To expand search, see Tilings and Symmetry. Laterally related topics: Frieze Patterns, Plane Patterns, Bichromatic Strip Patterns, Five Fold Symmetry, Pattern, The Regular Solids, Double Frieze Patterns, Two Sided Frieze Patterns, Rotational Symmetry Groups (Rosettes), Bichromatic Plane Patterns, and Dynamic Symmetry.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Chorbachi, W. K. In the tower of Babel: beyond symmetry in Islamic design. Symmetry 2: unifying human understanding, Part 2. Comput. Math. Appl. 17 (1989), no. 4-6, 751--789. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A30 92K99), MR: 91a:01058c.
An interesting and personal account of how the author discovered geometric manuscripts written for Islamic artisans. With this discover, the author gives a new historical and scientific basis to the study of certain kinds of Islamic art. Much work preceding the author's had focused on religious, mystical, or perceptual interpretations of the work. Many ideas were primarily hypothetical, such as the (incorrect) idea that all Islamic art derives from the circle. The author suggests that many religious and mystical interpretations of Islamic geometric art should not be regarded as being historically based. Instead, the author shows how some Islamic art is highly mathematical, showing concerns with such topics as Pythagorean triangles and the notion of similarity (he gives an example where a shape appears in three different scales, each similar shape being derived from the last by a clever process). Much of the article discusses these in the context of a cyclic quadrilateral appearing in Islamic art with sides 1, 2, 2, 71/2. The author even noted an Islamic anticipation of a shape used to produce Penrose tilings. The author suggests that symmetry groups, while useful, can not alone give a full understanding of Islamic art. Closely related topics: The Islamic World, Art, Plane Patterns, Pythagorean Triangles and Triples, Religion, and Mathematics and Mysticism.Modify notes on this entry Modify bibliography entry Make comment on this entry
Grünbaum, Branko. The emperor's new clothes: full regalia, G-string, or nothing? With comments by Peter Hilton and Jean Pedersen. Math. Intelligencer 6 (1984), no. 4, 47--56. (Reviewer: H. S. M. Coxeter.) SC: 01A15 (01A60 05B45 20F32 52A45), MR: 86d:01004.
Grünbaum's article: The author discusses the common misconceptions that the Egyptians and the artists of the Alhambra had used all 17 types of plane patterns. In fact, the Egyptians appear to have missed the five symmetry groups which have three-fold rotations. The sources for these misconceptions are discussed as well. The author has done fairly extensive research on the subject, and has concluded that two of the four plane patterns missing from the Alhambra seem not to appear at all in Islamic art (these are pg and pgg; the two missing at the Alhambra but present elsewhere are p2 and p3m1). A final theme of the author's is that the language of symmetry groups may at times be inadequate to discuss patterns, and can also be misleading in connection with the intentions of the artists themselves.The response by Peter Hilton and Jean Pedersen: The author's acknowledge Grünbaum's correction about the Egyptians. The authors note that the Egyptians and Moore's between them only missed one symmetry group, p3m1. They comment briefly on Chinese and Japanese designs, and quote Schattschneider, who notes that Chinese and Japanese artwork features rotations and glide reflections much more strongly than Islamic art does. Schattschneider also cites an illustration from a Japanese book that seems to suggest that underlying lattices of squares, equilateral triangles, rhombuses, and parallelograms were consciously used in developing symmetry patterns. The authors acknowledge the limitations of group theory in discussing symmetry, but also emphasize its usefulness. Closely related topics: Plane Patterns, Ancient Egypt, The Islamic World, Japan, and China.
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Jablan, Slavik. Ornament today. Geometry in the pre-scientific period; ornament today, 33--65, Hist. Math. Mech. Sci., 3, Math. Inst., Belgrade, 1989. SC: 01A10, MR: 92g:01008.
The author discusses how a wide variety of mathematical notions can be used to help describe and understand the patterns occurring in art. One of the most important is, of course, the notion of symmetry, including those in the rotational symmetry patterns, frieze patterns, plane patterns, and their bichromatic (or antisymmetry) variants. More complex types of patterns also occur in art, and as Grünbaum, Grünbaum, and Shephard observed in their article Symmetry in Moorish and other ornaments, many of the problems originating from these are still unsolved. Examples are given from the Paleolithic to the 20th century. The author touches on (to give a few examples) interlace patterns (often considered to be connected with weaving), similarity symmetry, symmetries in higher dimensional spaces, and on some of the ideas of the theory of tilings, including Penrose tilings and hyperbolic tilings. The author also gives examples from the work of artists including M. C. Escher, B. Riley, and R. Neal. A fine article. A fine article. It could easily take a class an entire semester to examine in detail all the ideas presented. Closely related topics: Art, Pattern, Symmetry, Frieze Patterns, Plane Patterns, Bichromatic Strip Patterns, Bichromatic Plane Patterns, Rotational Symmetry Groups (Rosettes), Weaving, Similarity, and M. C. Escher.Modify notes on this entry Modify bibliography entry Make comment on this entry
Schattschneider, Doris. The fascination of tiling. The Visual Mind, 157--164, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
As the author notes, "interlocking shapes displayed in majolica tile, inlaid wood, brickwork, carved stucco, stone pavement, sewn patchwork or printed fabric hold a special fascination for many people that goes far beyond the aesthetic pleasure that these patterns provide." The author gives the example of the artist M. C. Escher, who "often described regular divisions of the plane as the 'richest source of inspiration I have ever struck.'" This article is an excellent introduction to some of the mathematical problems that arise in the study of tilings. The author discusses for example uniqueness of tilings, Penrose tilings, Conway Criterion tilings, rep-tiles, and some of the issues that arise in the classification of tilings. She uses the example of rep-tiles to give a hint on how one can prove that certain tilings are aperiodic. She uses an artwork of M. C. Escher to illustrate three methods of classifying tilings. She mentions some other issues in tiling as well, such as tilings of non-planar surfaces; the references should help track these issues down further. Very clearly written. One minor comment is that a couple of times in the article she tells us that there is "no test or algorithm" that will answer a certain kind of question; it seems that she may actually mean only that there is no test or algorithm currently known. Closely related topics: Tilings, M. C. Escher, and Art.Modify notes on this entry Modify bibliography entry Make comment on this entry
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