Filling Space with Tetrahedra and Half-Size Octahedra
Four tetrahedra are arranged around a half-size octahedron. This arrangement is repeated to form a space filling structure with alternating layers of tetrahedra and octahedra.
Contributed by: Sándor Kabai (June 2016)
with additinal contributions by Dániel Erdély
Open content licensed under CC BY-NC-SA
Dániel Erdély told me about the tetrahedron arrangement; his father made a model of it years ago. Without octahedra, the space filling ratio of tetrahedra is 2/3, as explored by a number of authors including John Conway and Klee Irwin.
 J. H. Conway and S. Torquato, "Packing, Tiling, and Covering with Tetrahedra," Proceedings of the National Academy of Sciences, 103(28) 2006 pp. 10612–10617. doi:10.1073/pnas.0601389103.