  Several discrete geometry problems are closely related to the arithmetic
theory of elliptic curves defined on the rational fields $\mathbb{Q}$. In this
paper we consider the $\theta$-congruent number for $\theta=\frac{\pi}{3}$ and
$\frac{2\pi}{3}$ and tiling number n. For the case that $n\geqslant 2$ is
square-free odd integer, we determine all $n$ such that the Selmer rank of
elliptic curve $E_{n,\frac{\pi}{3}}:\ y^2=x(x-n)(x+3n)$ or/and
$E_{n,\frac{2\pi}{3}}:\ y^2=x(x+n)(x-3n)$ is zero. From this, we provide
several series of non $\theta$-congruent numbers for $\theta=\frac{\pi}{3}$ and
$\frac{2\pi}{3}$, and non tiling numbers n with arbitrary many of prime
divisors.
