To expand search, see Physics. Laterally related topics: Dynamics, Force, and Motion and Statics.
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.
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, Astronomy, Information Theory, and Arithmetic.
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, and Astronomy.