Let there any platonic solid originally having $\color{blue}{V_{o}} \space \color{blue}{\text{no. of vertices}}$, $\color{blue}{E_{o}}\space \color{blue}{\text{ no. of edges}}$ out of which $\color{blue}{e_{o}\space \text{no. of edges meet at each vertex}}$ & $\color{blue}{F_{o}}\space \color{blue}{\text{ no. of faces}}$.These are co-related by Euler's formula as $F_{o}+V_{o}=E_{o}+2$ .
Now, it is truncated at each vertex to the mid-points of its edges to generate a new solid having $\color{blue}{V}$ no. of vertices , $\color{blue}{E}$ no. of edges & $\color{blue}{F}$ no. of faces then in general we have
$$\color{blue}{V= (\text{no. of original edges meeting at each vertex})\times (\text{no. of original vertices})-(\text{no. of original edges}) }$$$$\color{blue}{=e_{o} V_{o}-E_{o}}$$
$$\color{blue}{F=(\text{no. of original vertices})+(\text{no. of original faces}) }$$$$\color{blue}{= V_{o}+F_{o}}$$
Now, applying Euler's formula we get
$$\color{blue}{E}=F+V-2$$$$=e_{o} V_{o}-E_{o}+V_{o}+F_{o}-2$$
$$\color{blue}{=(e_{o}+1)V_{o}+F_{o}-E_{o}-2}$$
A regular tetrahedron originally has $\color{blue}{V_{o}=4} \space \color{blue}{\text{no. of vertices}}$, $\color{blue}{E_{o}=6}\space \color{blue}{\text{ no. of edges}}$ out of which $\color{blue}{e_{o}=3\space \text{no. of edges meet at each vertex}}$ & $\color{blue}{F_{o}=4}\space \color{blue}{\text{ no. of faces}}$ Now by truncating it to the mid-points of the edges, a new solid is obtained which has $$V=3(4)-6=6$$ $$F=4+4=8$$ $$E=8+6-2=12$$ Thus, the solid obtained has 6 vertices, 12 edges & 8 (triangular) faces. Hence it is an $\color{blue}{\text{Octahedron}}$
Similarly by truncation of regular hexahedron (cube) & octahedron to the mid-points of the edges will generate a solid called $\color{blue}{\text{Cuboctahedron}}$ having 12 vertices, 24 edges and 8 equilateral triangular faces & 6 square faces.
Similarly by truncation of regular dodecahedron & icosahedron to the mid-points of the edges will generate a solid called $\color{blue}{\text{Icosidodecahedron}}$ having 30 vertices, 60 edges and 20 equilateral triangular faces & 12 regular pentagonal faces.