From the emergence theory object library files…
Both the vertex-sharing and edge-sharing 3-fold clusters have the same face plane classes and share the same relative face angles. In a way, they embed each other. Here is an overlay of the two together to show the relationship. The edge sharing cluster is scaled to be double the edge lengths of the vertex-sharing cluster.
In this video we are rotating the structure around in Mathematica, viewing it from various angles, and changing the rotation angle as well as the transparency of the tetrahedra.
To rotate and manipulate this object yourself, you can download the Mathematica file associated with this object here:
You can download the free Wolfram Mathematica file player here.