Emergent physical phenomena and broken symmetries can be linked through one-dimensional objects and their dot or cross products. Eight kinds of one-dimensional (1D) objects (four; vector-like, the other four; director-like) and their dot/cross products are defined in terms of symmetry. The dot or products form certain mathematical groups. Those 1D objects are associated with characteristic physical phenomena, and when a 3D system has identical or lower (but not higher) symmetries than an 1D object with particular phenomena, the 3D system can exhibit the phenomena. Using this straightforward concept, we can understand and also predict numerous new emergent phenomena in known materials or new complex materials for desired phenomena.