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

By Sherman Stein, Sandor Szabó

Usually questions about tiling house or a polygon bring about questions touching on algebra. for example, tiling via cubes increases questions on finite abelian teams. Tiling by means of triangles of equivalent components quickly consists of Sperner's lemma from topology and valuations from algebra. the 1st six chapters of Algebra and Tiling shape a self-contained therapy of those subject matters, starting with Minkowski's conjecture approximately lattice tiling of Euclidean house by means of unit cubes, and concluding with Laczkowicz's contemporary paintings on tiling by way of comparable triangles. The concluding bankruptcy offers 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 ordinary higher point algebra classes, yet academics, researchers mathematicians will locate the e-book both beautiful.

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Sei. Hung. 16 (1965), 329-373. 18. R. M. Robinson, Multiple tilings of η-dimensional space by unit cubes, Math. Zeit. 166 (1979), 225-264. 34 ALGEBRA AND TILING 19. T. Schmidt, Uber die Zerlegung des n-dimensionalen Raumes in gitterformig angeordnete Würfeln, Sehr. math. Semin. u. Inst, angew. Math. Univ. Berlin 1 (1933), 186-212. 20. S. K. Stein, Algebraic tiling, Amer. Math. Monthly 81 (1974), 445-462. 21. S. Szabo, A reduction of Keller's conjecture, Periodica Math. Hung. 17 (1986), 265-277.

0 ) . , 0), being t h e difference of vectors in L, is also in L. In fact, it is in Κ since it is in t h e same cylinder as ( 0 , . . , 0 ) . T h u s y - z is an integer a n d therefore ( z , 1 , 0 , . . , 0) differs from (y, 1 , 0 , . . , 0) by an element of M. Consequently their components in Τ are identical. Moreover y = z + u, where u is an integer. T h e first coordinate of the translation of (y, 1 , 0 , . . , 0) is then 2 2 2 2 z + u + X2161 -I V x b, 2 2r r (1) a fact that will b e used in a m o m e n t .

This is a tiling problem in which we wish to tile a particular b o u n d e d region with congruent copies of a very simple cluster. In this case we allow rotations of the clusters. T h e solution illustrates one of the simpler algebraic techniques for analyzing tiling problems. 37 Cubical Clusters 1. Reductions T h e m e t h o d for altering a tiling to one with simpler translating vectors rests on a certain equivalence relation defined on t h e set of translation vectors. It turns out that the clusters that correspond to an equivalence class form a cylinder.

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