Here's one way of thinking about the monodromy representation associated to the irreducible triple cover of **S**^{1}.
disclaimer
(You may have to rotate the picture a bit -- just click and drag --
until you can see what's happening.)
Abstractly, we know there ought to be a group homomorphism

&pi_{1}(*S*,*s*)-> Aut(*T*_{s})
The fundamental group of the circle is isomorphic to **Z**. In
fact, let *g* denote the loop corresponding to ``walk
counterclockwise once around the circle''; it's a generator of
&pi_{1}(*S*,*s*).
The fiber *T*_{s} consists of three points, which we'll arbitrarily
number. In this way, what we really have is a representation

**Z** -> Sym( 1
2
3 )
Dragging the slider up the screen shows that going around the circle
cyclically permutes the fibers; under this homomorphism
*g* -> (1
2
3)
and the image of the monodromy representation is the cyclic subgroup {(1
2
3), (1
3 2), id } of Sym( 1
2
3 ).
The observation that *going around the circle three times returns to the initial state* reflects the fact that the image of *g* has order three.

This is the picture behind some of my mathematics.

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