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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.
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), Penrose Tilings, Weaving, and Similarity.Modify notes on this entry Modify bibliography entry Make comment on this entry
Mamedov, Kh. S. Crystallographic patterns. Symmetry: unifying human understanding, II. Comput. Math. Appl. Part B 12 (1986), no. 3-4, 511--529. SC: 00A69 (01A99 20H15 51F15), MR: 87e:00008.
This article discusses how crystallographic patterns "and their distribution and connection with natural phenomena and subjects of pure and applied art." It is written as an essay from a personal point of view. As the author tells us "I have made no effort to restrict the style of my meditations. I have presented a flow of free and sincere statements, and have not attempted to impose on them a style which might conceal their individuality. A great advantage of such statements is that one's 'falsehoods' are merely considered to be delusions, thus somehow mollifying the anger of those strict critics who feel obliged to adhere to absolute truths." The author himself is a chemist, so it is not surprising that there is some discussion of how crystallographic patterns in art are similar to those in chemistry. However, his observations on art from his own background in a nomadic family from Azerbaijan may be at least as valuable. The author notes that M. C. Escher is often identified with the applied art of crystallographic patterns, but these ideas are common in many cultures. Crystallographic patterns involving elements such as colored symmetry "are very characteristic of ancient and medieval decorations of Siberia, Kazakhstan, Central Asia, Azerbaijan, and Asia Minor." Quite a few examples of the art in this article use Islamic khufic script, and as he notes it is common to attribute the rise of patterned art rather than representational art to religious demands. The author does not seem entirely sympathetic with this idea, writing "The the problem was 'explained with God's help.' It is evident that in such cases it is much easier for the representatives of some other tradition to invent a new explaining theory than to examine the artwork using the language of its own traditions." The author gives some examples of crystallographic patterns in his own art and that of his associates. Interesting and enjoyable article. Closely related topics: Symmetry, Plane Patterns, Religion, and Language and Literature.Modify notes on this entry Modify bibliography entry Make comment on this entry
Nagy, Dénes. Symmet-origami (symmetry and origami) in art, science, and technology. Symmetry Cult. Sci. 5 (1994), no. 1, 3--12. SC: 00A69 (01A99), MR: 1 309 239.
Discusses the history and philosophy of origami and then (in a little more depth) discusses some of its applications. The author discusses applications in math and science education, and also in art, design, and technology. A particularly interesting application of paper-folding and the theory of polyhedra is in music education, where one researcher devised "a 'tower' of five octahedra, to illustrate some basic concepts in musicology. His inspiration was from a work by Möbius written in 1861. Ganter's compound polyhedron illustrates geometrically the following concepts and their connections: the vertices correspond to the notes of the chromatic scale, the edges corresponds to the thirds and fifths, and the triangular faces correspond to the triads." He mentions that M. C. Escher was interesting in construction paper models (though it is not really clear how deep that interest lay). It is interesting that the well-known book by T. Sundara Row entitled Geometric Exercises in Paper Folding seems to be independent from the Japanese traditions. Closely related topics: Origami, Symmetry, Japan, Education, Music, and August Ferdinand Möbius (1790-1868).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, Penrose Tilings, and Art.Modify notes on this entry Modify bibliography entry Make comment on this entry
Schattschneider, Doris. The plane symmetry groups: their recognition and notation. American Mathematical Monthly 86 (1978), 439--450.
Discusses in detail the classification of plane patterns. Although the author avoids group-theoretic notation, she manages to bring out the group theoretic nature of the plane pattern groups more clearly than most other authors discussing these patterns. There is a very useful chart on the seventeen plane patterns that clearly labels the locations of the centers of rotation (with labels that distinguish the 2, 3, 4, and 6-fold centers), the axes of reflection, and the axes of glide-reflection. The chart may give a better understanding of the differences between the different symmetry groups than the flowcharts that appear in some other sources. The author discusses the generating regions for each of the plane patterns, and gives examples for each symmetry group of two set of generators of the group (except in the case of the pattern p1, where there is only one natural set of generators. She illustrates the plane patterns with lattices, most of which are from China. There are a couple of examples from the artwork of M. C. Escher as well. There is also a table cross-referencing notations used by different sources. There are six different notations in all; as the author notes, one the differences results from the common confusion between the groups p3m1 and p31m. Closely related topics: Plane Patterns, Group Theory, Art, and China.Modify notes on this entry Modify bibliography entry Make comment on this entry
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