Hyde S T, Ramsden S J, Robins V
Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, Australia.
Acta Crystallogr A Found Adv. 2014 Jul;70(Pt 4):319-37. doi: 10.1107/S205327331400549X. Epub 2014 May 28.
The concept of an orbifold is particularly suited to classification and enumeration of crystalline groups in the euclidean (flat) plane and its elliptic and hyperbolic counterparts. Using Conway's orbifold naming scheme, this article explicates conventional point, frieze and plane groups, and describes the advantages of the orbifold approach, which relies on simple rules for calculating the orbifold topology. The article proposes a simple taxonomy of orbifolds into seven classes, distinguished by their underlying topological connectedness, boundedness and orientability. Simpler `crystallographic hyperbolic groups' are listed, namely groups that result from hyperbolic sponge-like sections through three-dimensional euclidean space related to all known genus-three triply periodic minimal surfaces (i.e. the P, D, Gyroid, CLP and H surfaces) as well as the genus-four I-WP surface.
orbifold的概念特别适合对欧几里得（平坦）平面及其椭圆和双曲对应平面中的晶体群进行分类和枚举。本文使用康威的orbifold命名方案，阐述了传统的点群、带饰群和平面对称群，并描述了orbifold方法的优点，该方法依赖于计算orbifold拓扑的简单规则。本文提出了一种将orbifold简单分类为七类的方法，这些类别通过其底层拓扑连通性、有界性和可定向性来区分。列出了更简单的“晶体学双曲群”，即通过与所有已知的 genus-three 三重周期极小曲面（即P、D、螺旋面、CLP和H曲面）以及genus-four I-WP曲面相关的三维欧几里得空间中的双曲海绵状截面得到的群。
