78. Let the number of faces, edges, and vertices of one of two mutually regular polyhedra be $\mathscr{G}, \mathscr{P}$, and $\mathscr{B}$, respectively, and let $n$ edges meet at each vertex of this polyhedron. Find the number of faces, edges, and vertices of the corresponding quasi-regular polyhedron.
78. Let each face of the polyhedron $T$ have $k$ sides. Since at each vertex of the polyhedron $T$ there are $n$ edges converging, each face of the polyhedron $T^{\prime}$, reciprocal to $T$, has $n$ sides, and the number of these faces is $\mathscr{B}$. By the remarks made on page 35, the almost regular polyhedron obtained from $T$ and $T^{\prime}$ will have $\mathscr{G}$ $k$-sided and $\mathscr{B} n$-sided faces, i.e., a total of $\mathscr{B}+\mathscr{G}$ faces.

Furthermore, since on each edge of the polyhedron $T$ there is one vertex of the almost regular polyhedron, the latter must have a total of $\mathscr{P}$ vertices.

Finally, since at each of the $\mathscr{P}$ vertices of the almost regular polyhedron, four edges converge, and each edge connects two vertices, the total number of edges of the almost regular polyhedron must be $\frac{4 \mathcal{P}}{2}=2 \mathcal{P}$ edges.