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In this paper, we introduce a geometric design and fabrication framework for a family of interlocking space-filling shapes which we call bi-axial woven tiles. Our framework is based on a unique combination 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

Bi-Axial Woven Tiles: Interlocking Space-Filling Shapes Based on Symmetries of Bi-Axial Weaving Patterns