Icosahedral crystals: Where are the atoms

The positions of atoms in icosahedral crystals are shown to correspond to arrangements of three-dimensional hypersurfaces in six-dimensional periodic lattices. The six-dimensional space-group symmetry allows for crystals which cannot be formed by space-filling decorated tilings (quasicrystals). Structures with continuously varying ''unit cells'' are possible. The connection between tilings, projections, cuts, and density-wave descriptions is pointed out