Monohedrally knotted tilings of the 3-space
A tiling by congruent copies of a single tile is called a monohedral tiling. The following theorem is from an unpublished example of P. McMullen. Theorem 1. For any polyhedral knot \(K\) in \({\mathbf E}^3\) and any \(\varepsilon > 0\) there exists a tile \(M\) which is isotropic to an \(\varepsilon\)-neighborhood of \(K\) such that \(M\) tiles \({\mathbf E}^3\) monohedrally. Zaks gives an elegent, one page construction that proves the following generalization. Theorem 2. Let \(G\) be a connected graph and \(f : G\rightarrow {\mathbf E}^3\) any piecewise linear embedding of \(G\) into \({\mathbf E}^3\). For any \(\varepsilon > 0\) there exists a tile \(M\) which is isotropic to an \(\varepsilon\)-neighborhood of a set \(H\), isotropic to \(f(G)\), such that \(M\) tiles \({\mathbf E}^3\) monohedrally.
scientific article; zbMATH DE number 1110036