Algebra and Tiling: Homomorphisms in the Service of Geometry (Carus Mathematical Monographs, Series Number 25) by Sherman Stein (PDF)

Description

Often questions about tiling space or a polygon lead to questions concerning algebra. For instance, tiling by cubes raises questions about finite abelian groups. Tiling by triangles of equal areas soon involves Sperner’s lemma from topology and valuations from algebra. The first six chapters of Algebra and Tiling form a self-contained treatment of these topics, beginning with Minkowski’s conjecture about lattice tiling of Euclidean space by unit cubes, and concluding with Laczkowicz’s recent work on tiling by similar triangles. The concluding chapter presents a simplified version of Rédei’s theorem on finite abelian groups. Algebra and Tiling is accessible to undergraduate mathematics majors, as most of the tools necessary to read the book are found in standard upper level algebra courses, but teachers, researchers and professional mathematicians will find the book equally appealing.

User’s Reviews

Editorial Reviews: Review ‘Algebra and Tiling is perfect for bringing alive an abstract algebra course. Intuitive but difficult problems of geometry are translated into algebraic problems more amenable to solution. Full of nice surprises, the book is a pleasure to read.’ Choice Book Description A concise investigation into the connections between tiling space problems and algebraic ideas, suitable for undergraduates. About the Author Sherman Stein received his PhD from Columbia University. His research interests are primarily algebra and combinatorics. He has received the Lester R. Ford prize for exposition. He is now retired from teaching at the University of California, Davis.Sandor Szabó received his PhD from Eötvös University. He currently teaches in the Institute of Mathematics and Informatics at the University of Pécs, in Hungary. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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.

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