Algebra and Tiling: Homomorphisms in the Service of Geometry by Sherman Stein, Sandor Szabó

By Sherman Stein, Sandor Szabó

Frequently questions on tiling house or a polygon bring about questions touching on algebra. for example, tiling via cubes increases questions about finite abelian teams. Tiling by means of triangles of equivalent components quickly includes Sperner's lemma from topology and valuations from algebra. the 1st six chapters of Algebra and Tiling shape a self-contained therapy of those issues, starting with Minkowski's conjecture approximately lattice tiling of Euclidean house by means of unit cubes, and concluding with Laczkowicz's fresh paintings on tiling by means of related triangles. The concluding bankruptcy provides a simplified model of Rédei's theorem on finite abelian teams. Algebra and Tiling is on the market to undergraduate arithmetic majors, as many of the instruments essential to learn the ebook are present in commonplace higher point algebra classes, yet lecturers, researchers mathematicians will locate the e-book both beautiful.

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Example text

A n 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 η-space and assume that L is a set of vectors in η-space such that the set of translates {v + C: ν e L] tile η-space. ) 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 Ζ-tiling (Q-tiling). If L is a lattice we speak of lattice tiling by C.

More about the algebraic version of Minkowski's conjecture Let G b e a n abelian group and let Αχ,... ,A„ be subsets of G. If each element g of G is uniquely expressible in the form 9 = αχ·α , η αχ £ Αχ,... , A a n d that the product Αχ·· A is & factorization of G. ) This concept is a natural extension of the concept of direct product of subgroups a n d we will write G = ΑχΑ · • • A . We will assume throughout that each Ai is normalized to contain t h e identity element of G. Let a b e an element of the abelian group G a n d r a positive integer.

If | - bi\ = 1, | a - 621 < 1,· · •, |a„ - 6 „ | < 1 we call the two cubes adjacent. Geometrically speaking, t h e two cubes are adjacent if their intersection is an (τι - l)-dimensional subset that is part of a face of b o t h cubes perpendicular to the first coordinate axis. Figures 1 (a) and (b) exhibit nonadjacent cubes in 2-space and Figures 1 (c) and (d) exhibit adjacent cubes in 2-space. η α ι n 2 Exercise 2. Verify that t h e geometrical and arithmetical descriptions of adjacency are equivalent.

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