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The Plaid Model

Richard Evan Schwartz
Princeton University Press
Publication Date: 
Number of Pages: 
Annals of Mathematics Studies
[Reviewed by
Meghan De Witt
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Schwartz has put together a book intended as an introduction to a new construction in geometry which he calls the Plaid Model.  The construction produces a cube filled with polyhedral surfaces.  Slicing this cube in different manners yields, alternately, information about the outer billiards orbits of kites and about the Truchet tile system.  The book is not geared towards a particular class commonly taught in the undergraduate curriculum, or towards individuals of a specific mathematical level.  Based on the level of mathematics covered, it might be accessible to an advanced undergraduate seeking a research topic, though the amount of background required and its minimal overlapping with a standard undergraduate curriculum would require such a project to be of some duration.  The topic might also be of interest to a graduate student or young researcher seeking a relatively new area to explore.  Or a mathematician who wishes a topic that has sub-projects easily accessible by undergraduate students may be interested.  For instance, with minimal background, an advanced undergraduate could begin physical explorations of the geometric models to discover various results, even if the theoretical underpinnings might be beyond their reach.
The book completely introduces this new model, though the order of presentation often presents main results before the background material and then goes back to explain them afterward.  This organization style means that it is not a quick read, as it requires much cross-referencing and digesting.  The book also comes with a companion computer program that illustrates the results and constructions and allows the reader to fully explore these constructions.
The model itself is largely a mixture of combinatorics, geometry, and elementary number theory, and thus might be of interest to a wide range of individuals seeking a new area of study that admits, fairly quickly, physical constructions to experiment and occupy oneself with.  The number of topics that the model relates to and provides connections between is large and varied and includes polytopes, renormalization, continued fractions, corner percolation, tile systems, spacetime diagrams, interacting particles, and outer billiards.
While the book covers a highly specialized and new topic, its connections to existing fields suggest it could be a fertile area of work.  It might also be suitable for a reading course among graduate students, or possibly advanced undergraduates, who are seeking a non-traditional topic to explore.  It was an enjoyable excursion into a new field.  


Meghan De Witt ([email protected]) is a professor at St. Thomas Aquinas College whose constantly shifting primary interests are currently group theory extensions, data applications, and mathematical outreach programs.