Modified Mosseri-Sadoc tiles from $D_6$

A modified set of Mosseri-Sadoc (MS) tiles tessellating 3D Euclidean space with icosahedral symmetry is introduced. The new set of tiles are embedded in dodecahedron with a threefold symmetric order. The modified Mosseri-Sadoc (MMS) tiles can be inflated by a new inflation matrix with positive eigenvalues $\tau^3$ and $\tau$ with the corresponding eigenvectors representing the volumes and the Dehn invariants of the tiles, respectively, where $\tau=\frac{1+\sqrt5}{2}$ is the golden ratio. The MMS tiles are obtained by projection of the 4D and 5D facets of the Delone cells tiling the $D_6$ root lattice in an alternating order. It is also proved that a subset of the lattice $D_6$ projects into the dodecahedron inflated by $\tau^n$ with an arbitrary integer $n$ and tiled by the MMS tiles.