A constructive framework for discovery, design, and classification of volumetric Bravais weaves.

Woven fabrics have a long history of study across the fields of art, mathematics, and mechanics. While weaves with symmetries in $R^2$ have been extensively formalized, classified, and characterized, a systematic framework for representing and designing weaves in $R^3$ remains absent. Despite their relevance to engineering applications-particularly in composite materials-volumetric weaves are often designed in an ad hoc manner, typically by stacking planar weaves and introducing trivial thread connections along the stacking axis. In this article, we establish a formal framework for volumetric weaves by defining them through the isometries of Bravais lattices and their corresponding Voronoi cells in $R^3$ . This approach provides a structured description of the design space for a specific family of volumetric weaves, which we call volumetric Bravais weaves. As an example of volumetric Bravais weaves, we analyze, cubic primitive weaves (cP-weaves) in detail as the simplest example within the volumetric Bravais weave framework. This example demonstrates how volumetric Bravais weave structures naturally emerge from 3D lattice isometries. Furthermore, we show that all possible cP-weaves can be systematically generated using a set of cP lattice isometries and cube isometries. Our findings reveal that even just the space of cP-weaves is at least one order of magnitude larger than that of conventional two-way 2-fold weaves, highlighting the potential of our approach for expanding the design space of volumetric weaves.