Geometry: Drawing from the Islamic Tradition

Transcription

1 Graduate Theological Union From the SelectedWorks of Carol Bier 2015 Geometry: Drawing from the Islamic Tradition Carol Bier, Graduate Theological Union Available at:

2 Geometry: Drawing from the Islamic Tradition Carol Bier Visiting Scholar, Center for Islamic Studies, Graduate Theological Union Research Associate, Textile Museum, George Washington University Abstract Getting students involved in careful observation and analysis and encouraging their exploration of cultural forms of expression is an excellent means of introducing mathematical ideas. Geometric patterns abound in Islamic art and architecture. Exhibiting great ingenuity over the centuries, Muslim artists and craftsmen created beautiful patterns to adorn architectural monuments and exquisite objects. The Alhambra in Spain and the Taj Mahal in India offer the most famous examples of extraordinary patterns using brick and glazed tile, or carved and inlaid marble. Other examples of patterns are made using metal, wood, and fiber. Students may gain conceptual and theoretical understanding of patterns through activities and hands-on exercises that relate circles to centers, axes and grids, symmetry and tessellations. Through folding and cutting, drawing and coloring, one learns about relationships among number, shape, and the nature of space, which so characterize geometric patterns seen in Islamic architecture, silk textiles, illuminated manuscripts, wood inlaid with mother-of-pearl, carved stone, and other materials, contributing to a beauty of form, pattern, and structure. Introduction Properties of the circle are fundamental to geometry. Circles and their centers underlie the formation of many geometric patterns in Islamic art across time and space from Spain in the west to India and the east, north into Central Asia and south of the Sahara in Africa. Not only in historic monuments, geometric patterns are also characteristic of Islamic art today around the globe where such patterns at times serve as cultural markers. Depending on location, students can take local walks, or they may use postcards or digital images to serve as a basis for analysis, or color line drawings to gain an understanding of two-dimensional patterns, which are based on points, lines, and angles in the plane. Whatever the activity, careful observation, reflection, and analysis draw students into geometry in an manner that is experiential and memorable, and may also instill an awareness of cultural issues and sensitivity. Sources to Study Islamic Geometric Patterns Print resources (see References: Books and Articles, below) provide numerous photographs of Islamic geometric patterns, and some include diagrams and analyses, along with discussion (Critchlow 1976; El-Said and Parman 1976; El-Said 1993; Broug 2013). Websites and online databases offer easy access to images (see References: On-Line Resources, below). Islamic geometric patterns abound on historic monuments in many locations, including Cordoba and Granada in Spain, Fez and Casablanca in Morocco,

3 Cairo in Egypt, Damascus and Aleppo in Syria, Istanbul and Konya in Turkey, Bukhara and Samarkand in Uzbekistan, Kabul and Ghazna in Afghanistan, Ahmedabad and Delhi in India, Kuala Lampur in Malaysia, to name only some cities and towns. There are many more, as well as monuments in Islamic style throughout Europe, UK, and North America, and architecture in South and Central America in which Islamic influence is present. Add to this range of opportunities the rich collections of Islamic art in major museums throughout the world (see References: Museum Collections, below), which provide many examples of small scale Islamic geometric patterns to view up close, some so tiny they benefit from the use of a magnifying glass! Cultural Forms of Expression Although geometry in Islamic art is far too often portrayed as a response to a presumed prohibition against figural imagery, this is not borne out by the historical evidence of pictorial narratives that exist in book illustrations and miniature paintings, wall paintings and story cloths, coins, metalwork, and ceramics. (See, for example, Freedberg 1989). The predilection for geometric patterning is likely the result of intensive explorations into the nature of two- and three-dimensional space that comprised scientific endeavors supported by royal and princely patronage in both mathematics and architecture. The misunderstanding of Western perceptions may be linked to the 19 th -century promotion of Islamic geometric patterns as decorative and ornamental, without regard for cultural origins (Jones 1856). Historically, such patterns have deep cultural roots as emergent forms of visual expression in architecture and the arts in many media. They fit within a larger rubric that one might call Post-Euclidean mathematics, which includes, for example, advances in the development of algebra, trigonometry, spherical geometry, number theory, and algebraic geometry. Islamic Geometric Patterns: Activities and Exercises Four simple sets of exercises serve as an easy way to draw students into the world of Islamic geometric patterns, providing opportunities for experiential learning while sharing cultural values and introducing basic principles of geometry. 1. Establishing Grids The first exercise challenges students to tight-pack pennies (or any round coins of equal dimension). The pennies will tight-pack in only two ways one yields centers that when connected form a square grid; one yields centers that connect to form a triangular grid. Ask students to compare the spaces of the interstitial areas between the pennies, to determine which is the tighter packing. Islamic geometric patterns are often derived from a grid generated by connecting the centers of tight-packed circles. 2a. Design Basics The second exercise also involves pennies. Ask students How many pennies fit around a penny? Surprisingly, I ve never been in a class of either students or adults who readily can answer this question, and yet it is basic and

