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Mathematics and geometric ornamentation in themedieval Islamic world.

Jan P. Hogendijk

Abstract. We discuss medieval Arabic and Persian sources on the design and construc-tion of geometric ornaments in Islamic civilization.

2010 Mathematics Subject Classification. Primary 01A30; Secondary 51-03.

Keywords. Islamic mathematics, tilings, pentagon, heptagon

1. Introduction

Many medieval Islamic mosques and palaces are adorned with highly intricategeometric ornaments. These decorations have inspired modern artists and art his-torians, and they have been discussed in connection with modern mathematicalconcepts such as crystallographic groups and aperiodic tilings. The Islamic orna-mental patterns can certainly be used to illustrate such modern notions.

Medieval Islamic civilization has also left us an impressive written heritage inmathematics. Hundreds of Arabic and Persian mathematical manuscripts havebeen preserved in libraries in different parts of the world. These manuscriptsinclude Arabic translations of the main works of ancient Greek geometry suchas the Elements of Euclid (ca. 300 BC) and the Conics of Apollonius (ca. 200BC), as well as texts by medieval authors between the eighth and seventeenthcenturies, with different religious and national backgrounds. In what follows I willrefer to Islamic authors and Islamic texts, but the word Islamic will have acultural meaning only. Most Islamic mathematical texts were not related to thereligion of Islam, and although the majority of Islamic authors were Muslims,substantial contributions were made by Christians, Jews and authors with otherreligious backgrounds who lived in the Islamic world.

Many Islamic texts on geometry are related to spherical trigonometry and as-tronomy, and most Islamic scholars who studied the Elements of Euclid were study-ing in order to become astronomers and possibly astrologers. Yet there are alsoIslamic works on geometrical subjects unrelated to astronomy. In almost all me-dieval Islamic geometrical texts that have been published thus far, one does notfind the slightest reference to decorative ornaments. This may be surprising be-cause the authors of these texts lived in the main Islamic centers of civilizationand may have seen geometric ornaments frequently.

2 Jan P. Hogendijk

In this paper we will see that the Islamic geometric ornaments were in generaldesigned and constructed not by mathematician-astronomers but by craftsmen(Arabic: s.unnac.) Our main question will be as follows: what kind of mathematicalmethods, if any, did these craftsmen use, and to what extent did they interactwith mathematician-astronomers who were trained in the methodology of Greekgeometry? We will discuss these questions on the basis of the extant manuscriptmaterial, which is very fragmentary. In sections 2-5 we will discuss four relevantsources, and we will draw our conclusions in the final section 6. For reasons ofspace, we will restrict ourself to plane ornaments and pay no attenion to decorativepatterns on cupolas and to muqarnas (stalactite vaults).

2. Abul-Wafa

We first turn to the book on what the craftsman needs of the science of geom-etry1 by the tenth-century mathematician-astronomer Abul-Wafa al-Buzjan.This work contains some information on the working methods of the craftsmen,which will be useful for us in Section 4 below. Abul-Wafa worked in Baghdad,one of the intellectual centers of the Islamic world. He dedicated his booklet toBaha al-Dawla, who ruled Iraq from 988 to 1012, and who apparently employedmathematicians as well as craftsmen at his court. Almost all of the booklet consistsof ruler and compass-constructions belonging to plane Euclidean geometry. Theyare explained in the usual way, that is, by means of geometric figures in which thepoints are labeled by letters, but without proofs. Abul-Wafa says that he doesnot provide arguments and proofs in order to make the subject more suitable andeasier to understand for craftsmen [1, 23].

The booklet consists of eleven chapters on (1) the ruler, the compass and thegonia (i.e., a set square); (2) fundamental Euclidean ruler-and-compass construc-tions, and in addition a construction of two mean proportionals, a trisection ofthe angle, and a pointwise construction of a (parabolic) burning mirror; (3) con-structions of regular polygons, including some constructions by a single compass-opening; (4) inscribing figures in a circle; (5) circumscribing a circle around figures;(6) inscribing a circle in figures; (7) inscribing figures in one another; (8) divisionof triangles; (9) division of quadrilaterals; (10) combining squares to one square,and dividing a square into squares, all by cut-and-paste constructions; and (11)the five regular and a few semi-regular polyhedra. Abul-Wafa does not mentiongeometric ornaments.

