模拟相邻拉链和拼接的多项式
Polyominoes Simulating Arbitrary-Neighborhood Zippers and Tilings
摘要：
本文提供了一个经典的瓦片理论和复杂的邻域自组装的情况下，在实践中存在的桥梁。平面上某个位置的邻域是与该位置相邻的坐标集，包括大小为4的经典邻域和任意复杂的邻域。一个广义的瓦片系统包括一组瓦片、一个邻域和一个关系，该关系规定了哪些是给定瓦片的“可接受的”相邻瓦片。因此，在正确形成的组件中，瓦片根据该关系被分配平面的位置。我们证明，在给定但任意的邻近区域（拉链）中定义的任何有效的瓦片路径都可以通过简单的微瓦片“带”来模拟。带状物是一种特殊的多角形，由平面上的非自交叉的瓷砖序列组成，其中连续的瓷砖沿着其相邻的边缘粘附。最后，我们将这种结构扩展到传统的平铺的情况下，证明了我们可以模拟任意邻域平铺的简单邻域平铺，同时保持其一些基本属性。
英文摘要：
This paper provides a bridge between the classical tiling theory and the complex neighborhood self-assembling situations that exist in practice. The neighborhood of a position in the plane is the set of coordinates which are considered adjacent to it. This includes classical neighborhoods of size four, as well as arbitrarily complex neighborhoods. A generalized tile system consists of a set of tiles, a neighborhood, and a relation which dictates which are the "admissible" neighboring tiles of a given tile. Thus, in correctly formed assemblies, tiles are assigned positions of the plane in accordance to this relation. We prove that any validly tiled path defined in a given but arbitrary neighborhood (a zipper) can be simulated by a simple "ribbon" of microtiles. A ribbon is a special kind of polyomino, consisting of a non-self-crossing sequence of tiles on the plane, in which successive tiles stick along their adjacent edge. Finally, we extend this construction to the case of traditional tilings, proving that we can simulate arbitrary-neighborhood tilings by simple-neighborhood tilings, while preserving some of their essential properties.
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