The Creation of Woven Tiles from the Union of Delaunay Lofts

Abstract

In this work, we build upon the previously established foundations of Delaunay Lofts, which are a new class of shapes inspired by biological phenomena known as scutoids. Delaunay Lofts naturally provides edge collapse and vertex split processes, which make them highly complex in terms of topological structure. Delaunay lofts are obtained by layer-by-layer Voronoi decomposition of points. Therefore, any cross-section of a Delaunay Loft is always a convex polygon. Moreover, the only topology change from one layer to another is the change in the number of sides of cross-section polygons. Through our methodology, we obtain woven tiles as the union of Delaunay Lofts. This extension provides two advantages over the original Delaunay Lofts that can lead to woven tiles: (1) Each cross-section polygon can be non-convex, and (2) we can connect and disconnect polygons. To create woven tiles, we use unions of a linear pattern of points that can produce patterns similar to woven textiles. We have only investigated two widely used woven patterns: plain and twill. Plain woven tiles provide an approach to construct generalized Abeille’s flat vaults, which was introduced by French engineer and architect Joseph Abeille. Twill patterns provide even further generalizations of Abeille’s vaults, thus demonstrating the existence of a general class of woven tiles. By taking the union of several individual Delaunay Lofts, woven tiles appear in the resulting structure. Using this method, we are able to create non-convex polygons, as well as connecting and disconnecting the polygons in any manner we desire. Additionally, we can design the resulting shapes on each layer since we have control of the Voronoi Site locations. In developing this approach, we created a procedural framework that allows for the manipulation of rules and guidelines for the creation of the Delaunay Loft. As a result, we are able to generate a wide variety of Woven Structures based on Delaunay Lofts from tweaking various parameters. Using this method, we 3D printed some Woven Structures in order to help analyze symmetrical, structural, and tiling properties.