Book contents
- Frontmatter
- Preface
- Contents
- Chapter 1 Minkowski's Conjecture
- Chapter 2 Cubical Clusters
- Chapter 3 Tiling by the Semicross and Cross
- Chapter 4 Packing and Covering by the Semicross and Cross
- Chapter 5 Tiling by Triangles of Equal Areas
- Chapter 6 Tiling by Similar Triangles
- Chapter 7 Rédei's Theorem
- Epilog
- Appendix A Lattices
- Appendix B The Character Group and Exact Sequences
- Appendix C Formal Sums
- Appendix D Cyclotomic Polynomials
- Bibliography for Preface
- Supplement to the Bibliography
- Name Index
- Subject Index
- Symbol Index
Chapter 2 - Cubical Clusters
- Frontmatter
- Preface
- Contents
- Chapter 1 Minkowski's Conjecture
- Chapter 2 Cubical Clusters
- Chapter 3 Tiling by the Semicross and Cross
- Chapter 4 Packing and Covering by the Semicross and Cross
- Chapter 5 Tiling by Triangles of Equal Areas
- Chapter 6 Tiling by Similar Triangles
- Chapter 7 Rédei's Theorem
- Epilog
- Appendix A Lattices
- Appendix B The Character Group and Exact Sequences
- Appendix C Formal Sums
- Appendix D Cyclotomic Polynomials
- Bibliography for Preface
- Supplement to the Bibliography
- Name Index
- Subject Index
- Symbol Index
Summary
In Chapter 1 we were concerned with the way translates of a single cube fit together to tile space. In this chapter we examine tilings by translates of a finite collection of cubes, which we will call “clusters.” Chapters 3 and 4 will treat a special family of clusters that exists in all dimensions. Before we can state the main results of this chapter, we need some definitions.
As in Chapter 1, we assume a fixed coordinate system. We continue to identify each unit cube whose edges are parallel to the axes with its vertex that has the smallest coordinates. An n-dimensional cluster C is the finite union of unit cubes whose edges are parallel to the axes and which have integer coordinates. A cluster is not necessarily connected
Let C be a fixed cluster in n-space and assume that L is a set of vectors in n-space such that the set of translates {ν + C:ν ∈ L} tile n-space. (For a given cluster there may be no such lattice.) If all the coordinates of all the vectors in L are integers (rational numbers), we speak of an integer tiling (rational tiling) by C, or simply a Z-tiling (Q-tiling). If L is a lattice we speak of lattice tiling by C. Combining the two notions, we speak of a Z-lattice tiling and a Q-lattice tiling.
We will prove the following theorems, all of which concern tilings by translates of a cluster.
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- Algebra and TilingHomomorphisms in the Service of Geometry, pp. 35 - 56Publisher: Mathematical Association of AmericaPrint publication year: 2009