Bi-Axial Woven Tiles: Interlocking Space-Filling Shapes Based on Symmetries of Bi-Axial Weaving Patterns Vinayak R. Krishnamurthy * J. Mike Walker ’66 Department of Mechanical Engineering Texas A&M University Ergun Akleman † Departments of Visualization & Computer Science and Eng., Texas A&M University Sai Ganesh Subramanian ‡ J. Mike Walker ’66 Department of Mechanical Engineering Texas A&M University Katherine Boyd § Department of Visualization Texas A&M University Chia-An Fu ¶ Department of Visualization Texas A&M University Matthew Ebert ‖ J. Mike Walker ’66 Department of Mechanical Engineering Texas A&M University Courtney Starrett ** Department of Visualization Texas A&M University Neeraj Yadav †† Department of Architecture Texas A&M University (a) A set of curve segments that are closed under sym- metry operations. The yel- low curve shows the ba- sic element of the repeating curve segment for this case. (b) The curve segments in a 2.5D fundamental domain, which is rectangular prism. (c) Voronoi decomposition of fundamental domain using curve segments as Voronoi sites. Yellow tile in the center is a space filling tile. (d) Assembly of space filling tiles by its replicas. The yel- low tile is removed to show the inner structure. (e) Physical assembly of 3D printed tiles in a different configuration, where flexi- ble dark green piece plays the role of locking this con- figuration. Figure 1: The computational pipeline for the geometric design and fabrication of woven tiles is shown. This particular example illustrates the tiles generated using the plain weave symmetries filling 2.5D space. The Figure 1c shows the curves in fundamental domain. The yellow curve shows the basic element of the repeating curve segment. All other curve segments in the fundamental domain can be obtained by rotating and translating this yellow curve. The Figure 1d shows overall assembly by removing the tile that corresponds to yellow curve. We obtained the shapes of top surfaces also with Voronoi decomposition. Abstract In this paper, we introduce a geometric design and fabrication frame- work for a family of interlocking space-filling shapes which we call bi-axial woven tiles. Our framework is based on a unique combina- tion of (1) Voronoi partitioning of space using curve segments as the Voronoi sites and (2) the design of these curve segments based on weave patterns closed under symmetry operations. The underlying weave geometry provides an interlocking property to the tiles and the closure property under symmetry operations ensure single tile can fill space. In order to demonstrate this general framework, we focus on specific symmetry operations induced by bi-axial weaving patterns. We specifically showcase the design and fabrication of woven tiles by using the most common 2-fold fabrics called 2-way genus-1 fabrics, namely, plain, twill, and satin weaves. * e-mail: vinayak@tamu.edu † e-mail: ergun.akleman@gmail.com ‡ e-mail: sai3097ganesh@tamu.edu § e-mail: katherineboyd@tamu.edu ¶ e-mail: sqree@tamu.edu ‖ e-mail: matt_ebert@tamu.edu ** e-mail: cstarrett@tamu.edu †† e-mail: nrj31y@tamu.edu Index Terms: Human-centered computing—Visualization—Visu- alization techniques—Treemaps; Human-centered computing— Visualization—Visualization design and evaluation methods 1 Introduction 1.1 Motivation Space-filling shapes have applications in a wide range of areas from chemistry and biology to engineering and architecture [48]. Using space-filling shapes, we can compose and decompose complicated shell and volume structures for design and architectural applications. Space-filling shapes that are also tileable, can be further provide an economical way for constructing structures because they can be mass-produced. Despite their practical importance, the variety of 2.5D and 3D space-filling tiles at our disposal are quite limited. The most commonly known and used space-filling shapes are usually reg- ular prisms such as rectangular bricks since they are relatively easy to manufacture and are widely available. However, reliance on regu- lar prisms, significantly constrains our design space for obtaining reliable and robust structures [16, 45, 58, 70, 71], particularly when current additive manufacturing techniques are gradually becoming more affordable across engineering and construction domains. In this paper, we introduce a geometric design and fabrication frame- work for a new class of interlocking space-filling shapes which we call bi-axial woven tiles. Systematic design of modular, tileable and, simultaneously inter- locking building blocks is a challenging task. We find that there is Graphics Interface Conference 2020 28-29 May Copyright held by authors. Permission granted to CHCCS/SCDHM to publish in print and digital form, and ACM to publish electronically.