Auxetic Dihedral Escher Tessellations

The auxetic structure demonstrates an unconventional deployable mechanism, expanding in transverse directions while being stretched longitudinally (exhibiting a negative Poisson's ratio). This characteristic offers advantages in diverse fields such as structural engineering, flexible electronics, and medicine. The rotating (semi-)rigid structure, as a typical auxetic structure, has been introduced into the field of computer-aided design because of its well-defined motion patterns. These structures find application as deployable structures in various endeavors aiming to approximate and rapidly fabricate doubly-curved surfaces, thereby mitigating the challenges associated with their production and transportation. Nevertheless, prior designs relying on basic geometric elements primarily concentrate on exploring the inherent nature of the structure and often lack aesthetic appeal. To address this limitation, we propose a novel design and generation method inspired by dihedral Escher tessellations. By introducing a new metric function, we achieve efficient evaluation of shape deployability as well as filtering of tessellations, followed by a two-step deformation and edge-deployability optimization process to ensure compliance with deployability constraints while preserving semantic meanings. Furthermore, we optimize the shape through physical simulation to guarantee deployability in actual manufacturing and control Poisson's ratio to a certain extent. Our method yields structures that are both semantically meaningful and aesthetically pleasing, showcasing promising potential for auxetic applications.

We thank all the anonymous reviewers for their valuable comments and constructive suggestions. This work is supported by the grant No.61972232 from National Nat- ural Science Foundation of China (NSFC) and the Key Research and Development Plan of Shandong Province of China (No.2020ZLYS01).