiles which they call "girih" tiles.

**Figure 3:** Periodicgirih pattern from the Seljuk Mama Hatun Mausoleum in Tercan, Turkey (~1200 CE), with girih-tile reconstruction overlaid at bottom.

Art historians have until now assumed that the intricate tile work had been created using straight edges and compasses, but the study suggests that Muslim artisans were using a basic toolkit of girih tiles made up of shapes such as the decagon, pentagon, diamond and hexagon.

"Straight edges and compasses work fine for the recurring symmetries of the simplest patterns we see, but it probably required far more powerful tools to fully explain the elaborate tiling with decagonal [10-sided] symmetry," P. J. Lu said, quoted by the journal The Independent on 25 February 2007. He adds that "individually placing and drafting hundreds of decagons with a straight edge would have been exceedingly cumbersome. It's more likely these artisans used particular tiles that we've found by decomposing the artwork".^{[5]}

The scientists found that by 1453, Islamic architects had created overlapping patterns with girih tiles at two sites to produce near-perfect quasi-crystalline patterns that did not repeat themselves. "The fact that we can explain so many sets of tiling, from such a wide range of architectural structures throughout the Islamic world with the same set of tiles, makes this an incredibly interesting universal picture," P. J. Lu said.^{[6]}

With this result, despite the debate that surrounded it in scholarly circles, we can see that very important discoveries in the Islamic scientific tradition are still to come, and that with the continuing research in different sources, including those of material remains of Muslim civilisation, the picture of our knowledge may be enriched and even changed dramatically.

As a background to the present day research on Islamic architecture as a conjunction of mathematics, arts and practical knowledge, we can mention the ongoing work on the *Muqarnas*. A *Muqarnas* is a type of corbel used as a decorative device in traditional Islamic architecture. The term is the Arabic word for stalactite vault, an architectural ornament developed around the middle of the 10th century in north eastern Iran and almost simultaneously, but apparently independently, in central North Africa. A *Muqarnas* is a three-dimensional architectural decoration composed of niche like elements arranged in tiers. The two-dimensional projection of a *Muqarnas* vault consists of a small variety of simple geometrical elements. Excellent examples can be found in the Alhambra in Granada, and in the mausoleum of Sultan Qaitbay in Cairo.^{[7]}

**Figures 4**: The iwans that surround the courtyard of Masjid-i-Jame in Isfahan consist of three Muqarnas, each giving different impressions:** (4A)** the southern Muqarnas, occupying a frontal position facing the courtyard, has an apex made up of 8 segments, suggesting a primitive strength; **(4B)**the eastern Muqarnas, with its 11-segmented apex, is complex and not aesthetically pleasing.

**(4C)** the western Muqarnas has a 5-segment apex, and displays an elegant form such as that seen in Hakuho sculptures.(Source).

The singular beauty of the *Muqarnas* has been reported by travellers throughout history. Their descriptions, however, are no more than brief introductions, and many details remain unclear. In his work on such architectural ornaments, Shiro Takahashi created exact drawings of many varieties of *Muqarnas*, classifying them into types in an attempt to clarify the formative styles of *Muqarnas*.^{[8]}

On the other hand, scholars from Heidelberg University in Germany, led by Yvonne Dold-Samplonius, designed The Muqaras Project aimed at the study of *Muqarnas* tradition in Islamic architecture. The project is entitled: Mathematical Concepts and Computer Graphics for the Reconstruction of Stalactite Vaults - *Muqarnas* - in Islamic Architecture.^{[9]}

The focus in this project is laid on two main points. One is that, from the late 11th century on, all Muslim lands adopted and developed the *Muqarnas*, which was widely used in constructions. The second and far more important point is that, from the moment of its first appearance, the *Muqarnas* acquired four characteristic attributes, whose evolution and characteristics form its history: it was three-dimensional and therefore provided volume wherever it was used, the nature and depth of the volume being left to the discretion of the maker; it could be used both as an architectonic form, because of its relationship to vaults, and as an applied ornament, because its depth could be controlled; it had no intrinsic limits, since not one of its elements is a finite unit of composition and there is no logical or mathematical limitation to the scale of any one composition; and it was a three-dimensional unit which could be resolved into a two-dimensional outline.

**Figures 5**: Muqarnas drawings in The Topkapi Scroll, the best preserved example of its kind, displaying geometry and ornament of Islamic architecture: **(5A)** Vault fragment with black-dotted polygonal grid lines, triangular one-twentieth repeat unit of a decagonal vault, and fan-shaped radial Muqarnas quarter vault; **(5B)** Fan-shaped radial muqarnas quarter vault, and shell-shaped radial muqarnas quarter vault; **(5C)** Fan-shaped radial muqarnas quarter vault, rhombodial one-eight repeat unit of an octagonal fan-shaped radial muqarnas quarter vault, fan-shaped radial muqarnas quarter vault, and rectangular repeat unit of stellate Muqarnas fragment. (Source).

"The *muqarnas* is a ceiling like a staircase with facets and a flat roof. Every facet intersects the adjacent one at either a right angle, or half a right angle, or their sum, or another combination of these two. The two facets can be thought of as standing on a plane parallel to the horizon. Above them is built either a flat surface, not parallel to the horizon, or two surfaces, either flat or curved, that constitute their roof. Both facets together with their roof are called one cell. Adjacent cells, which have their bases on one and the same surface parallel to the horizon, are called one tier." ^{[10]}The work of the group is based on the analysis made by Yvonne Dold-Samplonius of the mathematical work of the 15th century Timurid mathematician Ghiyath al-Din Mas'ud al-Kashi (ca. 1380-1429). Al-Kashi defines the *Muqarnas* as:

Building on the classification of different varieties of *Muqarnas* by al-Kashi, the project analyses the intermediate elements which connect the roofs of adjacent cells. In this sense, al-Kashi distinguishes four types of *Muqarnas*: The Simple *Muqarnas* and the Clay-plastered *Muqarnas*, both w