The first densest lattice of chemical supertetrahedra?
In one of the last issues of the Angewandte, Wang and coworkers  presented an extremely interesting structure, in which metal-organic polyhedra (MOP) are assembled in different ways. These polyhedra consist of polyoxovanadate metal clusters and bridging dicarboxylate linkers. The authors call them VMOP. The overall shape of the MOP can be described as tetrahedron, or to be precise, as truncated tetrahedron see Fig. 1.
Fig. 1: VMOP, in atomistic representation (left) and simplified as a truncated tetrahedron (right).
In fact, the authors obtained two different phases, which differ in the kind of packing: in the VMOP-alpha isomer (a very low-density phase, which is thermodynamically less stable) each truncated tetrahedron makes perfect contact with four neighbors via the (small) four trigonal faces (the truncated faces), thus leading to a dia-like framework (corner-connected tetrahedra).
However, in VMOP-beta the MOPs are packed in a corner-to-face fashion, i.e. each
truncated tetrahedron has contact with eight neighboring tetrahedra (trigonal-to-hexagonal face-to-face-connection), see Fig. 2.
Fig. 2: The eight direct neighbors of a central truncated tetrahedron (in green) in the packing of VMOP-beta.
This packing mode of regular (non-truncated) tetrahedra is known as the densest lattice packing (i.e. only translations are allowed) of tetrahedra, which was proven in 1969 by Hoylman . The resulting packing density is 18/49 ≈ 36.73 %. Here, each tetrahedron is in contact with 14 others (4 corners + 4 faces + 6 edges). However, in VMOP-beta the 6 edge-edge connections are a bit further apart.
I think, the structure of VMOP-beta is very remarkable and I am not aware of any analogous chemical structure – do you?
PS: I would like to thank Ahmad Rafsanjani Abbasi (ETH Zürich) for bringing this structure to my attention.
 Y. Gong, Y. Zhang, C. Qin, C. Sun, X. Wang, Z. Su, Angew. Chem. Int. Ed. 2019, 58, 780.
 D.J. Hoylman, Bull. Amer. Math. Soc. 1970, 76, 135.
What is the underlying net of VMOP_beta ? Once you have it we can check it on topcryst
Davide M Proserpio