Abstract：
A brief introductory review is provided of the theory of tilings of 3-periodic nets and related periodic surfaces. Tilings have a transitivity [p q r s] indicating the vertex, edge, face and tile transitivity. Proper, natural and minimal-transitivity tilings of nets are described. Essential rings are used for finding the minimal-transitivity tiling for a given net. Tiling theory is used to find all edge- and face-transitive tilings (q = r = 1) and to find seven, one, one and 12 examples of tilings with transitivity [1 1 1 1], [1 1 1 2], [2 1 1 1] and [2 1 1 2], respectively. These are all minimal-transitivity tilings. This work identifies the 3-periodic surfaces defined by the nets of the tiling and its dual and indicates how 3-periodic nets arise from tilings of those surfaces.
摘要：
简要介绍了3周期网和相关周期曲面的平铺理论。倾斜具有指示顶点的传递性[pqrs]，边缘,脸和瓷砖传递性。适当,描述了网的自然和最小传递性平铺。基本环用于找到给定网络的最小传递性平铺。平铺理论用于查找所有边缘和面部过渡平铺（q=r=1），并找到七个，一,一个和12个具有传递性的平铺实例[1111]，[1112],[2111]和[2112]，分别。这些都是最小传递性平铺。这项工作确定了由平铺网及其对偶网定义的3周期表面，并指出了3周期网如何从这些表面的平铺中产生。
