Algebra and Tiling: Homomorphisms in the Service of Geometry (The Carus Mathematical Monographs #25)

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Overview

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.
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Editorial Reviews

From the Publisher
'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
Choice
"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."
The Mathematics Teacher
"The students or mathematician whose area of interest is algebra should enjoy this text."
Booknews
The first six chapters treat algebraic implications of tiling space or polygons and investigate finite abelian groups, cyclic groups, and automorphisms of real or complex fields. The concluding chapter presents a simplified version of Redei's theorem on finite abelian groups. Can be used as the basis of an undergraduate or graduate seminar, and as a supplementary text for algebra and geometry courses. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780883850282
  • Publisher: Mathematical Association of America
  • Publication date: 1/1/1995
  • Series: The Carus Mathematical Monographs Series , #25
  • Pages: 224
  • Product dimensions: 5.84 (w) x 8.14 (h) x 0.80 (d)

Meet 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.

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Table of Contents

1. Minkowski's conjecture; 2. Cubical clusters; 3. Tiling by the semicross and cross; 4. Packing and covering by the semicross and cross; 5. Tiling by triangles of equal areas; 6. Tiling by similar triangles; 7. Rédei's theorem; 8. Epilogue; Appendices; References.
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Preface

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?
 Is it possible to tile a square with 30degree-60degree-90degree triangles?
 For positive integers k and n a (k,n)-semicross consists of kn+ 1 parallel n-dimensional unit cubes arranged as a corner cube with n arms of length k glued on to n non-opposite faces of the corner cube. (If n is 2, it resembles the letter L, and, if n is 3, a tripod.) For which values of k and n does the (k,n)-semicross tile space by translates?
 The resolution of each of these questions quickly takes us away from geometry and places us in the world of algebra.
 The first one, which grew out of our of Mikowski's work on Diophantine approximation, ends up as a question about finite abelian groups, which is settled with the aid of the group ring, characters of abelian groups, factor groups, and cyclotomic fields.
 Tiling by triangles of equal areas leads us to call on valuation theory and Sperner's lemma, while tiling by similar triangles turns out to involve isomorphisms of subfields of the complex numbers.
 The semicross forces us to look at homomorphisms, cosets, factor groups, number theory, and combinatorics.
 Of course, there is a long tradition of geometric questions requiring algebra for their answers. The oldest go back to the Greek: "Can we trisect every angle with straightedge and compass?" "Can we construct a cube with twice the volume of a given cube?" "Can we construct a square with the same area as that of a given disk?" These were not resolved until we had the notion of the dimension of a field extension and also knew that pi is transcendental.
 We consider only the algebra that has been used to solve tiling and related problems. Even so, we do not cover all such problems. For instance, we do not describe Conway's application of finitely presented groups to tiling by copies of a given figure. See [19] in the Bibliography on pp.200-201. Nor do we treat Barnes' use of algebraic geometry [1]. Thurston [22] has written a nice exposition of Conway's work, and providing the algebraic background for Barnes' work would take too many pages. The group with generators a and b and relations a-squared=e=b-cubed plays a key role in obtaining the Banach-Tarski paradox, which asserts that a pea can be divided into a finite number of pieces that can be reassembled to form the sun. A clear exposition of the argument was given by Meschkowski.
 We had two types of readers in mind as we wrote, the undergraduate or the graduate student who has had at least a semester of algebra, and the experienced mathematician. To make the exposition accessible to the beginner we have added a few appendices that cover some special topics not usually found in a typical introductory algebra course and also included exercises to serve as a study guide. For both the beginner and the expert we include questions that have not yet been answered, which we call "Problems," to distinguish them from the exercises.
 Now a word about the organization of this book and the order in which the chapters need to be read.
 Chapter 1 describes the history leading up to Minkowski's conjecture on tiling by cubes. We give the solution of that problem in the form of Redei's broad generalization of Hajos's original solution. Its proof, which is much longer than the proofs in the other chapter, is delayed until Chapter 7. (However, the proof uses only such basic notions as finite abelian groups, factor groups, homomorphisms and abelian groups into the complex numbers, finite fields, and polynomials over those fields).
 The beginner might start with Chapter 1, go to de Bruijn's harmonic bricks in Chapter 2, and then move on to Chapters 3 and 4, which concern the semicross and its centrally symmetric companion, the cross. After that, Chapter 5 and 6, which concern tiling by these chapters, the beginner would then be ready for the rest of Chapter 2 and the proof of Redei's theorem. The advanced reader may examine the chapters in any order, since they are essentially independent.
 We hope that instructors will draw on these chapters in their algebra courses, in order to bring abstract algebra ideas down to earth by applying them in geometric settings.
 The little that we include in the seven chapters is only the tip of the iceberg. The references at the end of each chapter and the bibliography at the end of the book will enable the reader to pursue the topics much further.  
 Several of these describe quire recent. In [14] Lackzkovich and Szekeres obtain the following result. Let r be a positive real number. Then a square can be tiled by rectangles whose width and length are in the ration r if and only if r is algebraic and the real parts of its conjugates are positive. This was done in 1990. Independently, Freiling and Rinne obtained the same result in 1994 by similar means. Kenyon [12] considers the question: Which polygons can be tiled by a finite number of squares? Gale [8] obtains a short proof using matrices by Dehn's theorem, which asserts that in a tiling of a unit square by a finite number of squares all the squares have rational sides.
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