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## Über dieses Buch

This book deals with the genre of geometric design in the Islamic sphere. Part I presents an overview of Islamic history, its extraordinary spread from the Atlantic to the borders of China in its first century, its adoption of the cultural outlook of the older civilisations that it conquered (in the Middle East, Persia and Central Asia), including their philosophical and scientific achievements - from which it came to express its own unique and highly distinctive artistic and architectural forms. Part II represents the mathematical analysis of Islamic geometric designs.
The presentation offers unlimited precision that allows software to reconstruct the design vision of the original artist. This book will be of interest to Islamic academics, mathematicians as well as to artists & art students.

## Inhaltsverzeichnis

### Chapter 1. A History

The bare historical facts concerning the origin of the religion of Islam are these—The Qur’ān, the Holy Book of Islam, began to be revealed to the Prophet Muhammad when he was forty years old, circa 610. These revelations continued until his death in 632, by which time their strict monotheistic message had become so widely accepted that the new religion had largely supplanted the ancient paganism of the Arabian Peninsula. The rapid, widespread acceptance of Islam also introduced a quite new sense of unity among the various Arabian tribes. For a list of key dates, see Appendix B.

### Chapter 2. The Scientific Contribution

From the time that Muslim forces overran the Byzantine provinces of Syria and Egypt in the mid 7th century it was inevitable that Islam would encounter the traditions of late-Classical science and philosophy that were still being taught in the Greek schools there. However, this process of cultural assimilation was slow to develop. The initial intellectual exchanges between the Muslims conquerors and the Christian majority were entirely dominated by religious disputation. Indeed, during the first Islamic dynasty, the Damascus-based Umayyads, there was little interest at all in the secular aspects of Classical culture.

### Chapter 3. The Religious Dimension

From its very inception the Islamic mission was dedicated to a revival and purification of the ‘religion of Abraham’, that is to say, the Judeo-Christian tradition. It adopted the strict, monotheistic credo of the Old Testament, and was hostile to the paganism that was still flourishing in Arabia. Muhammad’s first act, following his acceptance as a religious leader by the townspeople of Mecca, was to destroy the 360 idols around the Ka’ba. Up to this time Mecca had been an important centre of pagan pilgrimage.

### Chapter 4. The Evolution of Style

Islamic art has a recognisable aesthetic signature that somehow manages to express itself across an entire range of productions. The ‘language’ of this art, once established, was readily assimilated by each of the different nations and ethnicities that were brought within the Islamic sphere. Assimilated and built upon, because every region, at every period, produced its own versions of this super-national style.

### Chapter 5. Materials and Media

As indicated in Chap. 4, the decorative arts in Islam are marked by a remarkable degree of stylistic consistency which has been applied to a broad range of materials, each of which had a craft-base of its own, the history of which often traces back to pre-Islamic times. The stylistic coherence within the Islamic world and the many variations of its basic themes across time, influenced as they were by local artistic traditions, are all part of the fascination of this art. In Muslim architecture virtually any surface may be regarded as worthy of receiving elaborate decoration. This is particularly apparent in religious architecture (Fig. 5.1), but this principle extends to woodwork, ceramics, textiles, metalwork, books and many other art forms.

### Chapter 6. Countries and Regions

Egypt was brought into the Islamic sphere in the first wave of conquests, in 639. With the Tulunid assumption of power (868–905) it became an independent centre. Then, under the Fatimid dynasty (909–1171) it assumed its enduring role as the cultural focus of western Islam. The architecture of the Mamluk period (13th–16th) saw a continuity of earlier Egyptian traditions, but it also incorporated aspects of Iraqi and Syrian styles. Most of the buildings of these periods are of stone; in fact the general feeling conveyed by medieval Cairene architecture is that of a sombre stolidity.

### Chapter 7. Introduction

The script is spiritual geometry, although made perceptible by a physical instrument. Euclid, cited by Abu Baker as-Suli (Moustafa and Sperl 2014, Vol. 2, p. 287)
The quote above comes from the monumental work by Moustafa and Sperl on Arabic penmanship. Their result is a mathematical analysis of the Proportional Script—an elegant form of calligraphy. The analysis undertaken here is much simpler than the curves in calligraphy since we restrict our attention to geometric patterns using straight lines. The importance of geometry and proportion in art was clearly understood in the early years of the Islamic world.

