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"A heavy warning used to be given that pictures are not rigorous; this has never had its
bluff called and has permanently frightened its victims." – Littlewood's Miscellany
Report of
Science at Williams





2003 – 2004


Williamstown, Massachusetts
The underlying theme behind the images is the theory of particle collisions. The shapes show the different ways particles can move and collide on a circle and a line. Figure 1 shows two different ways of keeping track of three particles on a circle. It turns out that both of these ways, though visually different, yield the same mathematical object. Both objects are made up of 60 identical copies of a polyhedron known as the "associahedron,” an object that shows up in string theory.
Front Cover
The four figures in Figure 2 show different representations of the 4-dimensional version of the associahedron. Although the images look different, a simple argument shows that all four pictures are the same. The image surrounding each polytope, drawn with three black and four colored particles (distributed in different combinations) show the configuration space associated to each object. The black particles are fixed whereas the color particles are allowed to move and interact with each other.
The top of Figure 3 shows the arrangement of five 3-dimensional polyhedra gluing together in another configuration space. This space was discovered in Professor Satyan Devadoss’s SMALL 2003 summer undergraduate research group. The gluing that tells us how these objects should fit together is associated with the arrangement of four centrally symmetric particles on a line (three regular and one "thick" particle). This is shown in the bottom of Figure 3.
The Science Executive Committee wishes to express its gratitude to the extensive efforts of all the science departmental executive assistants in preparing contributions for this publication, and to
Kate Fletcher, administrative assistant at the Science Center, for assembling this material in its final form.

Editor: Dr. Bryce Babcock

This document is printed on recycled paper.
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