21 In a regular tetrahedron wooden block $ABCD$ with edge length 2, there is a point $P (AP<1)$ on the edge $AB$. A cross-section perpendicular to the edge $AB$ is to be sawed through point $P$. When the sawing stops at a certain position, it is measured that the saw cut $PM=1$ on the face $ABD$. The saw cut $PN=\frac{2}{3}$ on the face $ABC$, then $MN=$
21 1. Let the dihedral angle of a regular tetrahedron be $\alpha$, then
$$
\cos \alpha=\frac{\left(\frac{\sqrt{3}}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}-1}{2 \times \frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2}}=\frac{1}{3}
$$

Therefore,
$$
M N=\sqrt{1^{2}+\left(\frac{2}{3}\right)^{2}-2 \times 1 \times \frac{2}{3} \times \frac{1}{3}}=1
$$