November 2000
The tiniest dodecahedron. Scientists have achieved the
synthesis of the C20 fullerene with the geometry of
a dodecahedron. The C20 molecule has two other isomers, a
"bowl" and a "ring;" all three were synthesized. The work is
reported in the September 7 2000 Nature in a Letter by a
US-German team led by Horst Prinzbach of Freiburg. The molecule
was synthesized starting from C20H20,
dodecahedrane by first replacing the Hydrogen atoms with Bromine,
and then "debrominating." The substance existed in the gas phase
for at least 0.4 milliseconds. This report was picked up by ScienceWeek
in their October 13 issue.
Strange attractors onstage. In the October 19 2000
New York Times, Jennifer Dunning reviewed a dance presentation
at the Joyce
Theater: "Strange Attractors," choreographed by Stephen Petronio.
"The title ... comes from chaos theory and its
probing of how apparently random behavior can occur in a world
governed by deterministic laws. Program notes define a strange
attractor as `a moving and magnetic focal point in a seemingly
chaotic field.' Mr. Petronio has found the perfect analogy for his
own approach to choreography." Ms. Dunning gives us an idea of
how the mathematical abstraction works out on stage: "The protean
empty space between ranks and ensembles of dancers
seems the closest thing to a focal point ...,
drawing and pushing out dancers as it shifts about."
Incompressible is incomprehensible. Why are some things
so hard to understand? Jacob Feldman of the Rutgers Psychology
Department has an answer, reported
in the October 5 1000 Nature. He found in a large set of experiments
that for human learners, "the subjective difficulty of a concept is directly
proportional to its Boolean complexity (the length of the shortest logically
equivalent propositional formula)-that is, to its logical incompressibility."
For example a concept which encodes as (A and B) or (A and not B) is
equivalent to A and (B or not B), i.e. to A
and so can be compressed
to Boolean complexity 1. Whereas (A and B) or (not A and not B) cannot
be compressed and has complexity 4. Subjects were asked to
extract the concepts from sets of examples and non-examples. Main
conclusion: "For each concept, learning is successful to the degree
that the concept can be faithfully compressed." Feldman reflects
on his result: "In a sense, this final conclusion may seem negative:
human conceptual difficulty reflects
intrinsic mathematical complexity after all, rather than some
idiosyncratic and uniquely
human bias. The positive corollary though is certainly more
fundamental: subjective
conceptual complexity can be numerically predicted and perhaps explained."
A new Federal encryption algorithm was reported in the
October 20, 2000 Chronicle of Higher Education. The article,
by Florence Olsen, relates how the Commerce Department, after a
4-year search, has declared the new federal standard for
protecting sensitive information to be Rijndael, an algorithm
named after its inventors Vincent Rijmen and Joan Daemen. The two
Belgians beat out 20 other entries, including teams from
IBM and RSA. The new encryption algorithm, of which no
mathematical details were given, can be made stronger as more powerful
computer processors are developed. This was an entry requirement
for the competition. According to Raymond G. Kammer of NIST,
which managed the selection process, it
should be good for about 30 years, "that is, if quantum
computing doesn't manifest itself in five or six years."
The geometry of a fender-bender. "Dynamics of singularities
in a constrained elastic plate" is the title of a Letter in the October 12 2000
Nature, submitted by a French-Portuguese team led by
Arezki Boudaoud of the ENS. But at the end the secret comes out:
"The experiment
we have performed is typically a controlled version of what happens
when a car is bumped." The work involves geometry (surface curvature
with cone singularities) numerics (simulation of solutions to the
Föppl von
Kármán equations) and careful experiment.
-Tony Phillips
Stony Brook