Question

Show that five-fold rotational symmetry is not possible in a 2-D lattice

Answer 1

Since we deal with flat 2D lattice, we need to ask a question: what regular polygons can tile the 2D plane? These figures are triangles, some quadrilaterals and hexagons. Why? Because we the interior angle of a polygon with $n$ sides is

$$\theta=180^\circ-(360/n).$$

How to tile a 2D plane without holes? By following this condition:

$$m\theta=360^\circ,$$

which means that theta must divide 360 degrees into equal m integer parts. Solve these two equations and express $m(n)$:

$$m(n)=\frac{2n}{n-2}.$$

Only n equal to 3, 4 and 6 comply with this condition. That is why five-fold rotational symmetry is impossible in 2D.