Modified Mosseri-Sadoc tiles from $D_6$

Abstract

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 $τ^3$ and $τ$ with the corresponding eigenvectors representing the volumes and the Dehn invariants of the tiles, respectively, where $τ=\frac{1+\sqrt5}{2}$ 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 $τ^n$ with an arbitrary integer $n$ and tiled by the MMS tiles.

Figures (4)

• Figure 1: The Coxeter-Dynkin diagrams of $d_6$ and $h_3$ illustrating the symbolic projection.
• Figure 2: Dodecahedron of edge length 1 with vertices $\pm A_i$ and $\pm B_i$.
• Figure 3: Embedding of the MMS tiles in the dodecahedron.
• Figure 4: Threefold embedding of 7 dodecahedra $d\left(1\right)$ in the dodecahedron $d\left(\tau^2\right)$ along with the other tiles (not shown).