Symmetry is responsible in part for aesthetics. As a negative example, an assymmetric face would usually be perceived as ugly.
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symmetry principles always tell us something important. They often provide the most valuable clues toward deciphering the underlying principles of the cosmos, whatever those may be. In this sense, therefore, symmetry is certainly fruitful. Whether or not some all-encompassing symmetry is the grand principle that will necessitate our "theory-of-everything" is still to be determined.
Line symmetry.
z does not have a line of symmetry. z does not have a line of symmetry. z does not have a line of symmetry. z does not have a line of symmetry.
360 degrees, but this assumes that there are any angles. There need not be any angles - as illustrated by MC Escher in his set of Symmetry artwork.
Bilateral symmetry.