4 fundamental to the nature of circles of equal dimension. The answer, of course, is six. But then ask again, this time with reference to CDs, or jar lids, or whatever circles of equal dimension are readily at hand. Again, the answer is not always unanimous. Try again, with circles of larger dimensions, for example, paper plates, or rubber tires. The next step in this exercise is to have the students consider just why only six circles surround a seventh of equal dimension. 2b. Understanding Pi A follow up to this exercise is to ask students the meaning of pi. The answers I have received are a Greek letter, a mathematical symbol, , a never-ending decimal, an irrational number, cannot be expressed as fraction. But again, in all of the classes and lectures I ve taught, and workshops I ve conducted, never has the answer been a means of naming the invariable ratio of radius to circumference. Even when pressed to express the algebraic equations pertinent to circles, i.e. c = 2ᴨr = ᴨd, the students do not make the immediate connection between the equation and the invariable ratio as expressed by tight-packing pennies, disks, lids of yogurt containers, or paper plates. This exercise is also a means of addressing systems of proportion at variable scales, which are directly pertinent to patterns in architecture and the arts in all media. 3. Properties of Circles A third set of exercises is designed to move students further along to consider the properties of all circles by folding paper plates. These exercises were inspired by Brad Hansen-Smith s website which serves as an excellent resource for further exploration. Through this set of exercises, students explore properties of a circle and begin to identify these properties: circularity, uniform curvature, infinite radii and diameters, equidistance of circumference from center, ratio of radius to diameter. 3a. Establishing Diameters Everyone begins by marking two points on the circumference of the circle anywhere and then placing the points one upon the other. The process of discovery involves finding that everyone has made a semi-circle! And the line cutting the circle in half is a diameter! The students are asked to consider why this happens, and in every case, no matter where the points are marked on the circumference, a semi-circle is established. The answer has to do with creating a straight line that is a chord by placing the two dots one above the other, and every chord is bisected by a diameter when the paper plate is folded, one point above the other. 3b. Finding the Center Now everyone has established a diameter but where is the center? How can we locate the center of the circle? There are several answers: one may fold the diameter in half, one may mark another two points, or one may fold the plate in half again at any point. Students may then be asked to offer explanations (hypotheses) as to how

5 these actions locate the center of the circle, without measurement or reference to quantities (such as angles or dimensions). All diameters of a circle intersect at the center. 4a. Pentagons and Pentagrams A fourth set of exercises also draws upon the use of paper plates, focusing on the division of a circle into five (more or less) equal segments. This is accomplished by marking five points, approximately equidistant, on the circumference. The students are then asked to draw lines connecting these points to one another around the perimeter, creating five chords which together form a pentagon (five-sided polygon). Then the students are asked to draw the diagonals of the pentagon, connecting all of the points. The diagonals of a pentagon create a five-pointed star, or pentagram. What shape can be identified at the center of the criss-crossing diagonals? Within the fivepointed star, there is another pentagon! What is its orientation? Now connect the vertices of this pentagon to create another five-pointed star! Theoretically, this exercise could be extended either to the infinite (growing ever larger) or the infinitesimal (growing ever smaller). 4b. Divisions of the Circle and Diagonals of a Polygon The lesson continues How many diagonals does a pentagon have? (2 x 5 2 = 5) Why is this the case? (because from each vertex of this five-sided polygon, there are two diagonals and each diagonal connects to two other vertices).is there a formula we can develop that is general for all polygons? (n 3 2) Why is this the case? How many diagonals does a triangle have? 4c. Periodic Patterns and Quasiperiodicity By connecting points, one may also identify various triangles in any given polygon. Within pentagons and pentagrams (five-pointed stars), and color these in various ways to increase the educational value of this experiental exercise towards understanding mathematical principles that pertain ultimately to Penrose tiles and issues of quasiperiodicity in patterns with long-range global order but without periodicity: there is an absence of translational symmetry (Ajlouni 2012). Carrying these exercises further into the theoretical realm, students may begin to explore aspects of infinity, while experiencially examining the fractal dimension and self-similarity (Bonner 2003). More Exercises Many more exercises can be offered to introduce basic aspects of geometry based on patterns in Islamic art (Bier 2006; Bier 2007; D. Sutton 2007; A. Sutton 2009; Bier 2010). Such mathematical considerations underlie the play of pattern in Islamic art. The triangular grid lends itself to six-pointed stars and hexagons. By selecting centers in a triangular grid, one may establish a rhombic grid or a hexagonal grid. The square grid can be used to generate a pattern with eight-pointed stars and cross forms. Lines and angles may be established to create more relationships of stars and polygons. By connecting centers to form lines and angles, or highlighting curves, many seemingly

