To expand search, see Art. Laterally related topics: Symmetry, Perspective, Fractals in Art, Weaving, Renaissance Art, Tattoos, Pottery, Pattern, Architecture, Proportion and the Golden Ratio, Metal Work, Knots and Knotwork, Wood Carving, Bronzework, Needlework, Art History, Origami, and Mazes.
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.
Cromwell, Peter R. Celtic knotwork: mathematical art. Math. Intelligencer 15 (1993), no. 1, 36--47. SC: 01A07 (00A69), MR: 1 199 275.
Cromwell discusses a theory for the construction of Celtic knot friezes. These knot patterns may have been inspired by basketry (or maybe by textiles). He then analyzes the patterns in the knot friezes using a notion of a two-sided frieze pattern. There turn out to be 31 such patterns; 7 of these are the standard monochromatic strip patterns; 17 are exactly analogous to the bichromatic strip patterns; and 7 are like the monochromatic strip patterns but require the two sides to be identical. These last 7 "grey" patterns can't occur in knotwork, since the two sides of a crossing are not identical. Of the 24 monochromatic and bichromatic patterns, 12 cannot occur in Celtic knotwork because they would require strings that don't tie up, and 2 require a string straight through the centerline (and also don't occur). The other 10 can theoretically appear. Of these 10, two do not seem to occur at all, and one occurs but with an apparently different constriction technique (an example of this type is thought to be Scandinavian). The author is able to explain the rareness of these symmetry types in terms of the theory for their construction and from the fact that Celtic know friezes were generally finite and had their ends knotted together; these constraints require construction with an even grid, and the three problematic patterns require construction with an odd grid. This explains the type which does occur appears to use a different construction technique. In fact, the author found only one Celtic pattern that uses an odd grid. (And of course it can't be used in a bounded way, though it can be used in a kind of border.) All 7 of the monochromatic frieze patterns were apparently used in generating the existing know patterns, assuming the theory of construction is true (the author makes no claims that it is). The author includes examples of his own for the 3 problematic odd-grid know patterns. Excellent article. The author includes a good bibliography of related topics. It goes as far as Norwegian peasant art, for example. Not inordinately technical, in spite of the way it might sound. Closely related topics: The Celts, Knots and Knotwork, Two Sided Frieze Patterns, Frieze Patterns, Bichromatic Strip Patterns, and Weaving.
Gerdes, Paulus. Fivefold symmetry and (basket) weaving in various cultures. Fivefold symmetry, 245--261, World Sci. Publishing, River Edge, NJ, 1992. SC: 52B99 (01A07), MR: 1 178 750.
Gerdes suggests that five-fold symmetries arose from efforts to solve problems in basketweaving rather than in observations of five-fold symmetry in natural phenomena (such as starfish). One way five-fold symmetries can arise is by modifying the more obvious six-fold symmetries (such as those used by peasants in Mozambique) to fit a curved surface. The author reports that "these pentagonal-hexagonal baskets are, for instance, also woven by the Ticuna and Omagua Indians (northeastern Brazil), by the Huarani Indians, by the Kha-ko in Laos, and by the Menda in India. One sees them also in China, Japan, and Indonesia." The Malaysian sepak tackraw ball is similar to the soccer ball and is woven in the same way. The author reports that the peasants of the island Roti (Indonesia) may have discovered a way to fold a regular pentagon as a kind of a thimble. The author shows how a similar pentagonal weaving pattern is used in weaving brooms in Mozambique. (A near pentagram then appears inside the knot.) The author notes that a similar method is used in Angola to hold together the bars of a cage. The author in addition discusses how hat weaving techniques can lead naturally to three- and five-fold symmetries. The author's main example is with the hats of the Belu of central Timor, but he notes that related techniques are used in northern Mozambique, southern Tanzania, and by the Kuva of Congo. The author also shows a Chinese hat with five-fold symmetry. Two other particularly interesting examples are "a burden basket ... from the Papago Indians (Arizona) which combines beautifully a global sevenfold symmetry with local fivefold symmetry", and the "center of a Japanese basket, which combines global ninefold symmetry with local fivefold symmetry." Closely related topics: Five Fold Symmetry, Mozambique, Malaysia, and The Belu of Central Timor.
