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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.
Court, Nathan Altshiller. Mathematics in the History of Civilization. The Mathematics Teacher 41 (1948), 104--11.
How different concerns of society influenced mathematics. How the development of the concept of number is reflected in language. How the concept of how many led to arithmetic. How the concept of how much led to geometry. (Taxation and agriculture also contributed to both.) Efforts to keep time led to trigonometry. Navigation and associated astronomical problems led to logarithms [and more trigonometry]. Problems in artillery led to graphs. Both required an understanding of motion. Analytic geometry and calculus were invented in part to better understand motion. Statistics developed to understand problems in the social sciences. Also discusses the nature of mathematics: mathematics for its own sake and the axiomatic method. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Why Study History Of Math, Mathematics in Language, Number Systems, Arithmetic, Geometry, Taxation, Agriculture, Astronomy, The Reckoning of Time, Trigonometry, Artillery, Graphing, Navigation, Dynamics, Force, and Motion, Analytic Geometry, Calculus, Statistics, Social Science, and Proof.
Cox, Steven J. The shape of the ideal column. Math. Intelligencer 14 (1992), no. 1, 16--24. (Reviewer: Peeter Müürsepp.) SC: 01A99 (00A69), MR: 93a:01072.
Discusses the shape of the "ideal" column. Shows how the aesthetic and perceptual ideals of Greek and Roman times were relayed by Vitruvius and later by Alberti and others. Then shows how later scientists considered the problem from the point of view of structural strength instead. A key player in this new point of view was Lagrange. The author discusses mistakes in Lagrange's work and in the work of some later scientists and mathematicians. It is interesting that the author himself has made investigations in this area (together with M. L. Overton). The article Kirmser, Philip G. and Hu, Kuo-Kuang, The shape of the ideal column reconsidered is critical of these investigations, and includes a response by Cox. Closely related topics: The Column, Vitruvius, Leone Battista Alberti (1404?--1472), Statics, and Joseph Louis Lagrange.
Kirmser, Philip G. and Hu, Kuo-Kuang. The shape of the ideal column reconsidered. With a reply by Steven Cox. Math. Intelligencer 15 (1993), no. 3, 62--68. (Reviewer: Peeter Müürsepp.) SC: 73K05 (00A69 01A99 49N55 73H05), MR: 94e:73039.
This article criticizes some of the conclusions of Cox, Steven J., The shape of the ideal column, and contains a new derivation of the shape of the "ideal" column. In Cox's view the problem of the ideal column remains far from solved. Cox acknowledges some of the criticism, but in turn objects to the way Kirmser and Hu have had tacitly assumed the existence of a strongest column in order, which he considers far from clear. He says "Faced with their outright contempt for the question of existence of a strongest column, I find solace in L. C. Young's invocation of Perron's paradox." (This paradox starts "Let N be the largest positive integer", and then shows that there exists a larger number.) The mathematics involved is somewhat technical. Closely related topics: The Column, Statics, and Paradox.
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, Chemistry, Biology, Alan Turing, Computation, Fractals, and Architecture.
Nagy, Dénes. The 2,500-year old term symmetry in science and art and its "missing link" between the antiquity and the modern age. Symmetry: natural and artificial, 1 (Washington, DC, 1995). Symmetry Cult. Sci. 6 (1995), no. 1, 18--28. SC: 01A99, MR: 1 371 622.
Documents the evolution of the word symmetry from its beginnings in ancient Greece. As the author explains, the word originally had a somewhat different meaning: symmetry = syn together + metron measure, suggesting the notion of commensurability. The word was adopted into Latin but was apparently rare in the middle ages. It's reappearance can probably be credited to the importance to the Renaissance of the De architectura libri decem of Vitruvius (1st century BC). The author discusses the Hebrew, Indian, and Chinese words for symmetry as well. At the end of the article the author enumerates some modern generalizations and uses of symmetry. For example, the author mentions "Noether's theorems connecting symmetry transformations (invariances) and conservation laws", Gell-Mann and Ne'eman's classification of elementary particles, and "Graeser's reconstruction of Bach's Kunst der Fuge". Closely related topics: Symmetry, Language and Literature, Greece, Vitruvius, and Music.
Schroeder, Manfred R. Number theory and the real world. Math. Intelligencer 7 (1985), no. 4, 18--26. (Reviewer: M. Mendès France.) SC: 11-02 (00A69 01A99), MR: 87b:11001.
We learn in this interesting article that number theory has applications to, or at least connections, with the real world. The author begins with a discussion of the division of the scale into twelve equal semitones, and how this appears natural from the continued fraction representation of log23. Next, he discusses the acoustics of concert halls, and how ceilings designed with a knowledge of quadratic residues can better convert sound waves traveling longitudinally into lateral waves, and thereby produce a more accurate stereophonic effect. Another suggestion of the author on wave diffraction involves primitive roots. (If the reader wants to really understand this part of the article, some knowledge of physics will be necessary.) The author then discusses of applications of finite fields to error correcting codes and even a verification of Einstein's General Theory of Relativity (the slowing of electromagnetic radiation in a gravitational field, observed with radar echos of the planets Venus and Mercury). The applications of modular arithmetic to cryptography and fast methods of multiplication are more widely known, but will come as a pleasant surprise to the uninitiated. Many other applications are also briefly mentioned. The author has written a book Number Theory in Science and Communication: With Applications in Cryptography, Physics, Biology, Digital Information and Computing (Springer-Verlag, Berlin, 1984) that discusses these and other applications in more detail. Closely related topics: Number Theory, Music, Acoustics, Astronomy, Information Theory, and Arithmetic.
Smith, Thomas M. Some Uses of Graphing before Descartes. Mathematics Teacher 54 (1961), 565--67.
Briefly discusses how graphing was used before the 1600s. The De Configurationibus qualitatum of Nicole Oresme is particularly important in this regard. Oresme even points out that if the two axes represent time and velocity, then the enclosed area represents distance. Reprinted in Swetz, Frank J., From Five Fingers to Infinity. Closely related topics: Graphing, Nicole Oresme, Dynamics, Force, and Motion, and Calculus.
Taussky, Olga. From Pythagoras' theorem via sums of squares to celestial mechanics. Math. Intelligencer 10 (1988), no. 1, 52--55. (Reviewer: \v Stefan Porubsk\'y.) SC: 01-01 (01A99), MR: 89e:01002.
The author discusses parameterization of Pythagorean triangles, the law of quadratic reciprocity, representation of numbers in a fixed finite number of sums of squares numbers, quadratic forms, and connections with the complex numbers, quaternions, and Cayley numbers. The author tells that H. Levy and E. Isaacson observed the law of quadratic reciprocity in the study of water waves on a sloping beach (if sound waves behaved in an analogous way, would there be an applications in acoustics?). We see a surprising application of the parameterization of Pythagorean triangles in astronomy: E. Stiefel found observed that a straight line u1=c in the parameter plane (u1,u2) gives us triples (x,y,r) corresponding to a parabola, and if one moves along this line at a constant rate, one moves in a parabolic path according to Kepler's second law. Closely related topics: Pythagorean Triangles and Triples, Imaginary and Complex Numbers, Number Theory, Algebra, Acoustics, and Astronomy.