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Algebra and Tiling: Homomorphisms in the Service of Geometry (Carus Mathematical Monographs)

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In chapter 1 we briefly look at Minkowski's geometric theory of numbers, namely a theory of quadratic forms based on the geometry of lattices. In the course of these investigations Minkowski conjectured that in a lattice tiling of n-space by cubes some two cubes must share a complete face. Minkowski proved this for n=2,3. Our authors couldn't care less about his proofs however; instead we quickly move the the idea that worked for general n: Hajós reformulation of the problem as a simple statement about factorisation of abelian groups. The rest of the book is just more of the same, but without the heart and soul of classical mathematics. So, one may study tilings not only by cubes but by clusters of cubes (chapters 2-5), or one could try to tile some polygon by some triangles (chapters 5-6). The final chapter 7 presents Rédei's theorem, which generalises the group theoretic version of Minkowski's conjecture.