Non-orientable mechanics

Topology is a vast area in mathematics, with many unexplored aspects of how it can be applied in physics. In this thesis, we hope to address the question of “How can topology be utilized to enhance our understanding of a specific condensed matter system?” by utilizing metamaterials as our tool. Metamaterials offer significant design flexibility, enabling the realization of systems that are challenging to achieve in other condensed matter systems. Metamaterials also enable us to present outcomes in a more intuitive manner, accessible to audiences across diverse fields. In this thesis, we introduced a new concept --- non-orientable order, by using mechanical metamaterials. We discussed the connections between the non-orientable order and other existing topological fields, like topological defects and topological band theory. We demonstrated that the non-orientable order can lead to novel mechanical properties. Furthermore, we showed that this concept can be extended beyond mechanics.