Most of the information on the working methods of craftsmen is containedin Chapter 10. In that chapter, Abul-Wafa reports about a meeting betweengeometers and craftsmen in which they discussed the problem of constructing asquare equal to three times a given square (for an English translation see [16, 173-183]). The craftsmen seem to have had three equal squares in front of them andwanted to cut them and rearrange the pieces to one big square. The geometers

1Incomplete French and German versions are to be found in [21] and [20]. The completeversion in Arabic is in [1] and in facsimile in [18].

Islamic geometric ornaments 3

easily constructed the side of the required big square by means of Euclids Elements,but were unable to suggest a cut-and-paste construction of the big square fromthe three small squares. Abul-Wafa presents several cut-and-paste methods thatwere used by the craftsmen, but he regards these methods with some disdainbecause they are approximations. Abul-Wafa was trained in Euclids Elementsand therefore he believed that geometry is about infinitely thin lines and pointswithout magnitude, which exist in the imagination only. He complains that thecraftsmen always want to find an easy construction which seems to be correct to theeyesight, but that they do not care about a proof by what Abul-Wafa calls theimagination. He declares that the constructions that can be rigorously provenshould be distinguished from approximate constructions, and that the craftsmenshoud be provided with correct constructions so that they do not need to useapproximations anymore.2 We do not know how the booklet was received but the16th-century Persian manuscript which we will study in Section 4 contains a richvariety of approximate constructions.

3. The Topkap Scroll

The craftsmen themselves seem to have left us with very few documents about theiractivities in the field of geometric ornamentation. The most important publishedexample is the so-called Topkap Scroll, which is now preserved in the TopkapPalace in Istanbul, and which has appeared in the magnificent volume [14]. This29.5 m long and 33 cm wide paper scroll is undated and may have been compiledin Northwestern Iran in the 16th century, but the dating is uncertain. The scrollconsists of diagrams without explanatory text. Many of these diagrams are relatedto calligraphy or muqarnas and therefore do not concern us here. Some of thediagrams concern plane tilings. I have selected one non-trivial example in order todraw attention to the characteristic (and frustrating) problems of interpretation.The drawing on the scroll [14, p. 300] consists of red, black and orange lines, whichare indicated by bold, thin and broken lines respectively in Figure 1 (for a photoof the manuscript drawing see also [17]). The broken lines in Figure 1 define a setof five tiles, called gireh-tiles in the modern research literature, from the Persianword greh, which means knot. The thin lines form a decorative pattern whichcan be obtained by bisecting the sides of the gireh-tiles, and by drawing suitablestraight line segments through the bisecting points. It is likely that the pattern wasdesigned this way, but one cannot be sure because the scroll does not contain anyexplanatory text. The gireh tiles of Figure 1 have drawn recent attention becausethey can be used to define aperiodic tilings. In the absence of textual evidence, itis impossible to say whether the craftsmen had an intuitive notion of aperiodicity(for a good discussion see [8]).

2Note that Abul-Wafa presents an approximate construction of the regular heptagon by rulerand compass. Just like many of his Islamic contemporaries, he probably believed that the regularheptagon cannot be constructed by ruler and compass.

4 Jan P. Hogendijk

Figure 1. Drawing by Dr Steven Wepster.

4. An anonymous Persian treatise

One would like to have a medieval Islamic treatise, written by a craftsman, in whichthe design and construction of ornaments is clearly explained. Such a treatise hasnot been found, and thus far, only a single manuscript has been discovered in whichdiagrams on geometrical ornaments are accompanied by textual explanations. Inthis section we will discuss what this manuscript can tell us about the main questionin the beginning of the paper. The manuscript is a rather chaotic collection of 40pages of Persian text and drawings (for some photos see [14, 146-150]). The textconsists of small paragraphs which are written close to the drawing to which theyrefer, and although the texts and drawings appear in a disorganized order and maynot be the work of a single author, I will consider the collection as one treatise.3

It may have been compiled in the sixteenth century, although some of the materialmust be older as we shall see.

The treatise belongs to a manuscript volume of approximately 400 pages [5, 55-56]. Some of the other texts in the manuscript volume are standard mathematicalworks such as an Arabic translation of a small part of Eucl