Aperiodic tilings
Aperiodic tilings
复制标题
非周期性瓷砖
DOI：
10.1007/s002220050153
复制
发表时间：
1997
影响因子：
3.1
通讯作者：
Shahar Mozes
中科院分区：
文献类型：
--
作者：
Shahar Mozes
设X是一个黎曼流形。一个X-平铺系统，或者简单地说，一个理解X的平铺系统，是X的彩色紧实子集j的有限集合（称为tiles），以及一组有限的规则，这些规则控制着相邻瓷砖的有限（到有限大小）配置是允许的（“允许的”）。我们只考虑与X中的球同胚的瓦片C。A瓦片co是X的分割（镶嵌）形式为X= U Cg，其中每个Cg是集合j中某些瓦片的等距复制，因此任意两个瓦片Ci的内部是不相交的。每个Ci将以这样一种方式着色，即在平铺图中出现的所有有限构型都是可接受的。如果一个平铺是通过紧商流形M= F\X（其中F< Is (X)是一个无扭转的均匀晶格）的平铺来分解的，那么这个平铺就是周期的。如果一个平铺是通过一个固有商流形M= Z\X(其中Z< Is （X)是Is (X)的任意非平凡离散无扭转子群）的平铺来因式分解，则称为弱周期平铺。
Let X be a Riemannian manifold. An X-tiling system, or simply a tiling system where X is understood, is a finite collection J-of colored compact subsets of X called tiles together with a finite set of rules controlling which finite (up to a bounded size) configurations of adjacent tiles are allowed (" admissible"). We shall consider only tiles C which are homeomorphic to balls in X. A tiling co is a partition (tessellation) of X of the form X= U Cg where each Cg is an isometric copy of some tile of the set J-so that the interior of any two of these tiles Ci are disjoint. Each of the Ci's will be colored in such a way that all the finite configurations appearing in the tiling co are admissible. A tiling will be called periodic if it factors via a tiling of a compact quotient manifold M= F\X of X (where F< Is (X) is a torsion free uniform lattice). A tiling will be called weakly periodic if it factors via a tiling of a proper quotient manifold M= Z\X of X (where Z< Is (X) is any nontrivial discrete torsion free subgroup of Is (X)).