  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.
