By Claus M. Ringel

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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.