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Algebra and Tiling: Homomorphisms in the Service of Geometry

Sherman Stein and Sándor Szabó
Mathematical Association of America
Publication Date: 
Number of Pages: 
Carus Mathematical Monographs 25
[Reviewed by
Fabio Mainardi
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Mathematically speaking, a tiling of a set is a covering by a family of subsets with disjoint interiors. For example, a tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. A particular case is that of a lattice tiling, when all the subsets are translates of a single pattern. The existence of a tiling of a set with some prescribed properties is often a non-trivial matter and this book is an elementary introduction to this subject.

Instead of following a systematic approach, the authors choose representative problems; for instance:

  • is it possible to tile a square with an odd number of triangles, all of which have the same area?
  • if n-dimensional space is tiled by a lattice of parallel unit cubes, must some pair of them share a complete (n-1)-dimensional face?

(The answers turn out to be negative and affirmative, respectively).

The main theme of the book is to translate the existence of a given tiling (a geometrical problem), to an algebraic problem, typically related to group theory. Roughly speaking, the idea is that the coordinates of the vertices of the tiles (if they exist) are necessarily related to each other and must thus satisfy some algebraic relations.

The subtitle “homomorphisms in the service of geometry” means that algebra provides the study of tiling with powerful tools. However, the algebraic problems arising from a given tiling are often as interesting as the geometric counterpart. So one may also state: “geometry in the service of homomorphisms,” after all.

The prerequisites for this book are an acquaintance with the very basics of group and field theory. The appendices, on lattices, exact sequences, formal sums and cyclotomic polynomials, make this book quite self-contained and accessible to non-experts.

There are plenty of historical remarks, making the exposition lively, and there are dozens of illustrations. The exercises and problems are abundant and stimulating.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at