Non-periodic Tilings of R-n by Crosses

作者：Horak Peter^*; AlBdaiwi Bader

来源：Discrete & Computational Geometry, 2012, 47(1): 1-16.

DOI：10.1007/s00454-011-9373-5

• Tiling by n-cross
• Non-periodic tilings
• Enumeration of tilings

An n-dimensional cross consists of 2n+1 unit cubes: the "central" cube and reflections in all its faces. A tiling by crosses is called a Z-tiling if each cross is centered at a point with integer coordinates. Periodic tilings of a"e (n) by crosses have been constructed by several authors for all naN. No non-periodic tiling of a"e (n) by crosses has been found so far. We prove that if 2n+1 is not a prime, then the total number of non-periodic Z-tilings of a"e (n) by crosses is 2(N0) while the total number of periodic Z-tilings is only a"mu(0). In a sharp contrast to this result we show that any two tilings of a"e (n) ,n=2,3, by crosses are congruent. We conjecture that this is the case not only for n=2,3, but for all n where 2n+1 is a prime.

• 出版日期2012-1