To make this using Stella 4d (available here) I started with an icosahedron, placed a dodecahedron on each of its vertices, then rendered the central icosahedron invisible. The slight pulsating effect is caused by the program fitting the polyhedra tightly into each frame of the animation.
Software used: Stella 4d, available at www.software3d.com/Stella.php.
I made this using Stella 4d: Polyhedron Navigator. You can try this program for yourself at this website.
Each of these has a tetrahedron hidden from view in the center.
These were made using Stella 4d, which you may try for yourself here.
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