TAOCP 7.2.2.1 Exercise 357
A truncated octahedron has $6$ square faces and $8$ hexagonal faces, so a polysplatt is determined by a connected set of cells in the truncated-octahedral honeycomb.
Section 7.2.2.1: Dancing Links
Exercise 357. [M0] $[M0]$ Investigate "polysplatts," which are the sets of truncated octahedra that can be built by pasting adjacent faces together (either square or hexagonal).
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A truncated octahedron has $6$ square faces and $8$ hexagonal faces, so a polysplatt is determined by a connected set of cells in the truncated-octahedral honeycomb. The centers of these cells form the body-centered cubic lattice, and two centers are adjacent precisely when the corresponding truncated octahedra share a face. Therefore the enumeration of polysplatts reduces to the enumeration of finite connected subsets of this adjacency graph, with equivalence under the symmetries of the lattice.
A convenient representation assigns integer coordinates to the centers in the body-centered cubic lattice. Starting from one cell at the origin, each neighboring cell is reached by one of the face-adjacency vectors, and a polysplatt is a finite set of such coordinate points that is connected by these moves. The smallest examples are a single truncated octahedron, and larger examples are obtained by adjoining cells across either square or hexagonal faces. This gives the analogue of polyominoes, polyhexes, and polyspheres for the truncated-octahedral tiling: the objects are finite connected lattice animals in the truncated-octahedral adjacency graph.