A Keplerian Ag90 nest of Platonic and Archimedean polyhedra in different symmetry groups.

Polyhedra are ubiquitous in chemistry, biology, mathematics and other disciplines. Coordination-driven self-assembly has created molecules mimicking Platonic, Archimedean and even Goldberg polyhedra, however, nesting multiple polyhedra in one cluster is challenging, not only for synthesis but also for determining the alignment of the polyhedra. Here, we synthesize a nested Ag90 nanocluster under solvothermal condition. This pseudo-Th symmetric Ag90 ball contains three concentric Ag polyhedra with apparently incompatible symmetry. Specifically, the inner (Ag6) and middle (Ag24) shells are octahedral (Oh), an octahedron (a Platonic solid with six 3.3.3.3 vertices) and a truncated octahedron (an Archimedean solid with twenty-four 4.6.6 vertices), whereas the outer (Ag60) shell is icosahedral (Ih), a rhombicosidodecahedron (an Archimedean solid with sixty 3.4.5.4 vertices). The Ag90 nanocluster solves the apparent incompatibility with the most symmetric arrangement of 2- and 3-fold rotational axes, similar to the arrangement in the model called Kepler's Kosmos, devised by the mathematician John Conway.