Twain use the mirror in this fable as it to allude to other fables and fairy tales.
What is fable?
Fable is a kind of short story in which typically animals are the characters which are conveying a moral.
This fable can be interpreted as a social and literary parable. In this parable, Twain uses the mirror to emphasize the importance of interpretation in texts. He depicts creatures that are supposed to be staring at a piece of art. They can only see themselves, though. Twain is referring to how readers understand a text in this way. Even if several people are gazing at the same piece of art, they will each see something distinct because they will see themselves in it.
Hence he uses it to allude to other fables and fairy tales.
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