6. Draw 63 lines on the coordinate plane: $y=b, y=\sqrt{3} x + 2b, y=-\sqrt{3} x+2b$, where $b=-10,-9,-8, \ldots, 8,9,10$. These lines divide the plane into several equilateral triangles. The number of equilateral triangles with side length $\frac{2}{\sqrt{3}}$ is $\qquad$ $-$
6.660.

The six outermost straight lines determine a regular hexagon with a side length of $\frac{20}{\sqrt{3}}$. The three straight lines passing through the origin $O$ divide this hexagon into six equilateral triangles with a side length of $\frac{20}{\sqrt{3}}$. Since the side length of each large triangle is 10 times that of the small triangles, and each large triangle is divided into $10^{2}$ small triangles, the number of small triangles with a side length of $\frac{2}{\sqrt{3}}$ inside the hexagon is $6 \times 10^{2}=600$ (units). Additionally, there are 10 small triangles adjacent to each side of the hexagon (as shown in Figure 3). Therefore, the total number of equilateral triangles with a side length of $\frac{2}{\sqrt{3}}$ is $6 \times 10^{2}+6 \times 10=660$.