20. Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, the set of points on the surface of the cube that are a distance of $\frac{2 \sqrt{3}}{3}$ from point $A$ forms a curve (this curve may not lie in a single plane). Then the length of this curve is $\qquad$
20. $\frac{5 \sqrt{3} \pi}{6}$.

This curve is distributed across six surfaces. The parts within the plane $A C$, plane $A B_{1}$, and plane $A D_{1}$ are all circular arcs with $A$ as the center, a radius of $\frac{2 \sqrt{3}}{3}$, and a central angle of $\frac{\pi}{6}$. The total length of these three segments is $\frac{\pi}{6} \times \frac{2 \sqrt{3}}{3} \times 3=\frac{\sqrt{3} \pi}{3}$;

The parts within the plane $A_{1} C_{1}$, plane $D C_{1}$, and plane $B C_{1}$ are all circular arcs with a radius of $\frac{\sqrt{3}}{3}$ and a central angle of $\frac{\pi}{2}$. The total length of these three segments is $\frac{\pi}{2} \times \frac{\sqrt{3}}{3} \times 3=\frac{\sqrt{3} \pi}{2}$.
The total length of this curve is $\frac{5 \sqrt{3} \pi}{6}$.