To expand search, see Art. Laterally related topics: Symmetry, Perspective, Fractals in Art, Weaving, Renaissance Art, Basket Making, Tattoos, Pottery, Pattern, Architecture, 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.
Doczi, György. Seen and unseen symmetries: a picture essay. Symmetry: unifying human understanding, I. Comput. Math. Appl. Part B 12 (1986), no. 1-2, 39--62. SC: 92A27 (01A99 52-01), MR: 838 136.
Certainly an unorthodox essay. It may be hard to understand the author's terms dinegy and dinergic symmetry (involving the union of complementary opposites), at least in a concrete mathematical sense, but the discussion and pictures do emphasize how mathematical proportions can pervade both art and the natural world. Closely related topic: Symmetry.
Evans, Brian. Number and form and content: a composer's path of inquiry. The Visual Mind, 113--120, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
The author shows how the golden ratio occurs in music and art. His examples include Mozart's Symphony in G Minor, Grant Wood's American Gothic, Piet Mondrian's Composition with Blue, and some of his own musical and visual compositions. More controversial examples include the Great Pyramid in Egypt and Stonehenge, where the author shows how approximate values of both pi and the golden ratio can be found. The author mentions Luca Pacioli's statements on the golden ratio in De Divina Proportione and discusses other aspects of the philosophy of number and art as well. Closely related topics: Music, Art, Wolfgang Amadeus Mozart (1756-1791), Luca Pacioli, The Egyptian Pyramids, and The Stone Builders.
Fields, Margaret. Practical Mathematics of Roman Times. Mathematics Teacher 26 (1933), 77--84.
Surveys Roman mathematics. Some of the most interesting examples come from the De Architectura of Vitruvius, which discusses principles of symmetry and proportion and how to use them in architecture. Vitruvius goes as far as how to correct for an optical illusion on the capitals of columns. He also discusses geometric procedures to be used in laying out a town (to shut out winds), and various Roman instruments, including leveling instruments and an instrument for measuring distance called a hodometer. The hodometer is used for "telling the number of miles while sitting on a carriage or sailing by sea", and is particularly ingenious. Second to Vitruvius, the most important source on Roman engineering may be the Urbis Romae of Frotinus, which includes mathematical rules (not entirely successful) to determine the flow of an aqueduct. Surviving Roman bridges show a high level of skill; there were surely mathematical principles behind their design, but no detailed study has survived. Roman tunnels are equally impressive. Heron discusses how to use an instrument called the "dioptra" to survey for tunnels, measure the width of a river, and so on. Roman sundials were relatively unsophisticated. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Vitruvius, Architecture, Symmetry, Optics, Leveling, The Measurement of Distance, Frotinus, Heron, Surveying, and The Sundial.
Kemp, Martin. Spirals of life: D'Arcy Thompson and Theodore Cook, with Leonardo and Dürer in retrospect. Physis Riv. Internaz. Storia Sci. (N.S.) 32 (1995), no. 1, 37--54. SC: 01A99 (92-03), MR: 96j:01047.
Discusses theories of how art appears in biology. The author starts with St. Augustine, who concluded "If, then, we argue from the facts, first, that as everyone admits, not a single visible organ of the body serving a definite function is lacking in beauty, and, second, that there are some parts which have beauty and no apparent function, it follows, I think, that in the creation of the human body God put form before function." The author then discusses and compares the investigations of D'Arcy Thompson and Theodore Cook into the mathematical/biological manifestations of the spiral. Thompson and Cook agreed on many issues, though Thompson didn't approve of the "mystical conceptions" that he found in Cook's work. Specific topics discussed include the appearance of the golden ratio in biological systems (often in the guise of the Fibonacci series), turbulence, and transformations that take one biological object into a related one (one of Thompson's examples compares the skulls of Hyrachyus agrarius and Aceratherium tridactylum). In the process, the author touches on the work of Albrecht Dürer and Leonardo da Vinci (as the title suggests). Obviously, this article can not to be comprehensive, and the author himself tells us that the article is itself intended as a preface; it serves this function well. Both Thompson and Cook were well aware of the mathematical difficulties involved in thoroughly understanding the phenomena they wrote of. Cook wrote "It would only be possible to imagine life or beauty as being 'strictly' mathematical" if we ourselves were such infinitely capable mathematicians as to be able to formulate their characteristics in mathematics so extremely complex that we have never yet invented them." And Thompson wrote "And just as in the very simplest of actual cases we meet with a departure from such symmetry as could only exist under conditions of ideal simplicity, so do we pass quickly to cases where the interference of numerous, though still perhaps very simple, causes leads to a resultant which lies beyond our powers of analysis." The author writes that Thompson ended his book with "a plea for biological mathematicians and mathematical biologists to cultivate 'a field which few have entered and no man has explored'". He continues "Thompson's plea did not fall upon deaf ears, but it is only recently that new techniques of computer modeling have begun to realize something of the potential of some of his techniques." Closely related topics: Art, Biology, Spirals, Topology, Albrecht Dürer, and Leonardo da Vinci (1452-1519).
Loeb, A. L. The magic of the pentangle: dynamic symmetry from Merlin to Penrose. Symmetry 2: unifying human understanding, Part 1. Comput. Math. Appl. 17 (1989), no. 1-3, 33--48. (Reviewer: Marjorie Senechal.) SC: 01A99 (01A10 52-03), MR: 91a:01058a.
In this interesting and entertaining article, Merlin the magician assists Arthur and Key in exploring the secrets of dynamic symmetry (in a problem with four beetles in a square always walking towards each other), in the logarithmic spiral (the curve generated by the beetles), the golden rectangle (and its own associated spiral), and the Fibonacci numbers. The article closes with a discussion of the pentangle, which the author says "is central to the late fourteenth-century 'Sir Gawain and the Green Knight', to medieval sign theory as well as to recent research in quasi-periodic alloy crystals. The Socratic discussions here could be turned used as active learning exercises for talented students. Highly recommended. Closely related topics: England in the Middle Ages, Dynamic Symmetry, Spirals, Leonardo of Pisa (Fibonacci), The Pentagram, and Education.
Robins, Gay and Shute, Charles C. D. Mathematical bases of ancient Egyptian architecture and graphic art. Historia Math. 12 (1985), no. 2, 107--122. (Reviewer: Jens Høyrup.) SC: 01A15, MR: 87c:01002.
The authors discuss the slopes that occur in Egyptian pyramids and artwork. The discussion of Egyptian artwork is particularly interesting because of the Egyptian's conscious use of squared grids. The authors find no evidence of circles or the value of pi being used in to determine the overall dimensions of the pyramids, and similarly with the golden ratio. Similarly, the authors find no evidence of pi or the golden ratio being found in slopes of lines in Egyptian artwork. Nevertheless, the authors carefully discuss such claims rather than simply dismissing them out of hand. The authors do, however, find that certain "slopes" seem to have been preferred to others (as the authors note, the Egyptians seem to have preferred to measure slopes as run per unit rise rather than our rise per unit run). The authors buttress their arguments about the artwork through their use of new photographs; these carefully avoid distortion by means of a shift lens. The article is only moderately technical. Closely related topics: Ancient Egypt, The Egyptian Pyramids, The Circle, and Coordinates.