Showing papers in "The Mathematical Intelligencer in 2004"
543 citations
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TL;DR: In this paper, it was shown that hexagonal close packing is the densest packing possible, and the density is now 74.05 percent, which is the highest known density.
Abstract: If you pour unit spheres randomly into a large container, you will fill only some 55 to 60 percent of the space. If you shake the box while you are filling it, you will get a denser packing - something like 64 percent. What is the densest packing possible? In a little booklet which he published in 1611, Johannes Kepler claimed that the hexagonal close packing did the trick: pack one horizontal layer so densely that each sphere is surrounded by six spheres, then add the next layer by placing spheres into the dimples formed by the fnrst layer, etc. The density is now 74.05 percent. And so Kepler's conjecture was born...
77 citations
TL;DR: One need only glance at Archimedes's Proposition 8 above to see that On Floating Bodies is several orders of magnitude more sophisticated than anything else found in ancient mathematics as discussed by the authors.
Abstract: One need only glance at Archimedes’s Proposition 8 above to see thatOn Floating Bodies is several orders of magnitude more sophisticated than anything else found in ancient mathematics. It ranks with Newton’sPrincipia Mathematica as a work in which basic physical laws are both formulated and accompanied by superb applications.
34 citations
TL;DR: In this article, Hinke and Bernd developed an algorithm that grows a manifold in steps, starting from a small disc in the stable eigenspace of the origin and adding at each step a band of a fixed width.
Abstract: You have probably seen a picture of the famous butterfly-shaped Lorenz attractor — on a book cover, a conference poster, a coffee mug or a friend’s T-shirt. The Lorenz attractor is the best known image of a chaotic or strange attractor. We are concerned here with its close cousin, the two-dimensional stable manifold of the origin of the Lorenz system, which we call the Lorenz manifold for short. This surface organizes the dynamics in the three-dimensional phase space of the Lorenz system. It is invariant under the flow (meaning that trajectories cannot cross it) and essentially determines how trajectories visit the two wings of the Lorenz attractor. We have been working for quite a while on the development of algorithms to compute global manifolds in vector fields and have computed the Lorenz manifold up to considerable size. Its geometry is very intriguing and we explored different ways of visualizing it on the computer [6, 9]. However, a real model of this surface was still lacking. During the Christmas break 2002/2003 Hinke was relaxing by crocheting hexagonal lace motifs when Bernd suggested: “Why don’t you crochet something useful?” The algorithm we developed ‘grows’ a manifold in steps. We start from a small disc in the stable eigenspace of the origin and add at each step a band of a fixed width. In other words, at any time of the calculation the computed part of the Lorenz manifold is a topological disc whose outer rim is (approximately) a level set of the geodesic distance from the origin. What we realized then and there is that the mesh generated by our algorithm can directly be interpreted as chrochet instructions! After some initial experimentation, the first model of the Lorenz manifold was
34 citations
TL;DR: The Mathematical Communities Forum as discussed by the authors is a forum for discussion of mathematical communities throughout the world, and through all time, and includes mathematical schools, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one.
Abstract: This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of “mathematical community ” is the broadest. We include “schools ” of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
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TL;DR: The Opinion column as mentioned in this paper offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome, and opinions expressed here are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them.
Abstract: The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-in-chief, Chandler Davis.
TL;DR: The Mathematical Intelligencer as mentioned in this paper is a review site for the mathematical literature, where authors can submit an unsolicited review of a book of their choice, or if they would welcome being assigned a book to review, please write us, telling us their expertise and their predilections.
Abstract: Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.
TL;DR: The Mathematical Communities Forum as discussed by the authors is a forum for discussion of mathematical communities throughout the world, and through all time, and includes mathematical schools, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one.
Abstract: This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of “mathematical community” is the broadest. We include “schools” of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
TL;DR: In this article, the authors introduce a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on.
Abstract: This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
TL;DR: Coxeter was no revolutionary, and the non-Euclidean revolution was already part of history by the time he arrived on the scene as discussed by the authors, but he did experience the dramatic aftershock in physics.
Abstract: H. S. M. Coxeter, known to his friends as Donald, was not only a remarkable mathematician. He also enriched our historical understanding of how classical geometry helped inspire what has sometimes been called the nineteenth-century’s non-Euclidean revolution (Fig. 35.1). Coxeter was no revolutionary, and the non-Euclidean revolution was already part of history by the time he arrived on the scene. What he did experience was the dramatic aftershock in physics. Countless popular and semi-popular books were written during the early 1920s expounding the new theory of space and time propounded in Einstein’s general theory of relativity. General relativity and subsequent efforts to unite gravitation with electromagnetism in a global field theory gave research in differential geometry a tremendous new impetus. Geometry became entwined with physics as never before, and higher-dimensional geometric spaces soon abounded as mathematicians grew accustomed not just to four-dimensional space-times but to the mysteries of Hilbert space and its infinite-dimensional progeny.
TL;DR: The Mathematical Intelligencer as discussed by the authors is a review site for the mathematical literature, where authors can submit an unsolicited review of a book of their choice, or if they would welcome being assigned a book to review, please write us, telling us their expertise and their predilections.
Abstract: Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.
TL;DR: Weyl as discussed by the authors argued that for physics special relativity theory is of much greater consequence than the general theory, and that the reverse situation prevails with respect to mathematics: there special relativity theories had comparatively little, general relativity theory very considerable, influence, above all upon the development of a general scheme for differential geometry.
Abstract: There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had comparatively little, general relativity theory very considerable, influence, above all upon the development of a general scheme for differential geometry. —Hermann Weyl, “Relativity as a Stimulus to Mathematical Research,” pp. 536–537.
TL;DR: The Mathematical Communities Forum as mentioned in this paper is a forum for discussion of mathematical communities throughout the world, and through all time, including schools of mathematics, circles of correspondence, mathematical societies student organizations, and informal communities of cardinality greater than one.
Abstract: This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of “mathematical community” is the broadest. We include “schools” of mathematics, circles of correspondence, mathematical societies student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
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