Muslim artisans (craftsmen, artists, masons, and architects) produced extraordinary achievements in the field of geometric art (note that `ajib, extraordinary in Arabic, is a semi-aesthetical concept that may refer to the joy caused by art works). First, let us see some preliminaries of this Islamic "Renaissance" in art and in science:

- In the Middle East there was a great tradition of geometric design since ancient times. The climate required such buildings that provide protection against the sun. There are large surfaces of walls and roofs, as well as floors inside, that encourage the decoration. The invention of cylindrically-shaped seals (ca. 3500 B.C.), which were rolled out in clay, led to the mass production of patterns with repetitive motifs.

- The same region, from Babylonia to Egypt had its own traditions in practical geometry and astronomy, which were significantly enriched by the theoretical works of Greek scholars. Some of their schools survived in the Hellenized Orient (Alexandria, Antioch, Beirut), while the Persian Sasanid dynasty's Academy at Jundishapur (Gondeshapur) from the 3rd century B.C. gave a model for institutions where Greek, Hebrew, Hindu, and Syriac materials were also studied and translated. The region had an important influence from Hindu mathematics as it is reflected, among others, in the later spread of the Hindu-based "Arabic numerals". Alexandria remained the main center of Greek geometry for a long period, which is marked by the works of Euclid (3rd c. B.C.), Apollonius (3rd c. B.C.), Hero (1st. c. A.D.), Ptolemy (2nd c. A.D.), and Pappus (4th c. A.D.).

In short, we may say that there was a special "geometric sense" in the air in both art and mathematics. The Christian ethics of the Byzantine Empire and the Nestorians, a sect named after patriarch Nestorius (5th c. A.D.) in the Middle East, emphasized the importance of physical work (ora et labora, pray and work), which gave more esteem to artisans than the classical antiquity did. On the other hand, the Byzantine Emperor Justinian I, who supported the legal studies (Codex Justinianus), but opposed the "pagan" teachers, "exported" many Greek scholars to this region when he closed the Academy at Athens in 529.

The Islamic chronological era was started in 622, the year of the Hegira or Hijra, Mohammed's flight from Mecca to Medina. The rise of Islam was followed by its rapid expansion from Arabia for a large territory from Persia to North Africa (7th c.) and the Iberian peninsula (early 8th c.). The caliphs, Mohammed's descendants who became religious and political leaders, established new cultural centers, often on the basis of old cities, where both art and mathematics flourished.

- The existing geometric traditions in art had a further support from the religious sphere. The Islamic iconoclasm (or aniconism), the prohibition of figural images, is based on similar ideas that we have seen at the other two Western monotheistic religions, Judaism and Christianity: the teachings on one true God and the rejection of worshipping idols (idolatry). Although there is a rich tradition of figural representation in the Islamic art of illustrated books and some other fields, the prohibition of images remained strict in the religious sphere. Since figural painting and sculpture do not play a role in the mosques and other religious buildings, the artistic beauty is manifested by the geometric creativity in architecture and decorative art. Patterns that can be endlessly continued on large surfaces require a very high level of geometrical accuracy. Otherwise the local "defects" in the fittings would disturb the global structure. In some sense, Muslim artisans anticipated the later results of geometric crystallography in a visual form. Their work was not limited to 2-dimensional crystallographic patterns, but they extended their scope into two directions: they considered some aperiodic or quasicrystallographic patterns with fivefold symmetry and introduced new forms of 3-dimensional decoration. The muqarnas cornices and vaults, which were developed from the 10th century, resemble the honeycomb-cells or the stalactites hanging down in caves. The preference for repetition can be also associated with various oral and musical traditions: chanting the same verses, often from the Koran, or performing a musical piece with a given motif (maqam) and rhythmic pattern (iqa'at). We recommend a simple experiment in order to "feel" the mutual relationships between music and art: let us listen to Islamic music and parallel study an ornamental pattern from the same region visually, "jumping" from unit to unit according to the rhythm of the music. While the local arrangements in repetitive patterns are limited by mathematical rules (there are exactly 10 crystallographic rosette- or point groups that may appear in the 17 wallpaper groups), the rhythmic patterns in music, which are repeated through the whole piece, are determined by traditions. As a survey shows, there are 92 rhythmic patterns of various length from a few beats up to 50 [9]. Many of these have names and even mnemonic verses that help the learning process. (For example the 28-beat Turkish pattern remel is described as: dum-tek-ka, dum-tek-ka, tek-ka-dum, ...) We may also recognize a hidden "visual-mathematical" link of Islamic music if we carefully observe the musicians' hand at the fingerboard of the ud (lute): they perform various "finger modes" that are arranged according to circles, stars, and polygons. The geometrical definition of modes is obviously a helpful tool for memorization. Last, but not least, Arabic calligraphy is also a form of rhythmic art and its ornamental version frequently appear in architecture.

