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October 1999
``There are exactly 9, 117 and 926 topological types of, respectively, 4connected uninodal, binodal and trinodal networks, derived from simple tilings based on tetrahedra.'' This is from a letter ``Systematic enumeration of crystalline networks'' in the August 12, 1999 Nature. The authors, a UKUSGerman team of chemists, crystallographers and mathematicians, are applying algebra, topology and combinatorics to study how many different kinds of threedimensional crystals can exist. A network, here, is a periodic tiling of 3space that only uses tetrahedra. The tetrahedra can be irregular and may come in different sizes. 4connected means that each vertex is connected to four other vertices. A network is uninodal if every vertex ``looks like'' every other, binodal if there are exactly two types of vertices, etc. A companion piece in that issue of Nature, ``Crystal structures: Tiling by numbers'' by Michael O'Keeffe, explains the importance of this achievement, both as a guide to possible synthetic materials and, by furnishing theoretical trial structures, as a tool in the analysis of the crystal structure of new materials. Unexpected behavior from fractal drums. A team at the Université ParisSud in Orsay, France, has assembled fractal drums the size of a fingernail. This from a report by P. Weiss in the July 31, 1999 Science News, picking up on an article by Catherine Even et al. in the July 26 Physical Review Letters. The drumheads are made of membranes of liquidcrystal molecules. The news is that these drums vibrate in surprising ways: some shortwavelength oscillations get trapped in regions where there seemed to be no geometric obstruction to their escape.
Fermat's Last Theorem was just a special case of a new result that may have just been proved: the TaniyamaShimura conjecture. ``Fermat's Last Theorem Extended,'' a piece by Dana Mackenzie in the July 9, 1999 Science magazine, tells how a USFrench team (Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor) has announced a proof of this conjecture, ``a wonderful, major conjecture'' according to Ken Ribet of U.C. Berkeley, which had been unsettled since the early 1960's. The proof has not been checked by the experts yet, in fact it may not yet be all written down. Mackenzie quotes Conrad: ``I hope a complete draft will be ready by the end of the summer.'' No room on the family tree. ``About 20% of the people who lived 30 generations ago or earlier have no living descendants, whereas the remaining 80% are the ancestors of every person alive today.'' This would follow from a new mathematical model for the evolution of the human population, propounded by Damián Zanette, Susanna Manrubia and Bernard Derrida, all physicists, in a Physical Review Letters article picked up by Ana Berlin: ``Kissing Cousins'' in the September/October 1999 issue of The Sciences. Their methods also apply to other disordered systems. ``... although mathematics and physics are different, it is more a matter of degree than black and white.'' This is from a piece entitled ``Randomness Everywhere'' in the July 22, 1999 Nature, by C. S. Calude and Greg Chaitin. They explain recent results in algorithmic information theory due to Calude and to Theodore Slaman of U. C. Berkeley, building on earlier work of Chaitin and Robery Solovay. ``This work reinforces the message of algorithmic information theory that randomness is as fundamental and pervasive in pure mathematics as it is in theoretical physics.'' Get ready for experimental mathematics. Tony Phillips 
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