A fascinating new paper by Yang Jiao and Salvatore Torquato defines three dimensional packings of truncated tetrahedra, using a lattice vector deformation parameter $g$ that takes values in the interval $[0,2/9]$.
A trucated tetrahedron has four hexagonal faces and four triangular ones. The maximal $g = 2/9$ corresponds to the densest known packing of truncated tetrahedra, with packing fraction $207/208$. The packing has tetrahedra shaped holes. With Conway they have also constructed a packing with fraction $23/24$, and considered packings involving octahedra.
6 years ago