### Chapter 8. A Worked Example

In 1936 Maurits Escher visited the Alhambra with his wife Jetta and made a number of sketches (Schattschneider, M C Escher — Visions of Symmetry, 1990, p. 17) . One hundred years earlier, Owen Jones published his first work on the Alhambra (Jones and Goury, Plans, Elevations, Sections, and Details of the Alhambra, 1837). Both subsequently produced a drawing of the same pattern from that building (Locher 1992, pp. 41, 53; Victoria and Albert Museum collections website (719669) . Unlike Jones, we have the advantage of working from modern photographs, at least in this case Fig. 8.1.

### Chapter 9. An Octagonal Set

In this chapter, we are looking at not just a single tiling pattern, but a set of over 140 patterns. The set is based upon the khatem, the 8-pointed star so is called octagonal. Firstly, consider the three photos, in Figs. 9.1, 9.2 and 9.3. These three patterns all come from Spain or Morocco; indeed, the whole set of patterns might well also be called of Moroccan style, see Andrè Paccard (Traditional Islamic Craft in Moroccan Architecture, 1980) Paccard (1980).

### Chapter 10. Octagonal Tiles with a Large Star

In this chapter, we consider a similar set of octagonal tiles to those of in the last chapter, but used to form a pattern with a large a central star. Three cases are analysed according to the number of points to the large star.

### Chapter 11. Lines and Edges

The large set of patterns in Chap. 9 had the property edge-to-edge. This is the usual case, but there are exceptions (one example was shown in Fig. 5.​3). Further examples arise when a tile is not even a topological disc, but has a ‘hole’ as shown by the red tile in Fig. 11.1c. We call this a chelate. The ‘hole’ is actually a small polygon (wholly within another polygon) which always seems to be a kite. Three examples are shown in Fig. 11.1. (Note that Fig. 11.1a has a similar arrangement as Fig. 14.​5a.) All the examples here come from Cairo, and no example is known from the Maghreb. In spite of this irregularity, all the three figures could be coloured with just two colours.

### Chapter 12. Decagonal Patterns

Another large group of patterns similar to those introduced in Chap. 9 (octagonal patterns). A key property here is the presence of 10-pointed stars with a vertex angle of 72$$^\circ$$ or 108$$^\circ$$ as opposed to 8-pointed star with octogonal patterns. We saw that octogonal patterns came mainly from Spain and Morocco. Here, the patterns come mainly from Iran, Turkey or Egypt. As for the octagonal set, a few simple designs appear throughout the Islamic world.

### Chapter 13. What Is Correct?

The previous chapter has introduced patterns which are relatively straightforward so their representation in mathematical terms does not present any difficulties. Here we consider examples in which ‘Islamic style’ comes into play.

### Chapter 14. 6-Fold Delights

In this chapter we look at patterns whose symmetry is either or , that is, there is a rotation of order 6. Hence such patterns were not included in Chaps. 9, 10 or 12. We first consider the impact of earlier mathematical designs on those of Islam.

### Chapter 15. Two-Level Patterns

Two-level patterns are simply defined as patterns within patterns (Wichmann, The world of patterns, World Scientific, Singapore, 2001). In a detailed study, Peter Cromwell lists forty examples (Cromwell, Nexus Netw J, 18(1): 754, 2016); but in first part of this chapter we consider just the simpler ones. In the second part, we look at more complex examples and contrast the simple approach with the methods used by the designers in Iran.

### Chapter 16. Two Mamluk Masterpieces

The Mamluk dynasty which ruled Egypt and survived the Mongol depredations, produced some of the most impressive Islamic patterns particularly in wood on doors and minbars. Although our first example is in metal, it is an excellent example for which good photographs are available. It is Bourgoin’s Plate 137 and is from the Madrasah and Khanqa al–Azahr Barquq, Cairo, Egypt. This was drawn by Bourgoin, but we redraw it using a detailed analysis. A photograph of the complete door is shown in Fig. 16.1.

### Chapter 17. Conclusions

The aim of Part II of this book is to show how to construct Islamic geometric patterns in the optimal way using mathematical techniques. To survey the methods used, we also need to consider approximate methods (which may have been used throughout the past).

### Backmatter

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