The culture in which a piece of art is formed will always have a profound impact on the meaning, execution, and perspective of that work. It is easy to see this in contemporary art: even if a work comes from a perspective we are not familiar with, we still have at least a minimal level of insight into its context because we live in the same world in which it was created.
Religion is one of the easiest cultural influences to pin down, as most societies in human history have had one form or another of recorded religious practice. In ancient Greek theater, religion played much the same impactful role as it did in the average person's daily experience, though it was of course altered for art's sake and to make for better entertainment.
The gods play direct roles in Greek plays, influencing events through their powers and interacting with characters, but also creating mischief and participating in plot themselves. Though I'm fairly certain Zeus was not actually impregnating countless Greek...
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