2 Differentiation: Geometric Diversity and Design Classification

Bonner, Jay

doi:10.1007/978-1-4419-0217-7_2

Jay Bonner³

5327 Accesses
2 Citations

Abstract

The wide-ranging diversity of Islamic geometric patterns is a testimony to the degree of understanding that early Muslim pattern artists had of geometry and symmetry. Their inspired use of geometry led to the development of multiple varieties of pattern, symmetrical stratagems, and generative methodologies; the likes of which no other ancient culture came close to equaling in ingenuity and beauty. The diversity and complexity of this design tradition make it difficult to categorize, and indeed, no systematized method of comprehensive classification has been established. At best, writers and scholars addressing this subject employ descriptive analysis; for example, “The design … is a fully developed star pattern based upon a triangular grid. Its primary unit is a six-pointed star inscribed within a hexagon, which is surrounded by six five-pointed stars whose external sides form a larger hexagon.” However, detailed descriptions rarely elucidate beyond the visually obvious star types and square or triangular repeat units. Other fundamental features are frequently unaddressed when examining a given geometric pattern, including the symmetrical schema for more complex designs, the crystallographic plane symmetry group, the generative methodology, the incorporation of culturally associated additive features and treatments, and identification of the specific pattern family. The absence of appreciation for these less obvious, but nonetheless significant design features obscures the extraordinary scope of this design tradition, and it is only through a more nuanced and differentiated approach to this study, with its myriad cultural and geometric attributes, that a thorough understanding and appreciation of Islamic geometric patterns can be achieved.