6 complex patterns may be generated from simple algorithms. Further connecting lines, or deleting or erasing lines, many more patterns may be produced, including those with interlace. Students may also be asked to distinguish between theoretical mathematics and the practical applications of mathematics. In art, a line has two dimensions. Why is that? Execution of patterns in a medium, such as ceramics, fiber, or metal (indeed any material) brings theory into the context of materiality, introducing a third dimension, which is discounted as we analyze patterns in art visually, bringing our analysis back to theoretical consideration. Drawing upon the shared heritage of Greek geometry and Indian arithmetic, pattern play in Islamic art exploits the potential for an amazing array of geometric patterns in the plane, and introduces ambiguities among the relationships among numbers (arithmetic), shapes (geometry) and space (solid geometry). The mathematical principles of symmetry (a correspondence of points) are always combined with asymmetry (the absence of symmetry), and symmetrybreaking (physics) to produce decorative patterns in Islamic art. Such patterns captivate the mind at the same time as they please the eye. In addition to observing and analyzing patterns, or taking walks to locate tessellations, student assignments might include museum sketches, making tessellations, using a compass and straight edge to produce triangular and square grids, and an exploration of coloring patterns to affect our perceptions of two-dimensional space (Bier 2006). Other areas of exploration may compare patterns at different scales to experience the impact of systems of proportion, which are still based on circles. For example, many Islamic styles of calligraphy are based upon the width of the nib of a reed pen; looking at various Islamic calligraphic styles on coins, or three-dimensional objects, and on architecture, relates systems of proportion to the dimension of scale. The geometry remains constant. Whether constructed by folding and cutting, or approximated in ceramic materials such as brick, tile, and mosaic, or wood or stone, or textile materials such as silk, cotton, or wool, Islamic geometric patterns reflect an understanding of basic mathematics and relationships among number, shape and space. Looking at Islamic art and architecture offers significant opportunities for students to analyze patterns and learn about geometry while they put geometric principles into practice. References: Books and Articles Ajlouni, R. (2012): The Long-Range Global Order of Quasi-Periodic Patterns in Islamic Architecture, Acta Crystallographica A68, pp Berggren, J. L. (2007): Mathematics in Medieval Islam, in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, ed. V. J. Katz (Pp ). Princeton and Oxford: Princeton University Press.

