The symmetry of the whole experimental setups, including specific sample environments and measurables, can be compared with that of specimens for observable physical phenomena. We, first, focus on one-dimensional (1D) experimental setups, independent from any spatial rotation around one direction, and show that eight kinds of 1D objects (four; vector-like, the other four; director-like), defined in terms of symmetry, and their dot and cross products are an effective way for the symmetry consideration. The dot products form a Z₂ × Z₂ × Z₂ group with Abelian additive operation, and the cross products form a Z₂ × Z₂ group with Abelian additive operation or Q₈, a non-Abelian group of order eight, depending on their signs. Those 1D objects are associated with characteristic physical phenomena. When a 3D specimen has symmetry operational similarity (SOS) with (identical or lower, but not higher, symmetries than) an 1D object with a particular phenomenon, the 3D specimen can exhibit the phenomenon. This SOS approach can be a transformative and unconventional avenue for symmetry-guided materials designs and discoveries.

中文翻译：

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通过一维对象及其点/叉积连接涌现现象和对称性破缺

整个实验装置的对称性，包括特定的样本环境和可测量的，可以与可观察物理现象的标本进行比较。我们首先关注一维 (1D) 实验装置，独立于围绕一个方向的任何空间旋转，并展示八种一维对象（四种；矢量类，其他四种；导演类），定义在对称项及其点积和叉积是考虑对称性的有效方法。点积构成Z ₂ × Z ₂ × Z ₂群阿贝尔加法运算，叉积构成Z ₂ × Z ₂群阿贝尔加法运算或Q ₈，八阶非阿贝尔群，取决于它们的符号。这些一维对象与特征物理现象相关联。当 3D 样本与具有特定现象的 1D 对象（相同或更低但不高于对称性）具有对称操作相似性 (SOS) 时，3D 样本可以展示该现象。这种 SOS 方法可以成为对称引导材料设计和发现的变革性和非常规途径。