Neugebauer lecture: Mathematicians and Decorative Geometric Tilings in the Medieval Islamic World
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1 1 Neugebauer lecture: Mathematicians and Decorative Geometric Tilings in the Medieval Islamic World Jan P. Hogendijk Mathematics Department, Utrecht University Krakow, Sixth European Congress of Mathematicians, 2 July 2012
2 2 Otto Neugebauer ( ): personal memories ( ) Wilbour Hall, Brown University
3 3 Neugebauer s History of Mathematics Department at Brown Abe Sachs ( ), assyriologist David Pingree ( ), specialist in Sanskrit Gerald Toomer (born 1934), classicist
4 4 The Neugebauer School in History of Mathematics until ca Use manuscript sources in many languages; Akkadian (cuneiform), Arabic, Persian, Hebrew, Sanskrit, Ethiopean, etc. (Greek and Latin by default) 2. Study mathematics in contexts: astronomy, astrology, etc. 3. Technical analysis of sources leads to historical conclusions on transmission of mathematical knowledge between cultures.
5 5 Mathematical art in medieval Islamic civilization (ca ): 1. Plane tilings Darb-e Imam, Isfahan, Iran (15th c.)
6 6 2. Tiled cupolas Shrine of Shah Nematollah Vali, Mahan, Iran (15th c.)
7 7 3: Three-dimensional art: Muqarnas Shiraz, Iran
8 8 Questions What mathematical methods, if any, were used in the design and construction of the plane tilings? What was the interaction, if any, between the designers and the Islamic mathematician-astronomers?
9 9 Hundreds of Arabic and Persian manuscripts on mathematics have been preserved Most of the texts were written by mathematicians-astronomers, trained in Greek (deductive) mathematics. They contain almost no information on tilings.
10 10 Who designed and constructed tilings? Not mathematicians-astronomers but craftsmen (Arabic: ṣunnā c ). Sources are extremely scarce and fragmentary. We will discuss: 1. Abu l-wafā s booklet (ca. 990) on the geometry necessary for craftsmen (he was a mathematician-astronomer) 2. The Topkapi Scroll, a document by or for craftsmen 3. Anonymous medieval Persian manuscript, by or for craftsmen 4. A work by a mathematician (Omar Khayyām, ) on a problem inspired by craftsmen.
11 11 1. Abu l-wafā (10th-c.): booklet on the geometry necessary for craftsmen He says: Craftsmen (ṣunnā c ) use geometrical constructions but no proofs, and do not distinguish between exact and approximate constructions. They used ruler, compass (sometimes with a fixed opening), and a set-square ( gonia ).
12 2. The Topkapi Scroll: a document by/for craftsmen 12
13 Topkapi Scroll: a diagram 13
14 The dotted figures ( gireh tiles ) were probably used in the design process 14
15 15 3. Anonymous 16th c (?) Persian manuscript, Paris, Bibliothèque Nationale, Or pages, to be published in 2013.
16 16 Persian manuscript: 3.1. example of approximate construction with fixed compass-opening Construction of side of pentagon (DH) inscribed in the circle with center B and circumscribing set-square ADG with angles 30 o, 60 o, 90 o (in modern terms) G 6 4 D 15 Z B 9 E 15 H 4 5 manuscript has this figure with brief explanation A
17 17 Persian manuscript: 3.2. example of cut-and-paste construction Only figures, no text
18 18 Persian manuscript: 3.3. example of tiling with explanation C O G E B H D T L I M K N Z F S Make angle BAG three sevenths of a right angle. Bisect AG at point D. Cut off BE equal to AD. Produce line EZ parallel to AG. A
19 19 Explanation (continued) C O G E B H D T L I M K N Z F S Draw line TI parallel to BE, bisect TE at point H, and make TI equal to TH. Extend EI until it intersects AB at point K. A
20 20 Explanation (continued) C O G E B H D T L I M K N Z F S Produce KL parallel to BE. With center Z draw circular arc KMN in such a way that its part KM is equal to MN. On line AF take point S and that is the center of a heptagon. Complete the construction, if God Most High wants. A
21 Tiling realized in Isfahan, North Cupola of the Friday Mosque (ca. 1080) 21
22 22 The putting together of the tiling is not explained in the Persian manuscript P P P P H Q H P P P P
23 23 Persian manuscript 3.4: Twelve Kite Pattern Four kite pattern Isfahan, Hakim mosque Twelve kite pattern in Persian manuscript: 4 big kites, each subdivided into 2 smaller kites and 2 triangles.
24 24 Approximative construction of Twelve Kite Pattern in the Persian manuscript K 5 D H Z E Begin with little square, AD diagonal, GB = BA = AD, EZ = ZH = AG, KL//HG. L G B A
25 25 The Twelve Kite Pattern and a special right angled triangle (EZT ), also in the Persian manuscript. Q U E H B R S Z C T P altitude plus shortest side is hypotenuse (EZ + ZH = ET )
26 26 4. Mathematicians and the Twelve Kite Pattern. 4.1 Omar Khayyam in a letter on algebra (ca. 1100) To Construct a right-angled triangle such that: altitude + shortest side = hypotenuse (EZT such that EZ + ZH = ET, notation mine.) If EH = 10 and ZH= thing (modern: x) then a cube and two hundred things are equal to twenty squares plus two thousand in number modern x x = 20x segment with length x can be constructed by means of conic sections.
27 Anonymous construction of the triangle (EZT ) by means of conic sections, in an appendix to Omar Khayyam s text A Z T B H E M G EZ + ZH = ET. D Circle, center E, two perpendicular diameters AG, BD. Hyperbola through B, asymptotes AG and line through G parallel to BD. Z point of intersection
28 28 Conclusions 1: the mathematics of craftsmen This type of mathematics rarely appears in the works of the mathematician-astronomers. The craftsmen used geometric constructions without proofs, made no distinction between approximate and exact; used ruler, set-square, compass (with fixed opening), other instruments as well. Their approximate constructions were often very accurate Oral instruction must have been important
29 29 Conclusions 2: Interaction between mathematicians and craftsmen - as far as we can judge a. Abu l-wafa criticised the craftsmen b. Omar Khayyam and other mathematicians were inspired by the Twelve Kite Pattern c. The craftsmen respected the work of the mathematicians but did not understand the details.
30 Appendix: Destination of the Neugebauer Prize Continued collaboration, including student exchange, with the House of Mathematics, Isfahan ( 30
31 Mathematics and Art Conference, Isfahan,
32 Students and teachers from Utrecht, Leiden and Isfahan,
33 Workshop on Mathematics and Dutch and Islamic art, Isfahan,
34 Reviving the old crafts in a new building, Isfahan, House of Mathematics 34
35 35 Thank you! This presentation can be downloaded at A written version of this talk can be found on
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