Abstract
An anonymous Persian work on ornamental geometry, On interlocks of similar or complementary figures, appears to have been compiled by a scribe at a series of meetings between mathematicians and artisans. This unusual document, which contains 68 geometric constructions mostly with verbal explanations, can be dated by internal evidence to ca. 1300. Some of those constructions display the highest advancements attained by Muslim mathematicians, and thus represent the intimate link between theory and praxis that created the intriguing and awe-inspiring ornamental patterns. Three constructions are the solutions to problems that require cubic equations. At the time, mathematicians solved cubic equations by means of conic sections, but such solutions were only for demonstration purposes with no practical application. These three constructions in Interlocks of Figures are the cases of “moving geometry,” that is, mechanical procedures that are equivalent to the solutions for cubic equations.
First published as: Alpay Özdural , “The Use of Cubic Equations in Islamic Art and Architecture”. Pp. 165–179 in Nexus IV: Architecture and Mathematics, Kim Williams and Jose Francisco Rodrigues, eds. Fucecchio (Florence): Kim Williams Books, 2002.
Alpay Özdural (1944–2003).
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- 1.
- 2.
The only copy of this manuscript is preserved in Ms. Persan 169 in the Bibliothèque Nationale, Paris, a compilation of twenty-five works on mathematical subjects, mainly practical geometry.
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In this translation from the original Persian manuscript, simple restorations are added in square brackets; more detailed ones are explained in parentheses.
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In Arabic, ‛aqd literally means knot. Here it is used to mean the unit to be repeated to generate an interlocking ornamental composition. To convey both meanings, I translate it as “knot pattern.”
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It is assumed in the text that the length of segment GD is 1 and GA is x. Then perpendicular GE = 1, EK = 1, BD = x, BK = BG = 1 + x, EA = √(1 + x 2). Since AG:GE = AK : KB, we have x : 1 = [1 + √(1 + x 2)]:(1 + x). This equation can be reduced to x 3 + 2x 2−2x−2 = 0. The equation has one positive root, x = 1.1700865 accurate in seven decimals. I also compute the angles: tan ∠AEG = x, so ∠AEG = ∠B = 49.481553°, ∠A = 40.518447°, ∠AKG = ½∠AEG = 24.740777°.
- 6.
In the figure of the original manuscript, the points that belong to the T-ruler and those used to perform the moving geometry were distinguished by red ink. Some of these were identical to the ones that were used for the pattern itself; and no differentiation was made in the text. In order to avoid the confusion they create, the letters written in red ink are distinguished here by adding stars above, both in the figure and the text.
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In the original figure, the T-ruler is placed upon triangle GAB. The explanation in the text, however, makes sense only if the ruler is placed upon triangle GDB.
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Assuming AC = 1, the following values are also computed for later use: AB = 0.543689, BD = 0.4563109, CB = 0.83922867, CD = 0.7044022, and AD = 0.2955977.
- 9.
The following values are computed in addition to the ones in n. 16:GO = 0.7718445, CF = 0.5911954 = CB2.
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For my detailed argument on this point, see (Özdural 1998: 699–715). The photogrammetric drawing and the dimensions obtained from Rassad Survey Company, “Masjed-e Jame‘Esfahan” (paper presented to the International Committee for Architectural Photogrammetry at the Symposium on the Photogrammetric Survey of Ancient Monuments, Athens, 1974, pl. 13), are published in ibid., Fig. 32.5.
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Özdural, A. (2015). The Use of Cubic Equations in Islamic Art and Architecture. In: Williams, K., Ostwald, M. (eds) Architecture and Mathematics from Antiquity to the Future. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-00137-1_32
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