Three-Dimensional Penrose Tilings and Self-Similarity

We investigate a 3-dimensional analogue of the Penrose tiling, a class of 3-dimensional aperiodic tilings whose edge vectors are the vertex vectors of a regular icosahedron. It arises by an equivariant projection of the unit lattice in euclidean 6-space with its natural representation of the icosahedral group, given by its action on the 6 icosahedral diagonals (with orientation). The tiling has a canonical subdivision by a similar tiling ("deflation"). We give an essentially local construction of the subdivision, independent of the actual place inside the tiling. In particular we show that the subdivision of the edges, faces and tiles (with some restriction) is unique