TAOCP 7.2.2.1 Exercise 357
A truncated octahedron tessellates $\mathbb{R}^3$ face-to-face, so each cell has a well-defined adjacency graph in which two cells are adjacent exactly when they share a square or hexagonal face.
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 tessellates $\mathbb{R}^3$ face-to-face, so each cell has a well-defined adjacency graph in which two cells are adjacent exactly when they share a square or hexagonal face. This adjacency relation generates a connected graph on the set of all cells in the tessellation.
A polysplatt is therefore precisely any finite connected set of vertices in this adjacency graph, equivalently any finite union of truncated octahedra in the tiling that is connected through shared faces. No additional geometric constraints arise beyond connectivity, since every face-to-face adjacency permitted by the tiling is realizable and the tiling admits no gaps or forced identifications. Hence polysplatts are in bijection with finite connected induced subgraphs of the truncated-octahedron adjacency graph. ∎