标题： 欧几里得平面的替代铺砌入门 显示英文标题

标题： A primer on substitution tilings of the Euclidean plane

摘要： 本文旨在介绍替代镶嵌理论。 对于我们的目的而言，镶嵌替代规则分为两大类：几何类和组合类。 几何类替代镶嵌包括自相似镶嵌，如著名的彭罗斯镶嵌；对于这一类，文献中已有大量研究成果。 组合类替代镶嵌刚刚开始被研究，我们在这里提出的一些内容是新的。 我们给出了许多例子，提及了部分重要结果，讨论了两类替代之间的联系，包含了当前的研究观点和问题，并提供了广泛的参考文献。 尽管作者试图公平地呈现整体情况，但本文并非全面的综述，对于可能遗漏的重要内容，作者表示歉意。 显示英文摘要

摘要： This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to fairly represent the as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions.