# mathr

## Rectangles on a triangular lattice

Starting from this post I'll be using MathJax to display prettier equations. It does require Javascript, but it's much less awkward than the alternatives I've considered.

I was wondering about drawing rectangles on a regular triangular lattice. I found that the area is always a multiple of 4 cells.

Let $$\mathbf{u},\mathbf{v}$$ be the basis vectors of the triangular lattice, and $$\mathbf{x},\mathbf{y}$$ be the basis vectors of rectangular Cartesian coordinates. Specify a point on the lattice by $$p,q \in \mathbb{Z}$$, then:

$p \mathbf{u} + q \mathbf{v} = (p + \frac{q}{2}) \mathbf{x} + \frac{\sqrt{3}}{2} q \mathbf{y}$

with the squared Euclidean distance of the point from the origin being:

$\left|\left| p \mathbf{u} + q \mathbf{v} \right|\right| ^ 2= p^2 + q^2 + pq$

Now assume $$\gcd(p,q) = 1$$, to ensure that the line segment from the origin to the point passes through no other lattice points. A line $$(P,Q)$$ perpendicular to $$(p,q)$$ satisfies:

$(p + \frac{q}{2}) \mathbf{x} + \frac{\sqrt{3}}{2} q \mathbf{y} = \lambda ( -\frac{\sqrt{3}}{2} Q \mathbf{x} + (P + \frac{Q}{2}) \mathbf{y} )$

which reduces to

$\frac{P}{Q} = - \frac{p + 2 q}{2 p + q}$

Letting $$K = \gcd(p + 2 q, 2 p + q)$$, then $$P = -\frac{p + 2 q}{K}$$ and $$Q = \frac{2 p + q}{K}$$ and $$\gcd(P,Q) = 1$$.

The area of the rectangle can be found in terms of $$p$$ and $$q$$:

$A^2 = (p^2 + q^2 + pq) (P^2 + Q^2 + PQ)$

which reduces to

$A = \sqrt{3} \frac{p^2 + q^2 + pq}{K}$

Expressing this in terms of the smallest triangular cell on the lattice, which has area $$\frac{\sqrt{3}}{4}$$, gives a count of triangles:

$N = 4 \frac{p^2 + q^2 + pq}{K}$

Recall that $$\gcd(p,q) = 1$$ and $$\gcd(p + 2 q, 2 p + q) = K$$. This means that $$K \mid p + 2 q$$ and $$K \mid 2 p + q$$ which implies that $$K \mid 3 (p + q)$$. So $$K \mid 3$$ or $$K \mid p + q$$. When $$K$$ doesn't divide $$p + q$$ then $$K \in \{1,3\}$$. Suppose $$K \mid p + q$$, then $$K \mid p + 2 q$$ implies that $$K \mid q$$, and similarly $$K \mid 2 p + q$$ implies that $$K \mid p$$. But $$\gcd(p,q) = 1$$, therefore $$K = 1$$.

$$K = 1$$ implies that $$4 \mid N$$, so it remains to consider the other case when $$K = 3$$. Let $$p = 3 a + b$$ and $$q = 3 c + d$$ with $$b,d \in \{0,1,2\}$$. Now $$K \mid b + 2 d$$ and $$K \mid 2 b + d$$. Exhaustive case analysis shows that this holds only when $$b = d$$. Expanding

$p^2 + q^2 + pq = 3(\ldots) + b^2 + d^2 + bd = 3(\ldots) + 3 b^2$

shows that $$3 \mid p^2 + q^2 + pq$$. So $$\frac{p^2 + q^2 + pq}{K}\ \in \mathbb{Z}$$ and $$4 \mid N$$: the area of a rectangle on a triangular lattice is always a multiple of 4 cells.