In the Moldatu animation (see below), results are shown about measurements of the tetrahedral shape, in particular, the ‘endosphere‘, a sphere located in the central space of the tetrahedron, and the ‘exosphere‘, the sphere containing all four ‘mesospheres’ which make up the tetrahedron itself:
The main result is that the volume of the exosphere is equal to the volume of the central endosphere, plus the equivalent of 11 of the mesospheres. Since the tetrahedron itself is formed from 4 mesospheres, this leaves the remainder of the exosphere as consisting of the equivalent of 7 mesospheres.
Here is the document showing the proof of these measurements.
The Moldatu animation:
Sareoso 