- Parallel to these artistic traditions, there was a renewed interest in philosophy, science, and medicine. The religious sphere gave a strong support to mathematics and astronomy: the preparation of the theological calendar, the exact definition of praying times, and the determination of the directions to Mecca from all places were special tasks to these fields. The Abbasid caliphs in Baghdad headed a large empire from 750 until 945. (This year marked not only the end of their political power, but also the unity of the territories; on the other hand, they kept the religious authority in the fragmented Islamic lands until the Mongol invasion in 1258.) The caliphs and their regional leaders had a strong interest in practical questions that are useful for building a multi-ethnic Islamic Empire. The wealthy mercantile society needed more practical mathematics for trade and banking, the standardization of weights and measures, and navigation. The caliphs supported the translation of classical works into Arabic, a language with a special flexibility for coining new scientific terms. Moreover, they encouraged the participation of Christian, Jewish, and Persian scholars at the Bayt al-hikma (House of Wisdom, founded in 832) in Baghdad, which was modelled after the Persian Academy. The network of religious schools (madrasa) at mosques was extended with two additional types of institutions, the astronomical observatories and the libraries. The translated works gave a good basis for further achievements. Indeed, Muslim scholars not only transmitted and commented Greek mathematical works (many of them survive only in Arabic version), but also enriched this field. It is true that usually algebra is considered as their main field, which name is coming from the Arabic al-jabr (from the title of al-Khwarizmi's book, early 9th c.), but they also had an interest in geometry, especially in the application of algebraic methods for solving geometrical problems. Although this is an extreme example, the members of the group Ikhwan al-safa' (Sincere Brethren, 10th c.) went so far that they used geometry to define the perfect man in their encyclopedic work. The spread of papermaking gave a further support to the diffusion of scholarly works. The paper, which was invented for packing in China around the 2nd century B.C. and "rediscovered" for writing by an official in 105 A.D., reached Samarkand (now Uzbekistan) in 751 and then Baghdad in 793. It is true that the large-scale paper industry was started only much later, but the growing availability of cheap paper had an impact immediately.

Note that the fragmentation of the Abbasid Empire from 945 and even the Mongolian invasion and destruction of Baghdad in 1258 did not stop the patronage of science in some parts of the Islamic lands. Although a religious intolerance toward secular subjects and scientific ideas arised from the 11th century, especially in the religious schools, some local rulers had a special interest in scientific questions and maintained the traditions. For example, Yusuf ibn Hud, the King of Saragossa (Zaragoza, 11th c.), was himself a scholar and he compiled the Istikmal (Perfection, 11th c.), a comprehensive treatise on geometry. A large part of this work was reconstructed more recently on the basis of four anonymous manuscripts [10]. Ulugh Beg ("The Great Prince", 15th c.) from the Mongolian Timurid dynasty in Samarkand had a special interest in astronomy and invited many scientists to his madrasa, school of theology and science. The Nasrid dynasty (1230-1492) in the Kingdom of Granada, the only Muslim possession in the Iberian peninsula after the Christians recaptured the other parts in 1248, built a brilliant Islamic cultural center with fascinating achievements in architecture. This style had a significant influence from both Morocco and the nearby Christian territories. The Alhambra, the "Red" (al-hamra) Castle and Palace, became the symbol of Islamic ingenuity in art and geometry, and inspired generations of artists and scholars, from Owen Jones to M.C. Escher. The Iberian peninsula, where a large number of Christians, Muslims, as well as Jews lived together, became the center of translating scientific books from Arabic into Latin. These works, with a double function

