Theoretical Geometry and Artisanal Practice in the Islamic World May 2022 ‘In view of its origin, carpentry needs a good deal of geometry of all kinds. It requires either a general or specialized knowledge of proportion and measurement in order to bring forms from potentiality into actuality in the proper manner, and for the knowledge of proportions one must have recourse to the geometrician. Therefore the leading Greek geometricians were all master carpenters. Euclid, the author of the Book of Principles, was a carpenter, and known as such. The same was the case with Apollonius, the author of the book on Conic Sections, and Menelaus and others.’ - Ibn Khuldun from his Muqaddimah In the Introduction to his Classic, encyclopedic History of the World (the Muqaddimah, finished 1377 CE; 780 A.H.), the Islamic philosopher and social historian Ibn Khuldun makes various observations about geometry and the crafts, from which the above quotation is drawn. His comments are interesting, partly because of their apparent familiarity with various important figures of Classical mathematics, but also because they seem to reflect a rapport between the geometry practiced by architects and craftsmen and the tradition of academic geometry that Islam had inherited from the Greeks. This apparent common interest in matters Geometric has led many in more recent times to assume a rather closer connection than the evidence seems to support. There is certainly plenty of evidence for both Geometries in Islam but, when it comes down to it, little, actually, for their interconnection. The continuity of the Greek tradition of mathematics in the Islamic sphere, particularly of its Geometry, is however certainly impressive. Geometry was a favored subject from the beginning of the translation movement, largely because it had many obvious practical applications. Euclid’s Elements, which provided the foundations of plane geometry, was itself translated at an early stage and came to be greatly admired for its orderly, logical presentation of the subject. In fact, this work came to be valued not only as the foundation of theoretical Geometry but also as an exemplar of logical methodology. With its steady progression through a series of axioms, it set a benchmark for rigor in Islamic mathematical and scientific investigations of all kinds. Most importantly for the subject under discussion here, the new spirit of enquiry among the early Islamic translators and scholars, and their enthusiasm for Greek learning, seems to have exerted a broader influence on Islamic cultural attitudes. The well-known Islamic penchant for complex geometrical patterns, which can lay claims to being the most “scientific” of all art forms, can be seen as an expression of this shared enthusiasm and an indication of the extent to which Classical knowledge of mathematics had been absorbed in the Islamic world. This has led some commentators to assume a degree of collaboration between Islamic mathematicians and artisans in decorative panels that demonstrate a high degree of geometric complexity. In reality, however, there is little evidence that cross-disciplinary collaborations of this kind actually took place. There was certainly a familiarity with Geometry among mathematicians and among skilled artisans of various kinds. But their interests, skills, and aims were different (as was their standing in society). As the art historian Terry Allen has pointed out a propos of this subject, we cannot simply lump together every manifestation of interest in geometry (Allen, 2004). The theoretical geometry that is a branch of mathematics and the “hands-on” geometry used by artisans and architects were equally specialized, but quite distinct disciplines. The notion that complex geometries in decorative patterns derived from advanced understanding of geometrical theory is, for the most part, misconstrued. Moreover, although the “geometric mode” certainly became a ubiquitous form of decoration in Islamic art, this was not a manifestation of a more general interest in geometry in the Islamic world. There are no accounts of what the ordinary Muslim citizen thought of geometric ornament – (we can imagine that they would have been as impressed and intrigued as we are today) – but it doesn’t follow that they would have had any better idea than a modern non-Muslim with regard to their underlying principles of construction. These are more likely to have belonged to realm of specialized, insider knowledge. In fact, there are almost no references to decorative ornament in any surviving mathematical text from Islam’s medieval period. The geometric forms in its art and architecture art were created by artist/craftsmen, and in actuality, even the most complex decorative panels are relatively trivial from a purely mathematical point of view. It would of course have been perfectly natural for mathematicians of the time to take an interest in geometric designs – they still do today. And it is true that there are a few tantalizing glimpses of exchanges between what one might describe as “academic” geometers and artisans that used geometry, but again, there is no evidence of collaboration on actual projects. As far as can be judged (because the evidence of such encounters tends to be from the academic side), there was never a complete meeting of minds between the parties. The “advice to artisans” literature, such as it is, tends to consist of finger-wagging attempts to get the artist/craftsmen to construct their designs “properly,” i.e., according to purely Euclidean geometric principles. As indicated earlier, the inclusion of “approximations” (i.e., the inclusion of “not-quite” regular figures into a design) would have been anathema to any academic geometrician but were occasionally acceptable to designers. The widespread use of complex patterning in the Islamic world, over many centuries, clearly indicates that this mode of expression satisfied something integral to the Islamic ethos – not least because the use of the geometric mode is not expressed in any single unbroken tradition. Technically speaking, there are any number of ways to create high levels of complexity in Islamic geometric patterns, and in the end, it is only the broad geometric theme itself that is constant. That is to say that although there is tendency toward greater geometric complexity through time in the various Islamic regions, this is expressed in a wide range of different styles and approaches. It is however intriguing that these humble Muslim craftsmen, over time, managed to uncover and exploit for decorative purposes many of the possibilities allowed of formal symmetrical arrangement. There is little doubt that artist/craftsmen traveled (voluntarily or otherwise) across widely separated regions and that at different times and places pattern books and working drawings would have facilitated transmission. Given Islam’s fractured history, it is likely that traditions were broken, sometimes to be resumed in subtly different ways and that patterns would have been adapted from one medium to be used in another. Taken as a whole, this long tradition indicates that over time, experimentation vied with adherence to established forms. There is a thread of continuity in the use of decorative ornament in Islamic, but the real constant was a thorough appreciation of the underlying geometry of plane division and a mastery of what can only be described as artistic geometry. This involved a keen awareness of an unspoken set of rules which involved a strong sense of symmetry and a preference for the careful, balanced distribution of elements within framing panels. In the end, it is these criteria, rather than any mathematical formulae, that go a long way to make Islamic art, in all its diversity, so recognizable, coherent, and distinctive.