``There are exactly 9, 117 and 926 topological types of,
respectively, 4-connected 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 UK-US-German team of chemists,
crystallographers and mathematicians, are applying algebra, topology
and combinatorics to study how many different kinds of three-dimensional
crystals can exist. A network, here, is a periodic tiling of 3-space
that only uses tetrahedra. The tetrahedra can be irregular and may come
in different sizes. 4-connected 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
Paris-Sud 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
liquid-crystal molecules. The news is that these drums vibrate in
surprising ways: some short-wavelength 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 Taniyama-Shimura
conjecture. ``Fermat's Last Theorem Extended,'' a piece by Dana Mackenzie
in the July 9, 1999 Science magazine, tells how a US-French
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.
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