Could someone explain why the alpha angle in the chiral angle proof below is 120 degrees? Here's the question: Nanotube chiral angle as a function of $n$ and $m$
Can someone explain why the alpha angle in the chiral angle proof below is 120 degrees?
 A: The angle 120 degrees is calculated as
$$ \frac{360^\circ}{3} = 120^\circ$$
because the full angle 360 degrees in the hexagon is divided to three equal parts. One may draw the "Mercedes logo" triplet of arms into the hexagon network in several ways – one of the arms may be vertical; or one of them may be horizontal, and so on – and one of these ways explains why the angle is 120 degrees.
A: The internal angles of a regular hexagon are all $120^{\circ}$. In the picture below I have tried to make it clear why the angle in the linked question is therefore also $120^{\circ}$ in a different way than Luboš Motl.

The angle marked in red corresponds to the angle $\alpha$ you are asking about. Now, since all internal angles of a regular hexagon are $120^{\circ}$, the green angle is also $120^{\circ}$. Another way of seeing this is Luboš' method: recognize the Mercedes logo, it divides the $360^{\circ}$ of one full turn into three equal parts.
Since the angles of a triangle add up to $180^{\circ}$ (half a turn) and the angles in $D$ and $G$ in the triangle $D\hat{E}G$ are the same, both of them must be $30^{\circ}$. As you can see, the red angle minus the blue one is exactly $90^{\circ}$, so we can conclude that the red angle ($\alpha$) is equal to $120^{\circ}$.
Another way to see it (closely related to Luboš' answer and how I see it) is to realize that you can transform the internal hexagon angle in $D$ into the red angle $\alpha$ by just rotating the entire hexagon, meaning both angles ($\alpha$ and the internal hexagon angle) are the same.
