Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this article, we study counting problems that result from focusing on properties of the square torus one by one. After drawing insights from experimental evidence, we consider the implications between these properties and as well as the frequencies of these properties in each stratum of translation surfaces.

中文翻译：

---

正方形曲面的统计：对称和短环

正方形平铺曲面是一类平移曲面，在几何学和动力学方面特别受关注，因为作为正方形圆环的封面，它们共享一些简单性和结构。在本文中，我们研究了计数问题，这些问题是由一对一地专注于方形圆环的特性引起的。从实验证据中得出见解之后，我们考虑了这些特性之间的含义以及这些特性在平移表面每个层中的频率。