- partly transmitting the classical traditions in an enriched form,

- partly presenting new empirical challenges to those,

inspired the dawn of modern science in the West.

As we discussed earlier, there was no strong interaction between artists and mathematicians in the Hellenistic Age. The surviving Greek mathematical works from that time almost never deal with the questions of arts and crafts, excluding, of course, musicology that was considered as a mathematical science. One may say that the Greek treatises on mechanical devices, especially from the age of Archimedes, represent an interest that are also useful for craftsmen, but again we see rather theoretical works than practical manuals. What was the situation of the art-geometry relationships during the rise and flourishing of the Islamic culture? Did the artisans use the achievements of Muslim mathematicians? Of course the main focus of artisans was the actual object of art, not the documentation of their approach. Thus, many people believe, we hardly can find written sources. These challenges encouraged some 20th century authors to generate their own mystical-mathematical theories on Islamic art, while others, including the Dutch graphic artist M. C. Escher, were inspired to develop their form-language with a starting point in Islamic decorations. On the other hand, there are authentic sources on the geometry of Islamic art that need special attention. First let us see some documents from both sides, then the "missing links" that exist, but often ignored:

- Mathematics: Muslim scholars had an interest not only in the Greek tradition of Euclidean mathematics, or, using the modern terminology, pure mathematics, but also in a broader set of questions. Thus, the pre-Euclidean philosophical-scientific works encouraged some Muslim scholars to develop an encyclopedic interest. For example, Thabit ibn Qurra (836-901), a money-changer-boy in Harran (now in Turkey) turned polymath in Baghdad, whose native language was Syriac and also knew Greek and Arabic, translated or commented many Greek works (Plato, Aristotle, Euclid, Archimedes, Apollonius, Ptolemy) and wrote important books on mathematics, physics, astronomy, medicine, and philosophy. His further works deal with logic, ethics, politics, Syrian grammar, and the Sabian star-worshippers' religion, which was practiced by his ancestors. From our point of view it is significant that Thabit ibn Qurra dealt with many geometrical questions, including the measurement of plane and solid figures, geometrical constructions, conic sections, parabolic cupolas, and he is one of the pioneers of non-Euclidean geometry. Al-Farabi (ca. 870-950), another important scholar of this period, went for an eight-year period to Constantinople to study Plato's and Aristotle's works. Later he presented extensive commentaries to these and wrote philosophical, mathematical, and musicological works. Another important feature of the Greco-Muslim traditions is a focus on those few ancient treatises where applied mathematics is involved (Archimedes, Hero of Alexandria, Pappus). The Arabic translations and commentaries are especially rich in this field. Both the encyclopedic interest and the focus on applied questions contributed to a pragmatic interest in mathematics, which occasionally included problems related to arts. For example, there are various theoretical works that discuss geometrical figures and arrangements that are similar to those ones that are used by artisans. The influence of some of these works is going beyond the Islamic culture: the treatise by Abu Kamil (ca. 850-ca. 930) Kitab [...] al-mukhammas wa'l-mu`ashshar (Book on the Pentagon and the Decagon) survives not only in Arabic (Istanbul), but also in Hebrew and Latin translations (Munich and Paris), and inspired, among others, the book Practica geometriae by Fibonacci (Leonardo Pisano, 13th century). Interestingly, the extreme and mean ratio (golden section), which features on the intersecting diagonals of a regular pentagon everywhere, is mentioned just very briefly by Abu Kamil. I should admit his book focuses on the algebraic calculation of the geometric data: it is not useful for visual artists without appropriate commentaries. On the other hand, there are a few surviving mathematical works that give concrete advices to artists. Perhaps the most striking example is a textbook by Abu'l-Wafa' al-Buzjani (940-997/998) Kitab fima yahtaju ilayhi al-sani` min a`mal al-handasa (Book on What is Necessary from Geometric Constructions for the Artisans; various manuscripts versions in Arabic in Cairo, Istanbul, Milan, Paris, and Upsala, and in Persian in Mashhad, Paris, and Teheran). The author not only suggests his own methods, but also corrects the mistakes of artisans. In the case of the later periods, we may refer to an architectural treatise on "Measuring structures and buildings" in the framework of Al-Kashi's book Miftah al-hisab (The Key of Arithmetic, written in 1427; many manuscript versions survive that are located in libraries from Russia to India). It is not surprizing that later some of the geometrical methods of Muslim scholars were circulated among Renaissance artists. What about the contemporary Muslim artisans?

