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

By Sherman Stein, Sandor Szabó

Frequently questions about tiling house or a polygon result in questions touching on algebra. for example, tiling through cubes increases questions about finite abelian teams. Tiling by way 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 remedy of those subject matters, starting with Minkowski's conjecture approximately lattice tiling of Euclidean area via unit cubes, and concluding with Laczkowicz's contemporary paintings on tiling by means of comparable triangles. The concluding bankruptcy provides a simplified model of Rédei's theorem on finite abelian teams. Algebra and Tiling is offered to undergraduate arithmetic majors, as lots of the instruments essential to learn the booklet are present in typical top point algebra classes, yet academics, researchers mathematicians will locate the ebook both attractive.

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T h e algebraic version of Minkowski's conjecture, on t h e other hand, claims that in each factorization of G by cyclic subsets there is at least one subgroup among the factors. T h e next two exercises show that in Hajos's version we may assume that the cyclic subsets have prime orders. Exercise 25. ,a - } 2 r 1 C = {e,a ,a ,... r 2r ,α^ ^} 1 of cardinalities r and s respectively. ,α" } 1 and C' - {e, a ,a ,... s 2a , a ^ s } and so the factorization is not always unique. Exercise 26. ) (a) Prove that Β cannot b e a subgroup.

M o r e specifically, if Sk is a translate of a fixed set T, we say that Τ packs (covers, tiles) S by translates. " Chapters 1, 2 , 3 , 5 , and 6 concern only tiling, while C h a p t e r 4 treats packing and covering. In Chapters 1, 2 , 3 , and 4 S is η-space, while in Chapters 5 and 6 it is a polygon. As we have just seen, Minkowski was led by an algebraic question to consider tilings of space by translates of a unit cube. W h a t does a lattice of translates of a unit cube in 3-space look like?

Ein Satz über die lückenlose Erfüllung des 5- und 6-dimensional Raumes mit Würfeln,/. Reine Angew. Math. 177 (1937), 61-64. 8. J. Ε Lagarias and Α. M. Odlyzko, Solving low-density subset sum problems, JACM 32 (1985), 229-246. 9. J. F. Lagarias and P. W. Shor, Keller's cube-tiling conjecture is false in high dimensions, Bull. Amer. Math. Soc. 27 (1992), 279-283. 10. A. K. Lenstra, H. W. Lenstra, and L. Loväsz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515-534. 11.

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