6. Given that a regular tetrahedron can accommodate 10 small spheres with a radius of 1, the minimum edge length of the regular tetrahedron is $\qquad$
As shown in the figure, the edge length of the regular tetrahedron $S-ABC$ is 4. With its vertices and the midpoints of the edges as centers, a total of 10 small spheres with a radius of 1 can be made. $P-XYZ$ is a regular tetrahedron that is 1 unit away from each face of $S-ABC$. Noting that sphere $S$ is tangent to the three corresponding side faces of vertex $P$, we have $PS=3r=3$. Therefore, the height $h$ of the regular tetrahedron $P-XYZ$ is $4+\frac{4 \sqrt{6}}{3}$. Thus, the edge length $a$ of the regular tetrahedron $P-XYZ$ is $\frac{3}{\sqrt{2}} h=4+2 \sqrt{6}$.