Inflation and wavelets for the icosahedral Danzer tiling

The distribution of atoms in quasi-crystals lacks periodicity and displays point symmetry associated with non-crystallographic modules. Often it can be described by quasi-periodic tilings on R(3) built from a finite number of prototiles. The modules and the canonical tilings of five-fold and icosahedral. point symmetry admit inflation symmetry. In the simplest case of stone inflation, any prototile when scaled by the golden section number tau can be packed from unscaled prototiles. Observables supported on R(3) for quasicrystals require symmetry-adapted function spaces. We construct wavelet bases on R(3) for the icosahedral Danzer tiling. The stone inflation of the four Danzer prototiles is given explicitly in terms of Euclidean group operations acting on R(3). By acting with the unitary representations inverse to these operations on the characteristic functions of the prototiles, we recursively provide a full orthogonal wavelet basis of R(3). It incorporates the icosahedral and inflation symmetry.