Geometrical meaning of automorphisms of cyclic groups I'm looking for a geometrical interpretation of the action of automorphisms of cyclic groups. I'll take one particular example to make it clear :
I'm taking the cyclic group $\mathbb{Z}_{12}$, which has $\mathbb{Z}_2 \times \mathbb{Z}_2$ as it automorphism group. Considering the integers modulo 12, the automorphisms are the multiplications by 1, 5, 7 or 11.
I'm considering now the action of $\mathbb{Z}_{12}$ on a discretized circle of 12 points in a plane (an dodecagon). The action of multiplying by 11 (left figure below) corresponds to a geometrical reflection in the plane, and I recover the dihedral group of symmetries of the dodecagon.

However, multiplication by 5 or 7 sends shapes drawn on this circle to different shapes, and I find no obvious geometrical meaning to these. Is there any ? (For example in the right figure above, the action of multiplying by 5 on the purple triangle gives the green triangle, with no obvious relation between them) 
Since these automorphisms can be considered as affine transformations, and affine spaces can be considered as subspaces of projective spaces, I'm considering working in the projective line over the field of 13 elements but for now I'm getting nowhere...

Edit:
This post was edited as my original message was dealing with $\mathbb{Z}_8$. Interestingly, although $\mathbb{Z}_8$ has the same automorphism group, all multiplications correspond to geometrical reflections in the plane....
 A: You can't say that you are considering THE action of $\mathbb Z_8$ on the circle  because there are FOUR such faithful actions and three unfaithful actions. For the faithful ones, the generator, 1, can either be rotation by $\frac{2\pi}8=45^\circ$, rotation by $3\cdot\frac{2\pi}8=135^\circ$, rotation by $5\cdot\frac{2\pi}8=225^\circ$, or rotation by $7\cdot\frac{2\pi}8=315^\circ$. For the unfaithful ones, it can be rotation by $90^\circ$, $270^\circ$ or $180^\circ$. 
The first geometric interpretation of the four automorphisms of $\mathbb Z_8$ is that they allow you to pass from one choice of faithful action to another choice of faithful action. In general, the endomorphisms of $\mathbb Z_8$ (which are given by the monoid of $\mathbb Z_8$ considered multiplicatively) allow one to pass from one choice of action to another choice of action. This probably does not seem terribly satisfactory, so below is a second geometric interpretation. 
Each of the endomorphisms of $\mathbb Z_8$ corresponds (irrelevant of faithful action) to an endomorphism of the circle group, and endomorphisms of the circle group can be thought of as winding the plane around itself $x$ times.
So if you consider the set of vertices of your octagon, the geometric interpretation of an endomorphism $x\in\mathbb Z_8$ is that it acts on the vertices by winding the plane around itself $x$ times. This is why you the octagon becomes a star, one is the winding of the other.
The reason why the octagon is not preserved (but its set of vertices is) is that winding around actually changes the angles (e.g. $45^\circ$ become $3\cdot45^\circ=135^\circ$) and so distances must also change.
A: You can see the automorphisms of $Z_{12}$ as nice geometric transformations if you are willing to treat them separately.
-The identity (multiplication by 1) is always "geometric".
-Multiplication by -1 (or 11) is geometric in that it is realized by a reflection in the dodecahedron in your original post: the reflection fixes 0 and 6 = -6 (mod 12).
-Multiplication by 5 can be realized as follows.  First count the fixed points.  In other words solve $5x = x$ or $4x = 0$ in $\mathbb{Z}_12$.  The solutions are 0, 3, 6, and 9.  Now, view the group as the vertices of a rectangular hexagonal prism.  Let $P$ be the prism, and let $H_b$ and $H_t$ be the bottom and top hexagons.  Label the bottom vertices 0, 2, 4, 6, 8, 10 (cyclically) and label the top vertices 3, 5, 7, 9, 11, 1 so that 0 is joined to 3 by a vertical edge of the prism, 2 is joined to 5, etc.  The underlying graph is almost the Cayley graph of $Z_{12}$ with respect to the (additive) generating set $\{2,3\}$; add the additional edges for the generator 3 to complete the Cayley graph. Multiplication by 5 is realized by a reflection.  The fixed points are 0, 3, 6, and 9.
-Multiplication by 7 is realized in the same fashion.  The fixed points are the solutions to $6x = 0$, i.e. 0, 2, 4, 6, 8, and 10.  This time consider the Cayley graph with respect to the generating set $\{3, 4\}$.  There is an nice reflection of this graph with the desired fixed points.  (You should see three squares and 4 triangles; it can be nicely visualized on a torus, for instance.)
A: 
An automorphism for a cyclic group needs to take a generator onto a generator.
  An automorphism is also part of a symmetry group which for the bounded plane, can only be the identity, a rotation, or a reflection.  

