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Symmetry was not actually invented. It was just defined after the observation of similarly figured and shaped objects in nature.
Symmetry was not invented by man in the same way electricity, air, the concept of two, the ocean, a spiral, a circle, or movement were not invented by man. It is an observation of a natural (and man-made) pattern often seen. Since it's an observation (a description, if you will) it was not invented. As to a reasoning behind the observation, there most likely was none; it was simply observed and noted at some point in history.
Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection
line symmetry, rotational symmetry, mirror symmetry &liner symmetry
Asymmetry, Radial Symmetry, and Bilateral symmetry.