Dual pairs of polyhedra

What do the dodecahedron and icosahedron have in common?

These two polyhedra have the following fundamental parameters:

If we pay attention to the number of sides and vertices of these two polyhedra, they are the opposite.
What does this mean for us? Based on a dodecahedron, an icosahedron can be constructed.
To do this, on each of the 12 faces of the dodecahedron, we select the face center. By connecting all the centers, we obtain an icosahedron. This obtained icosahedron will be exactly inscribed in the original dodecahedron.

A similar procedure can be performed on an icosahedron and obtained from it a dodecahedron.

This property indicates that the icosahedron and the dodecahedron form a dual pair.

Video from our partners - the team "ART KOSEKOMA", clearly demonstrates this transformation.

An interesting feature in the transition from a dodecahedron to an icosahedron is the emergence of three polyhedra belonging to the class of semi-regular words, in other words, Archimedean solids.