From a tetrahedron to an octahedron
How to transform a tetrahedron into an octahedron: we make some successive sections with 4 planes that round off the vertices and are perpendicular to the rotations axes that go through the 4 couples of vertices (during the construction we get also a uniform polyhedron of type (3,6,6)). The dimensions of the polyhedra in the figure don't reproduce exactly this construction. Instead, the polyhedra in the figure represent the sections of a hypercube with 3-dimensional spaces perpendicular to a diagonal of the hypercube. In the same way, if we section a cube with planes perpendicular to a diagonal, we get triangles and hexagons.
immagine di Gian Marco Todesco