Three Regular Tessellations/Hexagons

Theorem
Regular hexagons form a regular tessellation:


 * RegularHexagonTessellation.png

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

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


 * $Q_{x,y} = s \tuple {x, \sqrt 3 y + \dfrac {\sqrt 3 \paren {3 - m} } 4}$

where:
 * $m = \begin{cases} 1 & : \textrm {for $x$ even} \\ -1 & : \textrm{for $x$ odd} \end{cases}$

From Regular Hexagon is composed of Equilateral Triangles, it follows that each hexagon can be triangulated into six congruent equilateral triangles.

Also, each of the six equilateral triangles has side length $s$, and $Q_{x,y}$ as one of its vertices.

By Three Regular Tessellations:Triangles, there exists a regular tessellation of equilateral triangles with side length s.

Set:


 * $i := 3 x$
 * $j := 2 y + \dfrac {1-m} 2$

The hexagon center $Q_{x,y}$ corresponds to the six vertices:


 * $A_{i+1, j}, A_{i+1, j+1}, B_{i-1,j}, B_{i-1,j+1}, C_{i,j}, C_{i,j+1} $

as they are labelled in the proof of Three Regular Tessellations:Triangles.

The six triangles in the triangulation of the hexagon correspond to the triangles of the regular tessellation:


 * $\triangle A_{i+1, j} B_{i+1, j} C_{i+1, j}$
 * $\triangle A_{i+1, j+1} B_{i+1, j+1} C_{i+1, j+1} $
 * $\triangle A_{i-1, j} B_{i-1, j} C_{i-1, j} $
 * $\triangle A_{i-1, j+1} B_{i-1, j+1} C_{i-1, j+1} $
 * $\triangle A_{i, j} B_{i, j} C_{i, j}$
 * $\triangle A_{i, j+1} B_{i, j+1} C_{i, j+1} $

as they are labelled in the proof of Three Regular Tessellations:Triangles.

For all $x' \in \Z$, there exist exactly one $x \in \Z$ and one $k \in \set {-1, 0, 1}$ such that:


 * $x' = 3 x + k$

For all $y' \in \Z$, there exist exactly one $y \in \Z$ and one $k \in \set {0, 1}$ such that:


 * $y' = 2 y + k$

which shows that each $\triangle A_{x', y'} B_{x', y'} C_{x', y'}$ is part of exactly one of the hexagons.

As the triangles form a regular tessellation of the plane, it follows that the hexagons also form a regular tessellation of the plane.