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Math in the Media 1199

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Tony Phillips' Take on Math in the Media
A monthly survey of math news

November 1999

The crumpling catastrophe

How to slice a tort?

Beautiful dynamics

The crumpling catastrophe: push a flat sheet of paper into a coffee filter holder. You will see ``conical dislocations in crumpling'' as described in an article with that title in the September 2, 1999 Nature by Cerda, Chaieb, Melo and Mahadevan, a Universidad de Santiago de Chile-MIT team. They report ``a quantitative description of the shape, response and stability of conical dislocations, the simplest type of topological crumpling deformation.'' Here is a picture of the distortion undergone by the paper:

Reprinted by permission from Nature Nature 401, 46 - 49 (1999)
� 1999 Macmillan Magazines Ltd.
It is interesting to compare this image with a picture of the ``cusp catastrophe:''

The ``cusp catastrophe." The orange-yellow plane is mapped to the blue-green plane by (x,y)-->(x³ - 2xy,y).
Folding occurs along the line y = (3/2)x², and crumpling at the point (0,0).
The study of conical dislocations gives a geometrical meaning, with curvatures related to the physical parameters of the experiment, to this purely topological concept.

How to slice a tort? A new algorithm devised by NYU politics professor Steven Brams and Union College math professor Alan Taylor gives a method for ``arbitrating any dispute in which goods are to be divided,'' according to Larissa McFarquhar in a Talk of the Town piece in the August 19 New Yorker. The patented algorithm allows a distribution of most of the goods in such a way as to divide evenly the satisfaction each disputant derives from the partition, with the round-off settled by cash if necessary. The New Yorker worried about how ``spite'' would perturb the calculation in a particularly acrimonious divorce, for example, but this worry seems to come from too material an interpretation of ``goods.'' For more information on Fair Division problems consult pages at University of Alabama -Center for Teaching and Learning, University of Colorado - Discrete Math Project, or Cedarville College.

Beautiful dynamics. "Persistent patterns in transient chaotic fluid mixing" in the October 21, 1999 Nature describes an elegant set of experiments in which a thin layer of fluid was stirred so as to create a time-periodic velocity field. Calculations had predicted ``the development of persistent spatial patterns, whose amplitude (contrast) decays slowly with time but without change of form.'' Here is a snapshot of one of the experiments:

Reprinted by permission from Nature 401, 770 - 772 (1999)
� 1999 Macmillan Magazines Ltd.
For more information see the Haverford College Nonlinear Physics and Fluid Dynamics Lab.

-Tony Phillips
SUNY at Stony Brook

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