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where can I find a proof that truncated octahedron tiles Euclidean 3D space?

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You mean references for the bitruncated cubic honeycomb? Sadly, there doesn't seem to be an online version of Grünbaum's paper... –  J. M. Dec 4 '10 at 0:02

1 Answer 1

To tile space, center them at lattice points and at the centers of lattice cubes (i.e. lattice points plus $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$). The eight faces of the octahedron face the neighboring 8 vertices of the "other" lattice (the one relatively shifted by $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$), while the six truncated corners face the six neighbor vertices on the "same" lattice.

Most sources (often crystallographic) consider this to be obvious enough that it needs no proof.

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