For more material on this topic, see subtopic Symmetry. To expand search, see Art and Geometry. For material on related topics, see Symmetry. Laterally related topics: Symmetry, Perspective, Fractals in Art, Weaving, Renaissance Art, Basket Making, Tattoos, Pottery, Architecture, Proportion and the Golden Ratio, Metal Work, Knots and Knotwork, Wood Carving, Bronzework, Needlework, Art History, Origami, Mazes, Analytic Geometry, Trigonometry, Geometric Theorems, The Pyramid, Similarity, The Triangle, The Method of Exhaustion, Projective Geometry, Algebraic Geometry, Non-Euclidean Geometry, The Parallel Postulate, The Regular Solids, Irrationals, The Pentagram, The Sphere, The Conic Sections, Polygons, Topology, Spirals, Line-Point Duality, Geometric Fixed Point Principles, The Cycloid, Tilings, The Square, Frieze Patterns, Plane Patterns, Bichromatic Strip Patterns, Five Fold Symmetry, Penrose Tilings, 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.
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, Symmetry, Frieze Patterns, Plane Patterns, Bichromatic Strip Patterns, Bichromatic Plane Patterns, Rotational Symmetry Groups (Rosettes), Penrose Tilings, Weaving, Similarity, and M. C. Escher.
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, Weaving, Basket Making, Pottery, and Tattoos.