  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.
