This part consists of all five Platonic Solids nested one inside another.
From largest to smallest, there is an icosahedron (20 triangular faces), a dodecahedron (12 pentagonal faces), a hexahedron (6 square faces), a tetrahedron (4 triangular faces) and an octahedron (8 triangular faces).
The overall shape is a Rhombic Triacontrahedron, a combination of an icosahedron and dodecahedron). These can be nested infinitely following the same order.
This tree-like surface was generated with a single mathematical expression using the implicit surfaces filter in MeshLab, and a shell was added to the isosurface as well in MeshLab. A hole was added (subtracted?) at the bottom to reduce the overall volume.