Example 4 Draw the following lines on the coordinate plane $y=k, y=\sqrt{3} x+2 k, y=-\sqrt{3} x+2 k$, where $k=$ $0, \pm 1, \pm 2, \cdots, \pm 10$. These 63 lines can divide the plane into several equilateral triangles. Find the number of equilateral triangles with side length $\frac{2}{\sqrt{3}}$.
(1994 12th American Invitational Mathematics Examination Problem)
Solve: The outermost straight lines form a regular hexagon with a side length of $\frac{20}{\sqrt{3}}$. The three lines passing through the origin, $y=0, y=\sqrt{3} x, y=-\sqrt{3} x$, divide this hexagon into 6 equilateral triangles with a side length of $\frac{20}{\sqrt{3}}$. Since the side length of each such large equilateral triangle is 10 times the side length of the discussed small triangles, each large triangle can be divided into 100 small triangles, thus the hexagon is divided into 600 equilateral triangles with a side length of $\frac{2}{\sqrt{3}}$. Additionally, there are 60 small equilateral triangles (with a side length of $\frac{2}{\sqrt{3}}$) located outside the hexagon and having one side on the edge of the hexagon (because there are 10 such small triangles outside each side of the hexagon), so the total number of equilateral triangles with a side length of $\frac{2}{\sqrt{3}}$ is 660.