Geometric or at least application view of a group with 3 elements? My teacher asked us to find applications in real life, or ways that a group with 3 elements migth show up in real problems, and the one I gave was about an watch for a planet where the entire day correspond to $3$ hours in the earth, and so the astronaut must create a watch that goes down to $0$ when it should reach $3$. Is is the group $Z_3$ with addition. However, turns out this is the only possible group, up to isomorphism, that has $3$ elements, so it's kinda hard to find another example to this exercise. Is there any geometric interpretation for a group with $3$ elements, or a real life problem that makes us think about groups of $3$ elements?
 A: This is a little bit 'constructed out of thin air', but I suppose many of such examples are:
In (western) music theory we divide one octave into 12 semitones: $\text{C,C#,D,D#,E,F,F#,G,G#,A,A#,B}$ (I'm only using sharps to keep it simple).
When we write a song, we at any given time usually work in some specific scale, for example $\text{C}$ major, $\text{G}$ minor, $\text{A#}$ mixolydian or whatever. Notice, how each scale has a special note called the tonic. ($\text{C}$ in the case of $\text{C}$ major)
Let's suppose for some reason we want to write a song in $\text{C}$ major, except we want to be able to change the tonic of our scale during the song according to certain rules: We may only go up or down a major third ($\text{M3}$, 4 semitones), except down a major third is (sort of) the same as up a minor sixth ($\text{m6}$, 8 semitones). We may also not change the tonic at all ($\text{P1}$). The possible scales we can have in our song are:
$\text{C}$ major, $\text{E}$ major, $\text{G#}$ major
This is an example of the group $Z_3 = \{0,1,2\}$ acting on itself. Let $S = \{\text{C, E, G#}\}$ and $I = \{\text{P1, M3, m6}\}$ with $\text{C}=\text{P1}= 0$, $\text{E} = \text{M3} = 1$, $\text{G#} = \text{m6} = 2$ and define $+ : I \times S \rightarrow S, (i,s)\mapsto i+s$. Of course we only gave new names to the elements of $Z_3$ but this is our idea:
Given an interval $i$ and our current scale $s$, we get a new scale $i+s$ by transposing $s$ by the interval $i$.
Note: $\text{P1,M3,m6}$ is notation for intervals used by musicians. Whether this musical notation makes a lot of sense is another story.