Gerdes, Paulus. Three alternate methods of obtaining the ancient Egyptian formula for the area of a circle. Historia Math. 12 (1985), no. 3, 261--268. (Reviewer: Richard L. Francis.) SC: 01A15, MR: 86k:01004.
Gerdes gives three possible methods that the Egyptians could have used in discovering their "value" of pi, which is in effect 4(8/9)2, or about 3.16. All methods are empirical. One is connected with how rope can be coiled, one is with how mats can be formed using concentric rings, and one with arranging small balls or cylinders in a circle (the Egyptians are known to have used such objects). In al cases, if it is desired that the size of the circle be chosen so as to obtain (in effect) a perfect square value for pi, the Egyptian value arises naturally. Closely related topics: Ancient Egypt and The Circle.
Gerdes, Paulus and Bulafo, Gildo. Sipatsi. Technology, art and geometry in Inhambane. Translated from the Portuguese by Arthur B. Powell and Gerdes. Instituto Superior Pedagógico, Ethnomathematics Research Project, Maputo, 1994. 102 pp. (Reviewer: J. S. Joel.) SC: 01A07 (00A08 00A69 01A13 51M20), MR: 95f:01002.
The authors discuss the construction and mathematical properties of the Mozambican sipatsi, which are essentially woven handbags. They are generally decorated with strip or frieze patterns, and in fact all 7 possible types of strip patterns occur in the sipatsi from Inhambane province in Mozambique. This book includes a description of the processes used to create the sipatsi, a catalog of the strip patterns found, and a chapter designed for people using the sipatsi to teach mathematics. The authors also give just a few examples of strip patterns on wooden spoons (also from Inhambane province) and on vases and pots (from Maputo). Closely related topics: Mozambique, Frieze Patterns, and Education.
Grünbaum, Branko; Shepard G.C. The geometry of fabrics. Geometrical Combinatorics, 77--97, Pitman, Boston, 1984.
Symmetry groups have already been used in discussing mathematical properties of textiles, but may not be appropriate for all kinds of fabrics. In this article, the author discusses primarily isonemal fabrics. In these, the symmetry group acts transitively on the strands of the fabric. Many fabrics that actually occur in nonmathematical discussions are actually isonemal. This article (and for example its classification of isonemal fabrics into 5 genera) could form part of the foundation for a new ehtnomathematical research area. In addition, a number of interesting combinatorial questions arise. The article focuses primarily on traditional fabrics, with two perpendicular layers. The terminology of the article may be less appropriate for the three layer fabrics that sometimes occur in basketweaving. "It is interesting to note that in the case of 3-way 3-fold fabrics some 'partial fabrics', that is parts of fabrics that do not hang together, are used in basketry... The classification of such fabrics seems to be a totally unexplored area of the subject." With regard to 3-way fabrics, he notes that "it seems that such fabrics are more stable under diagonal strain than 2-way 2-fold fabrics, and so have been used in such practical applications as parachutes." Similar concerns may of course account for their occurrence in basket making as well! Closely related topics: Weaving, Combinatorics, and Symmetry.
Wren, R. L. and Rossmann, Ruby. Mathematics Used by American Indians North of Mexico. School Science and Mathematics 33 (1933), 363--72.
Surveys the use of numbers and geometric shapes in various North American indigenous peoples. Includes sacred numbers, number words, including an unusual instance of subtractive number words in the Bellacoola of British Columbia, number systems, reckoning of time and seasons. Also includes geometric characteristics of dwellings and (briefly) textiles, basketry, pottery, and tattooing. Often pottery designs were borrowed from textile art. A common principle in weaving is that no line, curved or otherwise could intersect itself. (Is this principle partly responsible for the popularity of spirals?) Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Indigenous Mathematics of North America, Numerology, Number Words, The Bellacoola, The Reckoning of Time, Pattern, Weaving, Pottery, and Tattoos.