User:Anghel/Sandbox

Proof
Let $r \in \R_{>0}$ be the side length of the regular hexagons.

For all $x, y \in \Z$, let the center of each hexagon have Cartesian coordinates:


 * $r \tuple{ 3 x, \sqrt 3 y}$

If we label the vertices of each hexagon as $V_1, V_2, \ldots, V_6$ in anticlockwise direction with $V_1$ as the lowest left vertex, their coordinates are:

We show that $V_1, \ldots, V_6$ define the vertices of aregular hexagon.

Definition:Vector Length/Real Vector Space Distance Formula Angle Between Vectors in Terms of Dot Product

qed}}