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Math in the Media |
 Exits on the Pheromone Highway branch on average 53o from the "away" direction, so each intersection reads like an arrow pointing home. Image © Duncan Jackson, used with permission. |
Ant Geometry.
"Pheromone trails are used by many ants
to guide foragers between nest and food. But how does a forager that has
become displaced from a trail know which way to go on rejoining the trail?"
Richard Feynman (in "Surely you're joking ...")
considered this problem and speculated that a direction
might be written into the pheromone trail (e.g. A-B-space-A-B-space). In
fact, the ants use information encoded in the geometry of the (plentiful)
bifurcations along the trail. This has been conclusively shown by Duncan
Jackson, Mike Holcombe and Francis Ratnieks, a computer science-biology
team at the University of Sheffield; work reported in the December 16 2004
Nature. The Sheffield team worked with colonies of Pharaoh's ant
(Monomorium pharaonis). In one of their reported experiments,
they allowed individual ants to walk along experimental, straight trails:
"any reorientations occurring were as likely to be correct as incorrect
in relation to the polarity of the trail when it was originally formed."
In contrast, when meeting a trail bifurcation, "43% of fed ants [who
would presumably be heading home] made U-turns upon meeting the
bifurcation point when walking in the 'wrong' direction: that is, away from the
nest. Conversely only 8% of fed ants walking the 'correct' way made ...
corrections that led them to heading incorrectly away from the nest."
On the other hand, of the unfed ants, who presumably would be heading out for
food, "47% made course corrections at the bifurcation point when
moving the 'wrong' way" while "only 8% walking the 'correct' way made
incorrect course changes." The authors add: "Note that although only
45% of the ants corrected their orientation at a single bifurcation,
real networks contain many bifurcations and many opportunities for
course correction." [Interpreted as Markov chains, these numbers say
that
in the steady state 84% of the fed ants and 85.4% of the unfed
will be moving in
the appropriate direction. -TP]
The Magic of Math, in Queens. On November 24, 2004 the New York
Times ran "From Internet Arm Wrestling to the Magic of Math," Edward Rothstein's review of the new wing
of the New York Hall of Science, in Queens. Rothstein glances at the
high-tech baubles of the new installations, but saves most of his
admiration for the Mathematica
exhibition, which the Hall of Science
recently acquired from the California Science Center.
Mathematica was created for IBM in 1961 by the
celebrated design team of Charles and Ray Eames. Rothstein remembers
seeing it as a child: "I still recall wired
structures rising out of soapy liquid, their swirling
surfaces demonstrating solutions of mathematical problems;
the cubic array of bulbs that translated simple
multiplication into three-dimensional patterns of light;
the suspended Moebius strip - a surface with only one side
and one edge - on which a train continuously ran." And he
ponders the difference between this exhibition, assembled
at the apex of the post-sputnik wave of enthusiasm
for science, and the flashier but shallower
productions of today. "Mathematica samples varied branches of
mathematics, not blanching from explaining functions or
projective geometry; contemporary exhibitions set their
sights lower, restricting each display's focus.
Mathematica knows you won't fully understand it all
... . Contemporary displays are more concerned that you grasp a
single concept. They are play stations in a science lesson."
Father of fractals.
That's the title of Jim Giles' News Feature in the November 18 2004
Nature. Benoit Mandelbrot is the Father; the article is
illustrated with a large and spooky
image of part of "the set that bears his name."
Giles gives us a capsule intellectual history of
Mandelbrot, taking him from his 1963 paper on self-similarity in
graphs of cotton prices, through his years as an "academic
wanderer" and the 1982 publication of The Fractal Geometry of
Nature, when "the worlds of math and physics took notice."
After a brief and completely non-technical digression on fractals, we come
to the main point of the paper: Mandelbrot's attitude.
Apparently, he has not been very nice.
"As so often happens in academia, questions of precedence were central."
We hear reports of aggressivity and misbehavior at
conferences. Then
Giles focuses on Mandelbrot and Vilfredo Pareto (1848-1923), who had published
"similar studies on power laws in enonomics" many years before. Giles
claims that in the most recent reprinting of Mandelbrot's
1963 paper on cotton,
"many references to Pareto have been removed." And that one
paper by a third author, Mandelbrot and the
stable paretian hypothesis, appears in the same collection with
a new title: Mandelbrot on price variation. Are we supposed
to be horrified?
In fact Giles ends up
fairly conciliatory: "Even researchers who have been the subject of his
attacks praise his contributions to maths." [A deeper
analysis would examine the divisions in in post-war french society, politics
and science (even mathematics!), and how they played themselves out
in Mandelbrot's career. -TP]
-Tony Phillips
Stony Brook
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