Theoretical Geometry and Artisanal Practice in the Islamic World

David Wade

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 the Muqaddimah

In the Introduction to his encyclopaedic ‘History of the World’, the philosopher and social historian Ibn Khuldun1 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 little, actually, for their interconnection.

The continuity of the Greek tradition of mathematics in the Islamic sphere, particularly of its Geometry, is certainly impressive. Geometry was a favoured subject from the beginning of the translation movement, largely because it had many obvious practical applications2. Euclid’s Elements3, 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 bench-mark for rigor in Islamic mathematical and scientific investigations of all kinds4.

Most importantly for this account, 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 an expression of this shared enthusiasm, and an indication of the extent to which Classical knowledge of mathematics5 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 took place6. 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 Terry Allen has pointed out a propos of this subject We cannot simply lump together every manifestation of interest in geometry7. 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 derive 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.

In fact there are almost no references to decorative ornament in any surviving mathematical text from Islam’s medieval period4. 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 exercises, 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, were occasionally acceptable to designers, but would have been anathema to any academic geometrician.

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 tendency towards geometricism is not expressed in a 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 genre of geometric pattern itself that is constant. That is to say, that although there is tendency towards greater geometric complexity through time in the various Islamic regions this is expressed in a whole plethora of different styles.

There is little doubt that artist/craftsmen travelled (voluntarily or otherwise) across widely separated regions, and that at different times and places pattern-books and working drawings would have been available8. 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 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 constant was an understanding 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 that go a long way to make Islamic art, in all its diversity, so recognizable, coherent and distinctive.

  1. Ibn Khuldun 1377 CE/779 AH, was born in Tunis into an aristocratic family that had enjoyed a long history in Moorish Spain. He wrote the Muqaddimah (Prolegomena) to his Encyclopaedic History of the World in 1377 – a work that led him to be described as the first Social Historian.

  2. Such as land-surveying, town-planning, architecture, hydraulic engineering, ballistics etc. Interestingly, the Arabic term handasha, in its original usage, simply meant ‘geometry’, but it gradually broadened out to cover ‘engineering’ in a general sense, which is its meaning in modern Arabic.

  3. Euclid founded the famous school of mathematics at Alexandria in the early 3rd century BC. Little is known of his life, but according to Proclus he composed the ‘Elements’ by ‘collecting many of the theorems of Eudoxus, perfecting those of Plato’s pupil Theaetetus, and brought irrefutable demonstrations to much in geometry that had previously been loosely accepted by his predecessors’. The Elements were first brought into Arabic during the time of al-Mansur in the late 8th century, and received rather better translations in later periods. Interestingly, Procus’s own commentary on the Elements was translated into Arabic in the 10th century CE; in this he declares that geometric forms occupy an intermediate position between… higher realities and the material objects of the world of senses.

  4. Geometry enlightens the intellect and sets one mind right. It is hardly possible for errors to enter into geometric reasoning because it is well arranged and orderly. Thus the mind that constantly applies itself to geometry is not likely to fall into error.

    Ibn Khuldun, Muqaddimah
  5. Which although dominated by Geometry, included Arithmetic, Astronomy and Harmonics. The last of these, the study of Harmonics, derives from the Pythagoreans and was concerned with the investigation of the physics of sound, as well as the mathematical relations they found in both Music and Astronomy.

  6. Jan P. Hogendijk points out that although there are hundreds of medieval texts from the centres of Islamic civilisation on geometrical subjects, perhaps surprisingly, there is not the slightest mention of decorative ornament in any of them - Mathematics and Geometric Ornamentation in the medieval Islamic World; Mathematics Department; Utrecht University; 2010).

  7. In his ‘Islamic Art and the argument from Academic Geometry’, Solipsist Press, California, 2004. But see Postscript A.

  8. Qu’ran illumination may have been another means by which decorative styles were disseminated, and it is possible that many decorative tropes originated in this genre.