[Completion of a Tetrahedron to a Parallelepiped]

On the faces of a regular tetrahedron with edge $a$, equal regular pyramids are constructed. The plane angles at the vertices of these pyramids, opposite to the faces of the tetrahedron, are right angles. Consider the polyhedron formed by the tetrahedron and the constructed pyramids. How many faces does this polyhedron have? What is it called?
Consider the cube $A B C D A 1 B 1 C 1 D 1$. All edges of the triangular pyramid $A C B 1 D 1$ are equal. Therefore, this pyramid is a regular tetrahedron. On the lateral faces of the regular tetrahedron $A C B 1 D 1$ as bases, regular triangular pyramids $D A C D 1$ with vertex $D$, $A 1 A B 1 D 1$ with vertex $A 1$, $B A B 1 C$ with vertex $B$, and $C 1 C B 1 D 1$ with vertex $C 1$ are constructed. The planar angles at the vertices of these four pyramids are each $90^{\circ}$. Conversely, for any regular tetrahedron, there exists a unique cube, the intersecting diagonals of opposite faces of which are the edges of the given regular tetrahedron (it is sufficient to draw three pairs of parallel planes through the intersecting diagonals of opposite faces of the tetrahedron). On the lateral face of this tetrahedron as a base, an external unique regular triangular pyramid can be constructed, the planar angles at the vertex of which are right angles. It is clear that the lateral faces of such a pyramid lie in the faces of the cube.

## Answer

6 faces; cube.