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Notes

1.
This quotation references a geometric pattern used on a door at the Bimaristan al-Nuri in Damascus (1154). Tabbaa (2001), 88.
2.
Necipoğlu (1995), Chapter 4. Ornamentalism and Orientalism: the Nineteenth and Early Twentieth Century European Literature, 61–87.
3.
Jones (1856).
4.
Bourgoin (1879).
5.
The ongoing availability of Bourgoin’s work is due to its being kept in print as part of the Dover Pictorial Archive Series (printed without original text). In creating the illustrations for his book, Bourgoin does not appear to have used a traditional methodology for recreating the patterns in his collection. As a consequence, the proportions within many of his illustrations—especially those with greater complexity—are inaccurately represented, and have clearly discernable distortion. Being that this has been an artist’s reference for over 150 years, the direct copying of such problematic designs has occasionally promulgated these errors by their application within the applied and architectural arts.
6.
–Critchlow (1976).
–El-Said and Parman (1976).
–Wade (1976).
7.
Wade (1976), 63–79.
8.
Schneider (1980).
9.
Castéra (1996).
10.
Castéra (1996), 276–277.
11.
Bonner (2003).
12.
–Bonner and Pelletier (2012).
–Pelletier and Bonner (2012).
13.
–Müller (1944).
–Weyl (1952).
–Lalvani (1982).
–Lalvani (1989).
–Lovric (2003), 423–431.
14.
Abas and Salman (1995).
15.
These 2 fivefold rhombi are the same as those identified by Sir Roger Penrose in his groundbreaking research into aperiodic tilings. However, the application of the geometric patterns to the two rhombi in Fig. 9 does not include Penrose’s matching rules for forced aperiodicity and the design in Fig. 9c is therefore referred to herein as non-periodic rather than aperiodic. While never occurring within the historical record, it is certainly possible to populate these 2 fivefold rhombi with patterns that conform to the Penrose matching rules, thereby forcing the geometric design to be aperiodic [Figs. 480 and 482].
–Penrose (1974), 266–271.
–Gardener, Martin (January 1977), “Extraordinary nonperiodic tiling that enriches the theory of tiling,” Scientific America, pp. 110–121.
–Penrose (1978), 16–22.
16.
Pelletier and Bonner (2012), 141–148.
17.
Schneider (1980), pattern no. 416.
18.
Hankin (1925a), Figs. 45–50.
19.
Necipoğlu (1995), diagram no. 29.
20.
Necipoğlu (1995), diagram no. 90a.
21.
Necipoğlu (1995), diagram no. 35.
22.
Necipoğlu (1995), diagram no. 81a.
23.
Schnieder (1980), pattern no. 297.
24.
Bourgoin (1879), pl. 170.
25.
Schneider (1980), pattern no. 330.
26.
Necipoğlu (1995), diagram no. 61.
27.
Necipoğlu (1995), diagram no. 41.
28.
Necipoğlu (1995), diagram no. 69b.
29.
Necipoğlu (1995), diagram no. 59.
30.
Necipoğlu (1995), diagram no. 72c.
31.
Schneider (1980).
32.
Süleymanname: Presentations of gifts to Süleyman the Magnificent on the occasion of the circumcision of his sons Bayezid and Cihangir in 1530 by Ali b. Amir beg Sirvani. Topkapi Museum, Istanbul TKS H. 1517. See: Rogers and Ward (1988), 45c (f. 360a).
33.
Cairo, National Library, 7, ff. IV-2r.
34.
This Ilkhanid Quran is in the National Library in Cairo: 72, pt.19.
35.
Necipoğlu (1995), diagram no. 42.
36.
The wooden panel described and drawn by Herzfeld is no longer present at the Lower Maqam Ibrahim, and its current location is unknown. Herzfeld (1954-56), Fig. 56.
37.
–Fedorov (1891), 345–291.
–Schönflies (1891).
–Barlow (1894), 1–63.
–Pólya (1924), 278–298.
38.
Schattschneider (1990).
39.
–Müller (1944).
–Bixler (1980).
–Lalvani (1982).
–Grünbaum, Grünbaum, and Sheppard (1986), 641–653.
–Mamedov (1986), 511–529.
–Pérez-Gómez (1987), 133–137.
–Lalvani (1989).
–Chorbachi (1989), 751–789.
–Abas and Salman (1995).
–Lovric (2003), 423–432.
40.
Abas and Salman (1995).
41.
–Makovicky and Makovicky (1977), 58–68.
–Makovicky (1989), 955–999.
–Makovicky (1994), 1–16.
–Makovicky (1995), 1–6.
–Makovicky (1997), 1–40.
–Makovicky (1998), 107–127.
–Makovicky (1999), 143–183.
42.
This flowchart replicates that of Donald Crowe, Department of Mathematics, University of Wisconsin-Madison, and is included in his book on symmetry in cultural artifacts: See Washburn and Crowe (1988).
43.
The methodology behind the gathering of the data points for this statistical analysis of the distribution of the 17 symmetry groups within the tradition of Islamic geometric patterns is not provided in this study. See Abas and Salman (1995), 138.
44.
MS Persan 169, fol. 192a.
45.
The author is indebted to Professor Jan Hogendijk at the University of Utrecht for pointing out the connection between the panel with sevenfold symmetry at the Friday Mosque at Isfahan and the design from folio 192r in the anonymous manuscript at the Bibliothèque Nationale de France in Paris.
46.
Los Angeles County Museum of Art, the Madina Collection of Islamic Art, gift of Camilla Chandler Frost (M.2002.1.285).
47.
