jmc

The volume of any pyramid is $\frac{1}{3}$ the product of the base area and the height. However, determining the height of the purple tetrahedron is somewhat tricky! Instead of doing that, we observe that the total volume of the cube consists of the purple tetrahedron and four other "clear" tetrahedra. Each clear tetrahedron is formed by one of the black vertices of the cube together with its three purple neighbors. The clear tetrahedra are convenient to work with because they have lots of right angles.

Each clear tetrahedron has an isosceles right triangular base of area $\frac{1}{2}\cdot 6\cdot 6=18$, with corresponding height $6$ (a side of the cube). Thus, each clear tetrahedron has volume $\frac{1}{3}\cdot 18\cdot 6=36$.

The cube has volume $6^3=216$. The volume of the purple tetrahedron is equal to the volume of the cube minus the volume of the four clear tetrahedra. This is $216-4\cdot 36=\boxed{72}$.