Algebra and Tiling: Homomorphisms in the Service of Geometry

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 Redei’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.