In the collection of the Museum of Turkish and Islamic Arts, Istanbul, Turkey, accession no. 244.
48.
Topkapi Palace Museum Library MS H. 1956.
49.
Necipoğlu (1995), 37–38.
50.
The indicated numbers in this paragraph follow the numbering protocol in the Topkapi Scroll—Geometry and Ornament in Islamic Architecture. See Necipoğlu (1995).
51.
There are relatively few sources of photographs of the geometric patterns from the Tashkent scrolls. See
–Rempel’ (1961).
–Necipoğlu (1995), 12–13.
52.
British Library, London, BL Or. MS 848, ff. 1v-2.
53.
Hankin (1925a), 3–4, no. 15.
54.
Hankin (1905), 461.
55.
In addition to the two articles mentioned above, E. Hanbury Hankin, M.A., Sc.D., also published occasional articles concerning Islamic geometric pattern derivation in the Mathematical Gazette. I am indebted to Dr. Carl Ernst for first bringing the work of Hankin to my attention in 1980. See
–Hankin (1925b), 371–373.
–Hankin (1934), 165–168.
–Hankin (1936), 318–319.
56.
–Bonner (2003).
–Kaplan (2005).
–Lu and Steinhardt (2007a).
57.
In 1987 I had the good fortune to see and photograph the Topkapi Scroll while it was on temporary display at the Topkapi Museum in Istanbul. Other than the publications of Ernest Hanbury Hankin, this was my first corroboration that the polygonal methodology I had developed independently, and had been employing as an artist for many years, was, in fact, historical.
58.
Lee (1995), 182–197.
59.
Bonner (2000).
60.
Bonner (2003), 1–12.
61.
–Wade (1976) (after Hankin).
–Pander (1982).
–Makovicky (1992), 67–86.
62.
In 2007 Paul Steinhardt and Peter Lu published a paper citing their discovery of a set of “girih tiles” that share the inflation symmetry characteristics of the set of two prototiles with matching rules discovered by Sir Roger Penrose in the 1970s. The five “girih tiles” presented by the authors are, in fact, a subset of the ten polygonal modules with associated pattern lines detailed in my 2003 paper. Lu and Steinhardt’s pattern lines for each “girih tile” are identical to the pattern lines of the median pattern family (with characteristic 72° crossing pattern lines located at the midpoints of each polygonal edge) that is described in detail in my 2003 paper. See
–Bonner (2003).
–Lu and Steinhardt (2007a).
63.
–Bonner and Pelletier (2012), 141–148.
–Pelletier and Bonner (2012), 149–156.
64.
See footnote 241 from Chap. 1.
65.
Schneider (1980), pl. 22–23.
66.
Schneider (1980), pattern no. 253.
67.
In the collection of the Victoria and Albert Museum, London, acc. No. 437–1902.
68.
–Maheronnaqsh (1976).
–El-Said and Parman (1976).
–Critchlow (1976).
–Wade (1976).
–Bakirer (1981).
–El-Said (1993).
–Marchant (2008), 106–123.
–Broug (2008, 2013).
69.
–El-Said and Parman (1976).
–Critchlow (1976).
–Burckhardt (1976).
70.
From the forward by Titus Burckhardt: El-Said and Parman (1976).
71.
The popularized claims, advanced during the 1970s, that Islamic geometric patterns are inherently associated with perennial symbolisms have been convincingly refuted as ahistorical by several scholars of note: See
–Chorbachi (1989), 751–789.
–Necipoğlu (1995), 73–83.
72.
Christie (1910).
73.
The most comprehensive study of known pattern manuals and scrolls is that of Gülru Necipoǧlu. See Necipoğlu (1995).
74.
MS Persan 169, fol. 180a–199a. For a thorough account of the significance of this manuscript as one of the very few historical Muslim sources of geometric analysis and instruction for Islamic geometric designs, and for its place among other historical documents concerned with the practical application of mathematics, see
–Chorbachi (1989), 751–789.
–Chorbachi (1992), 283–305.
–Necipoğlu (1995), 131–175.
–Özdural (1996), 191–211.
–Necipoğlu [ed.] (forthcoming).
75.
–Necipoğlu (1995), 168–169.
–Özdural (1996), 191–211.
76.
Necipoğlu [ed.] (forthcoming).
77.
Özdural (1996), 192.
78.
Necipoğlu (1995), 169.
79.
–Chorbachi (1989), 776.
–Bulatov (1988), 52.
80.
Chorbachi (1989), 751–798.
81.
For example: “But we have found a technique of approximation (taqrīb) that, whenever we divide a right angle into nine equal parts, four parts of that angle are آ ح ب and five parts are ب و د. And this is the limit of approximation.” MS Persan 169, fol. 190a (upper right corner, diagonal text of four lines). Translation by Carl W. Ernst, Kenan Distinguished Professor of Religious Studies, The University of North Carolina at Chapel Hill.
82.
–“Some craftsmen (ṣunnā`) draw this problem in such a way that they take its height as seven portions and its width as six portions. The magnitude (`uẓm) is close.” MS Persan 169, fol. 187b (four lines of upside down text at the corner of the large rectangle). Translation by Carl W. Ernst, Kenan Distinguished Professor of Religious Studies, The University of North Carolina at Chapel Hill.