By applying these definition, we obtain a straightforward geometric visualization of $Aut[\mathbb Z_{12}]$.  
Start with your $t^1$ generator.


*

*Our first symmetry is identity or mapping everyone onto itself.  This is the trivial $1$-automorph which takes $t^1$ to $t^1$.  Alternatively, this can be seen as a $360^o$ rotation.

*Now, fix a reflection line from $t^0$ to $t^6$.  This reflects $t^1$ unto $t^{11}$.  Hence the $11$-automorph corresponds to a reflection along the vertical line.

*Next, fix a reflection line from $t^3$ to $t^9$.  This reflects $t^1$ unto $t^5$.  Hence the $5$-automorph corresponds to a reflection along the horizontal line.  

*Finally, rotate by $180^o$. This takes $t^1$ unto $t^7$.  One can alternatively view this as a compound transformation by first reflecting on the vertical (horizontal) and then reflecting on the horizontal (vertical).  Either way, this corresponds to the $7$-automorph.
To see this more clearly, map out the generators on your clockface and connect that dots.  You will have a rectangle joining $t^1$, $t^5$, $t^7$, and $t^{11}$.  Now you can clearly see that the v-reflection, h-reflection, $\pi$-rotation, and $2\pi$-rotation are the only transformations that preserve the rectangle, the only ways to map generators onto generators.

$11$-automorph defines inverses

Note that that the v-reflection, the 11-automorph,  defines the inverses of the elements of the group $\mathbb Z_{12} $:
\begin{matrix}  
 t^{11} = t^{-1}\\
 t^7 = t^{-5}  \\
t^{11} \not=  t^{-5} \not= t^{-7} \\
t^{7} \not= t^{-1} \not= t^{-11}\\
\end{matrix}  
We can verify this by considering the automorphisms of the $\mathbb Z_{12}$ subgroups, which should be invariant under the parent group automorphism.  
Subgroups $\mathbb Z_{1}$ and $\mathbb Z_{2}$ have trivial $Aut[\mathbb Z_1]$.  
Subgroups $\mathbb Z_3, \mathbb Z_4,$ and $\mathbb Z_6$ have non-trivial $Aut[\mathbb Z_2]$.  
Now map $Aut[\mathbb Z_{12}]$ onto the non-trivial subgroups and notice that you will always see the $1$-automorph opposite the $11$-automorph and the $5$-automorph opposite the $7$- automorph.  You will never see the $1$- and $11$- or the $5$- and $7$- automorphs mapped onto the same the subgroup automorph.  
This mapping also gives insight into what it means to be have trivial $Aut[\mathbb Z_1]$.  Do the mapping procedure on subgroups $\mathbb Z_1$ and $\mathbb Z_2$ and you'll note that in both cases, $Aut[\mathbb Z_{12}]$ maps to the same generator.

Additive and multiplicative interpretation of $Aut[\mathbb Z_{12}]$

Another observation is that interpreting $Aut[\mathbb Z ]$  as an additive group gives you the generators while interpreting it as a multiplicative group gives you the permutation symmetries.  I'm not sure about the wording of the latter but as an example, in the case of $\mathbb Z_{12} $ the multiplicative interpretation means that the ${1,5,7,}$ and $11$-automorphs correspond to:
\begin{matrix}  
x \rightarrow x \\  
x \rightarrow 5x  \\
x \rightarrow 7x  \\
x \rightarrow 11x \\
\end{matrix}  

Finally, all of this is a refection (pun intended) of the fact that
  \begin{matrix}
Aut[\mathbb Z_{12} ] = Aut[\mathbb Z_{8} ]= \mathbb Z_2 \otimes \mathbb Z_2 \end{matrix}
  which is the Klein 4-Group with two reflections ($\mathbb Z_2$) plus two rotations ($\mathbb Z_2$)

