In this paper, we present a new approach to obtain symmetric tiles with curved edges. Our approach is based on using higher-order Voronoi sites that are closed under wallpaper symmetries. The resulting Voronoi tessellations provide us with symmetric tiles with curved edges. We have developed a web application that provides real-time tile design. Our application can be found at https://voronoi.viz.tamu.edu. One of our key findings in this paper is that not all symmetry operations are useful for creating curved tiles. In particular, all symmetries that use mirror operation produce straight lines that are useless for creating new tiles. This result is interesting because it suggests that we need to avoid mirror transformations to produce unusual space-filling tiles in 2D and 3D using Voronoi tessellations.

翻译：本文提出了一种获得具有曲线边界的对称瓷砖的新方法。该方法基于使用在墙面对称群下封闭的高阶Voronoi站点。由此产生的Voronoi剖分提供了具有曲线边界的对称瓷砖。我们开发了一个提供实时瓷砖设计的网络应用程序，访问地址为https://voronoi.viz.tamu.edu。本文的核心发现之一是：并非所有对称操作都适用于创建曲线瓷砖。特别地，所有包含镜像操作的对称都会产生无法用于创建新瓷砖的直线。这一结果颇具意义，因为它表明在利用Voronoi剖分生成二维及三维空间中的非常规空间填充砖时，需要避免使用镜像变换。