–“Masters perform a test of the proportion of this problem, and Abu Bakr al-Khalil has performed the test by several methods (wajh, lit.“face”) and has achieved it. One of those [methods] is the following, which has been commented upon.” MS Persan 169, fol. 189a. (bottom three lines of main text). Translation by Carl W. Ernst, Kenan Distinguished Professor of Religious Studies, The University of North Carolina at Chapel Hill.
83.
This 200-year discrepancy diminishes arguments for the importance of this treatise to the development of this geometric idiom.
84.
MS Persan 169, fol. 192a.
85.
Hogendijk (2012), 37–43.
86.
MS Persan 169, fol. 193a.
87.
MS Persan 169, fol. 195b.
88.
Necipoğlu (1995), diagram no. 44.
89.
MS Persan 169, fol. 196a.
90.
Several scholars have suggested the possibility that some of the illustrations in Interlocking Figures may date to the Timurid period. Gülru Necipoğlu has specifically referenced the final illustration, with its distinctive swastika aesthetic, as likely of Timurid origin, but goes on to mention the possibility of an earlier origin: “even this last pattern is not inconsistent with an earlier medieval repertory,” Necipoğlu (1995), 180 [Part 4, note 113].
–MS Persan 169, fol. 199a.
–Bulatov (1988).
–Golombek and Wilber (1988).
91.
–MS Persan 169, fols. 188a, 189b, and 19a.
–Jan Hogendijk refers to this motif as the 12 kite pattern. See: Hogendijk (2012), 37–43.
92.
Wasma'a Khalid Chorbachi compares the rotating kite designs from the anonymous manuscript to multiple historical examples, including the door from Mosul. See Chorbachi (1989), 751–789.
93.
MS Persan 169, fol. 191a.
94.
Hogendijk (2012), 37–43.
95.
The one exception to the rule that all right triangles will produce rotating kite designs is in the case of the isosceles triangle with equal 45° acute angles. When this is mirrored along its long side it produces a square rather than a kite.
96.
Cromwell and Beltrami (2011), 84–93.
97.
Chorbachi (1989), 769.
98.
–Chorbachi (1989), 765.
–Özdural (1996), 191–211.
–Hogendijk (2012), 37–43.
99.
Some scholars who have written on the significance of the fourfold rotating kite designs in Interlocking figures have failed to differentiate between the geometric proportions of the examples from this treatise and the proportions of the examples from the architectural record. By conflating all rotating kite designs into a single complex construct requiring conic sections for mathematically accurate construction, the need for mathematicians to assist artists in the construction of simplified approximations is corroborated. See Chorbachi (1989), 751–789.
100.
“Producing a triangle such as this is difficult, and it falls outside of the Elements of Euclid. It belongs to the science of conics (makhrūṭāt) and it is produced by the action of moving the ruler (misṭara). When the height of the vertical is postulated (mafrūḍ) as in this example, postulated as half of segment آ ب, it produces the square آ ح ه و.” MS Persan 169, fol. 185b. (Bottom three lines of diagonal text in upper right). Translation by Carl W. Ernst, Kenan Distinguished Professor of Religious Studies, The University of North Carolina at Chapel Hill.
101.
Figure 70a illustrates the repetitive application of this construction, and, except for color, is identical to the prior representation by Alpay Özdural. See Özdural (1996), Fig. 7.
102.
MS Persan 169, fol. 194b.
103.
MS Persan 169, fol. 195b.
104.
MS Persan 169, fol. 193b.
105.
MS Persan 169, fol. 190b.
106.
This manuscript includes a second design (fol. 191r) that places six-pointed stars in 90° rotation around the vertices of the square repeat unit. However, the construction of the six-pointed stars is problematic in that it does not provide for the desired sixfold rotational symmetry. MS Persan 169, fol. 194a.
107.
This experimental change to the original design is the work of the author, but was inspired by an observation by Jan Hogendijk.
108.
MS Persan 169, fol. 189a. (bottom three lines of main text). Translation by Carl W. Ernst, Kenan Distinguished Professor of Religious Studies, The University of North Carolina at Chapel Hill.
109.
Özdural (1996).
110.
MS Persan 169, fol. 196a.
111.
Castéra (1996).
112.
Castéra (1996), 99. (note: this quotation is from the English edition of 1999).
113.
This photograph shows the hand of Jean-Marc Castéra using the orthogonal grid to construct the design by drawing freehand.
114.
Schneider (1980), pattern no. 426.
115.
Schneider (1980), pattern no. 425.
116.
Creswell (1969), 75–80, Figs. 12 and 15.

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Jay Bonner

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Bonner, J. (2017). 2 Differentiation: Geometric Diversity and Design Classification. In: Islamic Geometric Patterns. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0217-7_2

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DOI: https://doi.org/10.1007/978-1-4419-0217-7_2
Published: 19 August 2017
Publisher Name: Springer, New York, NY
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Online ISBN: 978-1-4419-0217-7
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