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

Often questions about tiling space or a polygon lead to other questions. 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: if such a group is factored as a direct product of subsets, each containing the identity element, and each of prime order, than at least one of them is a subgroup. Algebra and Tiling is accessible to undergraduate mathematics majors, as most of the tools necessary to read the book are found in standard upper division algebra courses, but teachers, researchers and professional mathematicians will find the book equally appealing. The Carus Mathematical Monographs Are An Expression Of The Desire Of Mrs. Mary Hegeler Carus And Of Her Son, Dr. Edward H. Carus, To Contribute To The Dissemination Of Mathematical Knowledge By Making Accessible At Nominal Cost A Series Of Expository Presentations Of The Best Thoughts And Keenest Researches In Pure And Applied Mathematics. The Publication Of The First Four Of These Monographs Was Made Possible By Mrs. Carus As Sole Trustee Of The Edward C. Hegeler Trust Fund. The Sales From These Have Resulted In The Carus Monograph Fund, And The Mathematical Association Of America Has Used This As A Revolving Book Fund To Publish The Succeeding Monographs. The Expositions Of Mathematical Subjects That The Monographs Contain Are Set Forth In A Manner Comprehensible Not Only To Teachers And Students Specializing In Mathematics, But Also To Scientific Workers In Other Fields. More Generally, The Monographs Are Intended For The Wide Circle Of Thoughtful People Familiar With Basic Graduate Or Advanced Undergraduate Mathematics, Encountered In The Study Of Mathematics Itself Or In The Context Of Related Disciplines, Who Wish To Extend Their Knowledge Without Prolonged And Critical Study Of The Mathematical Journals And Treatises. -- From Dust Jacket. Minkowski's Conjecture -- Cubical Clusters -- Tiling By The Semicross And Cross -- Packing And Covering By The Semicross And Cross -- Tiling By Triangles Of Equal Areas -- Tiling By Similar Triangles -- Rédei's Theorem -- Epilog -- Lattices -- The Character Group And Exact Sequences -- Formal Sums -- Cyclotomic Polynomials. Sherman K. Stein, Sándor Szabó. Includes Bibliographical References And Index. There is a long tradition of geometric questions requiring algebra for their answers. The oldest go back to the Greeks, such as "Can we construct a square with the same area as that of a given disk?" many of which were not resolved until mathematicians concluded that [lowercase Greek]Pi is transcendental. There are many questions that appear to focus on geometry, but which require algebra to solve. This book looks at the algebra used to solve tiling and related problems. One purpose of the text is to bring abstract algebra ideas down to earth by applying them in geometric settings. The text is intended for undergraduate and graduate students with at least a semester of algebra, and experienced mathematicians. For both beginners and experts, there are questions that have not yet been answered, which have been labeled "Problems" to distinguish them from the exercises The origins of most of the tiling problems we will explore go back just a short time.