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What is the relationship between an octahedron and a cube?
These two polyhedra have the following fundamental parameters:
Octahedron 
Hexahedron 

The form of each side 
Triangle 
Square 
Number of sides 
8 
6 
Number of vertices 
6 
8 
If we pay attention to the number of sides and vertices of these two polyhedra, then they are the opposite.
On the basis of a hexahedron, an octahedron can be constructed. To do this, we select the center of the face on each of the 6 faces of the hexahedron. By connecting all the centers together we obtain an octahedron. This obtained octahedron is exactly inscribed in the original hexahedron.
A similar procedure can be done on an octahedron and get a cube out of it.
This property indicates that the octahedron and the hexahedron form a dual pair.
Video from our partners  the team "ART KOSEKOMA", clearly demonstrates this transformation.
An interesting feature in the transition from a cube to an octahedron is the appearance of three polyhedra belonging to the class of semiregular ones, in other words Archimedean bodies.
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