Dodecahedral Structures with Mosseri-Sadoc Tiles (2009.07048v3)

Abstract: 3D-facets of the Delone cells representing the deep and shallow holes of the root lattice D6 which tile the six-dimensional Euclidean space in an alternating order are projected into three-dimensional space. They are classified into six Mosseri-Sadoc tetrahedral tiles of edge lengths 1 and golden ratio (tau) with faces normal to the 5-fold and 3-fold axes. The icosahedron, dodecahedron and icosidodecahedron whose vertices are obtained from the fundamental weights of the icosahedral group are dissected in terms of six tetrahedra. A set of four tiles are composed out of six fundamental tiles, faces of which, are normal to the 5-fold axes of the icosahedral group. It is shown that the 3D-Euclidean space can be tiled face-to-face with maximal face coverage by the composite tiles with an inflation factor tau generated by an inflation matrix. We note that dodecahedra with edge lengths of 1 and tau naturally occur already in the second and third order of the inflations. The 3D patches displaying 5-fold, 3-fold and 2-fold symmetries are obtained in the inflated dodecahedral structures with edge lengths tau to the power n with n equals 3 or greater than 3. The planar tiling of the faces of the composite tiles follow the edge-to-edge matching of the Robinson triangles.