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

PNAS Nexus, vol. 4, pp. pgaf219

Abstract

Abstract Woven fabrics have a long history of study across the fields of art, mathematics, and mechanics. While weaves with symmetries in R2 have been extensively formalized, classified, and characterized, a systematic framework for representing and designing weaves in R3 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 R3. 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.

Authors 4

1. Tolga Yildiz

   Texas A&M University

   Affiliation as printed

   Department of Computer Science and Engineering, Texas A&M University , College Station, TX 77840 ,

2. Alice C. Niemeyer Aachen Chair of Algebra and Representation Theory

   RWTH Aachen University

   Affiliation as printed

   Chair of Algebra and Representation Theory, RWTH Aachen University , Pontdriesch 10-16, 52062 Aachen ,

3. Vinayak Raman Krishnamurthy

   Texas A&M University

   Affiliation as printed

   Department of Computer Science and Engineering, Texas A&M University , College Station, TX 77840 ,

   J. Mike Walker’66 Department of Mechanical Engineering, Texas A&M University , College Station, TX 77840 ,

4. Ergün Akleman

   Texas A&M University

   Affiliation as printed

   Department of Computer Science and Engineering, Texas A&M University , College Station, TX 77840 ,

   Visual Computing and Computational Media Section, School of Performance, Visualization and Fine Arts, Texas A&M University , College Station, TX 77840 ,