I'm working on problem 1-6 in Exercises for the Feynman Lectures and the question asks: Can you explain why there are no crystals that have the shape of a regular pentagon? (Triangles, squares, and hexagons are common in crystal forms).
This is what I have so far:
The interior angle of a polygon with equal $n$-equal sides can be given by:
$$\theta_n = 180° - \frac{(360)}{n}$$
The crystal structure must have equally spaced angles so $\theta_n $ must be an integral number of 360°. Therefore the integral factor $m$ is the number of times $\theta_n$ must be multiplied to span the the entire 360°.
$$360/m = 180 - \frac{360}{n}$$
$$ m = \frac{2n}{n-2}$$
From this it is easy to see that $n = 5$ would not give an integer however the only values of $n$ for $n \ge 3$ that seem to give an integer are 3, 4 and 6. I want to try to prove that this holds for any $n > 6$ so that there are no integer values of $m$ for $n > 6$.
