|Dr. habil. Ilka Agricola|
The five nested tetrahedra intersect precisely in an icosahedron; this is useful for the description of the symmetry group of the icosahedron, since all its elements have the property to permute the five tetrahedra.
For details, I refer to our book
The star may be found in Section 3.8, while the the Platonic bodies used for the wallpaper background are from Section 2.5. Both pictures are from the cover of the afore mentioned book, designed by Peter Sykes (AMS). My warmest thanks to him for preparing these files!