7 Bier, C. (2015): Geometry in Islamic Art, Encyclopedia of the History of Science, Technology, and Medicine in Non-Western Cultures, ed. H. Selin. 3 rd ed. New York: Springer Verlag. Bier, C. (2010): "CarpetMath: Exploring Mathematical Aspects of Turkmen Carpets," Journal of Mathematics and the Arts 4.1 (Pp.29-47). London. Bier, C. (2009): Number, Shape, and the Nature of Space: Thinking Through Islamic Art, Oxford Handbook of the History of Mathematics, ed. E. Robson and J. Stedall (Pp ). Oxford: Oxford University Press. Bier, C. (2007): From Folding and Cutting to Geometry and Algorithms: Integrating Islamic Art into the Mathematics Curriculum, Bridges Donostia: Mathematics, Music, Art, Architecture, Culture, ed. R. Sarhangi and J. Barrallo (Pp ). San Sebastian: The Bridges Organization. Bier, C. (2006): Islamic Art: An Exploration of Patterns, Bridges London: Mathematics, Music, Art, Architecture, Culture, ed. R. Sarhangi and J. Sharp (Pp ). London: Tarquin Publications. Bonner, J. (2003): Three Traditions of Self-Similarity in Fourteenth and Fifteenth Century Islamic Geometric Ornament, Meeting Alhambra: ISAMA-Bridges Conference Proceedings, ed. J. Barrallo et al (Pp. 1-12). Granada: University of Granada. Bourgoin, J. (1973/1879): Arabic Geometric Design. New York: Dover. Broug, E. (2008): Islamic Geometric Patterns. London: Thames & Hudson. Broug, E. (2013): Islamic Geometric Design. London: Thames & Hudson. Castera, J.-M. (1999): Arabesques: Decorative Art in Morocco. Courbevoie: ACR. Critchlow, K. (1976): Islamic Patterns: An Analytical and Cosmological Approach. London: Thames & Hudson. El-Said, I. (1993): Islamic Art and Architecture: The System of Geometric Design, ed. T. El-Bouri and K. Critchlow. Reading: Garnet. El-Said, I. & A. Parman (1976): Geometric Concepts in Islamic Art. London: World of Islam Festival Publishing. Field, R. (1998): Geometric Patterns from Islamic Art & Architecture. Norfolk: Tarquin. Freedberg, D. (1989): The Power of Images: Studies in the History and Theory of Response. Chicago and London: University of Chicago Press. Grabar, O. (1992): The Mediation of Ornament. Princeton: Princeton University Press. Jones, O. (1856): The Grammar of Ornament. London: Day and Son. Katz, V. J. (2009): A History of Mathematics: An Introduction. Boston: Addison-Wesley.

8 Marks, L. U. (2010): Enfoldment and Infinity: An Islamic Genealogy of New Media Art. Cambridge: MIT Press. Necipoǧlu, G. (1995) The Topkapi Scroll: Geometry and Ornament in Islamic Architecture. Santa Monica CA: Getty Center for the History of Art and the Humanities. Singer, L., ed. (2008): The Minbar of Saladin: Reconstructing a Jewel of Islamic Art. London: Thames & Hudson. Sutton, A. (2009): Ruler & Compass: Practical Geometric Constructions. New York: Wooden Books. Sutton, D. (2007): Islamic Design: A Genius for Geometry. New York: Walker. References: Online Resources (Note: links accessed 29 May 2015) A Tiling Database (v37/2015) (Author: B. Wichmann) Arab Culture and Civilization (Middle East Policy Council) ArchNet (An international online community for architects, planners, urban designers, landscape architects, conservationists, and scholars, with a focus on Muslim cultures and civilizations) Harvard University: The Islamic Heritage Project (manuscript collection) Heilbrunn Timeline of Art History/The Nature of Islamic Art Oriental Carpets in Renaissance Painting (Wikipedia entry) Pattern in Islamic Art: The [David] Wade Photo-Archive, 2nd Edition Symmetry and Pattern in Oriental Carpets (Author: C. Bier, 1997) References: Museum Collections (Note: links accessed 29 May 2015) Athens, Greece, The Benaki Museum Baltimore MD, Walters Art Museum/Islamic Manuscripts Berlin, Germany, Museum für Islamische Kunst Brooklyn, NY, Collections: Arts of the Islamic World

9 Cleveland, OH, The Cleveland Museum of Art/Islamic Art Collections Copenhagen, Denmark, The David Collection Detroit MI, The Detroit Institute of Arts/Arts of Asia and the Islamic World Doha, Qatar, Museum for Islamic Art Honolulu HI, Doris Duke s Shangri La Kuala Lumpur, Malaysia, Islamic Arts Museum Kuwait City, Kuwait, Dar al-athar al-islamiyyah London, UK, Victoria & Albert Museum/Jameel Gallery of Islamic Art London, UK, The British Museum/Islamic Middle East st.aspx Los Angeles CA, Los Angeles County Museum of Art New York, NY, The Metropolitan Museum of Art/ Oxford, UK, The Ashmolean Museum / The Islamic Middle East Gallery Paris, France, The Louvre / Islamic Art Paris, France, L Institut du monde arabe Petersburg, Russia, The Hermitage/Oriental Art: Islamic Art (Near East) Singapore, Asian Civilizations Museum/West Asia, Islamic Gallery Toronto, Ontario, Canada, The Aga Khan Museum Washington DC, Freer Gallery of Art/Arthur M. Sackler Gallery Washington, DC, The Textile Museum