The icosahedral quasiperiodic tiling and its self-similarity

Abstract

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 orthogonal 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 this subdivision, independent of the actual position inside the tiling. In particular we show that the subdivisions of the edges, faces and tiles (with some restriction) are unique.