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Filling Space with Tetrahedra and Half-Size Octahedra

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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

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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.

Reference

[1] 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.

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