- Arts and crafts: Luckily, there are some surviving geometric works that were prepared by artisans. Thus, an anonymous Persian manuscript Fi tadakhul al-ashkal al-mutashabiha aw mutawafiqa (On Interlocking Similar and Congruent Figures, its original version was written sometimes in the 11th-13th cc.; Paris) presents various geometric constructions used by artisans. Its text and figures were published in a Russian book (Vil'danova and Bulatov) and attracted further studies more recently (Chorbachi, Özdural). The careful analysis of the work hints that the anonymous author was, with great probability, not a geometer, but an artisan. There are some interesting scrolls with artistic-geometrical patterns from later periods. These works are more practical: they present geometrical constructions in a visual form with no text. The fragmentary Tashkent Scrolls (16th and 17th cc.), now at the Institute of Oriental Studies in Tashkent (Uzbekistan), present the works of a master mason of the post-Timurid period. The scrolls were discovered in Bukhara in the 1930s and discussed in Russian papers (Gaganov, Baklanov). An exciting example of such scrolls was discovered more recently in the Topkapi Palace Museum Library (Istanbul) and published in a very attractive form with a brilliant survey on geometry in Islamic art in English (Necipoglu). This Topkapi Scroll (late 15th or 16th c.) is almost 30-meter long and includes 114 drawings of Timurid ornaments. We should not be surprized that both the Tashkent and Topkapi Scrolls are from the same region: this was the first place of the Islamic lands where the Chinese craft of papermaking spread, as we remarked earlier. Many modern examples show that the usage of pattern-scrolls survives among Muslim artisans until our time.

- The "missing link" between mathematics and art: Although the discussed works immediately hint a relationship between some mathematicians and artisans, we have a much stronger evidence for this. There are concrete documents on meetings and discussions between the two groups, which were often requested or even presided by the patrons or local rulers. Obviously not all mathematicians and artisans were involved in such meetings. The results, however, could have spread in wide circles in the form of such manuscripts and scrolls that we just discussed. Let us see some of these records. Abu'l-Wafa in his book on geometrical constructions remarks that he was "present at some meetings held among a group of artisans and geometers". The mathematician, philosopher, and poet Omar Khayyam (Umar al-Khayyami, 1048-1131?) is less clear, but he refers to a meeting where a ruler was present and some "simple ideas" were discussed. A recent study established a very convincing link between this mathematical treatise by Khayyam and the discussed Persian manuscript on interlocking figures (Özdural). The mathematician al-Kashi, in a letter to his father, described his debate with a master mason in the presence of "nearly five hundred people". We also learn from this letter that the master coppersmith was ordered to go to his house and to finish there an armillary sphere (astronomical instrument representing the celestial sphere). Such discussions were mutually useful to both sides: the artists learnt geometrical methods and the mathematicians dealt with new problems or applied their skills in new fields. Perhaps we should re-examine some old documents in order to find further clues. The topics of the informal meetings may provide a new insight into the "hidden secrets" of Islamic geometric art.