These twenty tetrahedrons share a common center point and represent the greatest number of tetrahedrons sharing a centerpoint within quantum-or-imperfect geometries.
We also call this gap geometry, imperfect geometry, quantum geometry or squishy geometry.
Fifteen objects share a center point using a five-octahedral cluster, five-tetrahedral cluster with another five-tetrahedral cluster on top. Within the octahedron there are four smaller octahedrons and eight tetrahedrons that perfectly share a common centerpoint.
Worksheet: An icosahedron with two groups of five tetrahedrons (in dots) facing each other and ten tetrahedrons circling the middle (solid orange line).
There is another way of looking at the twenty tetrahedrons with three groups of five tetrahedrons that leaves a small group of four and a single tetrahedron that are easily accounted.
Rotational symmetry. In any direction there is one face to face with a single edge exposed, then a convergence of five with a rotational symmetry as shown below.