Curved Space-Filling Tiles Using Voronoi Decomposition with Line, and Curve Segments Closed Under Wallpaper Symmetries

TL;DR

This paper introduces a method for creating symmetric, curved-edge tiles using Voronoi decomposition with higher-order sites, emphasizing the importance of avoiding mirror symmetries to generate novel space-filling patterns.

Contribution

The paper presents a new Voronoi-based approach for designing curved, symmetric tiles that are closed under wallpaper symmetries, along with a real-time web application for tile creation.

Findings

Mirror symmetries produce straight lines, not useful for curved tiles

Avoiding mirror operations leads to more innovative space-filling tiles

Voronoi tessellations can generate symmetric curved tiles

Abstract

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.