Problem 623

8. 32 .

Let the vertices of a regular tetrahedron be $A, B, C, D$. Then there are two types of planes whose distances to these four points are in the ratio $1: 1: 1: \sqrt{2}$.
(1) Points $A, B, C$ are on the same side of plane $\alpha$, there are two such planes;
(2) Points $A, B, C$ are on opposite sides of plane $\alpha$, there are six such planes.
By swapping points $A, B, C, D$, a total of $8 \times 4=32$ planes that satisfy the condition are obtained.