To expand search, see Science. Laterally related topics: Astronomy, Agriculture, Physics, and Chemistry.
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.
Byrne, Catriona. The left-handed Pythagoras. Math. Intelligencer 12 (1990), no. 3, 52--53. SC: 01A99, MR: 1 059 227.
The author notes an relief at Notre Dame de Chartres (dating from the 1100s) where Pythagoras is depicted as being left-handed. The author suggests that left-handedness is distinctly higher among mathematicians than in a random population. It would be interesting to know if any such association were perceived in the middle ages. Closely related topic: Medieval Europe.
Eglash, Ron. Fractal geometry in African material culture. Symmetry: natural and artificial, 1 (Washington, DC, 1995). Symmetry Cult. Sci. 6 (1995), no. 1, 174--177. SC: 01A13 (01A07), MR: 1 371 629.
This article is very brief, but mentions several tantalizing examples of fractals and recursive similarity in Africa. He gives an example of fractals in the layout of the settlement of Mokoulek in Cameroon. There are apparently also hints of fractal architecture in ancient Egypt. The author tells us that recursive scaling (infinite self-similar structures) is also seen in Ethiopian crosses, Egyptian cosmological icons, and Cameroon bronzeware. The author also tells us that "specific scaling techniques are particularly evident in Ghana, where the use of log spirals to represent self-organizing systems (biological morphogenesis and fluid turbulence is common", and that "binary recursion is used in Bambara sand divination" [in Mali]. Closely related topics: Fractals, Cameroon, Ethiopia, Ghana, Mali, and Ancient Egypt.
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, Spirals, Topology, Proportion and the Golden Ratio, Albrecht Dürer, and Leonardo da Vinci (1452-1519).
Mainzer, Klaus. Symmetry and beauty in arts and mathematical sciences. Physis Riv. Internaz. Storia Sci. (N.S.) 32 (1995), no. 1, 91--103. SC: 01A99 (00A69), MR: 96h:01043.
As this article explains, symmetry appears in a variety of disciplines over a variety of ages. The author begins by briefly discussing the natural and philosophical reasons for studying symmetry (starting in ancient Greek times). He then discusses the appearance of the 7 frieze groups and 17 ornamental groups of the plane and related groups in mathematics and crystallography. Next, he discusses appearances of symmetry and symmetry breaking in modern physics, in the theory of relativity, and in quantum mechanics and superstring theory. He finds that symmetry considerations are important in chemistry and biology as well: "In biochemistry macromolecules (for example L-amino acids or D-sugars) possess a characteristic homochirality ('dissymetry') which is assumed to be caused by parity violations of weak atomic forces." He also explains that "The emergence of pattern structure can be described by symmetry breaking not only in chemistry, but in biology. Since the pioneering work of the famous English logician and mathematician A. Turing on the chemical basis of morphogenesis in biology (1952), there has been an increasing interest in this topic." He then proceeds to discuss "Symmetry and Symmetry Breaking in the Computer World", focusing on dynamical systems. For example, he write, "Nevertheless the Feigenbaum diagram is self-similar. Every part of the tree contains the Feigenbaum diagram infinitely often like Russian dolls. It follows that mathematical chaos can be highly symmetric." He closes with a discussion of modern architecture, where he finds that symmetry concerns are important as well: "But the variety of historical reminiscences and asymmetrical elements in architecture does not mean a movement back to historicism or eclecticism. It is the expression of a sceptic and ironic view of the world which no longer believes in an omnipotent technical rationality and its claim to solve all human problems. It underlines individuality and the importance of accidental details, and has doubts about universal harmony and rationality. So it prefers symmetry breaking as a chance of variety, pluralism, and individual freedom." And this is a theme that nicely rounds of his article: "But variety and pluralism need not be in conflict with unity. It was Leibniz who suggested that the unity of the world can only be experienced by man under special aspects. So his motto was 'unity in variety.' It dates back to the old philosophical idea of Heraclitus that even symmetry breaking is related to a sometimes hidden symmetry." Interesting and thought-provoking article. Closely related topics: Symmetry, Philosophy, Greece, Physics, Chemistry, Alan Turing, Computation, Fractals, and Architecture.
Toth, Nicholas. The prehistoric roots of a human concept of symmetry. Symmetry in a kaleidoscope, 3. Symmetry Cult. Sci. 1 (1990), no. 3, 257--281. (Reviewer: J. S. Joel.) SC: 01A10 (00A99), MR: 93g:01005.
The author discusses how concepts of symmetry occur in Paleolithic artifacts such as stone tools and "Venus" figurines, and also in the roughly circular areas such as those used in a hut or even perhaps at Olduvai site DK 1 (some million years ago). The author has also noted some asymmetries in the making of flaked stone tools. "This slight but statistically significant patterning of asymmetry and possible preferential right-handedness between 1.9 and 1.5 million years ago may indicate a more profound specialization (lateralization) of the left and right hemispheres of the hominid brain by the early stone age." Closely related topics: The Paleolithic Era, Symmetry, and Archaeology.