Regular Polytopes, Root Lattices, and Quasicrystals

The icosahedral quasicrystals of five-fold symmetry in two, three, and four dimensions are related to the corresponding regular polytopes exhibiting five-fold symmetry, namely the regular pentagon (H2 reflection group), the regular icosahedron {3,5} (H3 reflection group), and the regular four-dimensional polytope {3,3,5} (H4 reflection group). These quasicrystals exhibiting five-fold symmetry can be generated by projections from indecomposable root lattices with twice the number of dimensions, namely A4→H2, D6→H3, E8→H4. Because of the subgroup relationships H2 ⊂ H3 ⊂ H4, study of the projection E8→H4 provides information on all of the possible icosahedral quasicrystals. Similar projections from other indecomposable root lattices can generate quasicrystals of other symmetries. Four-dimensional root lattices are sufficient for projections to two-dimensional quasicrystals of eight-fold and twelve-fold symmetries. However, root lattices of at least six-dimensions (e.g., the A6 lattice) are required to generate twodimensional quasicrystals of seven-fold symmetry. This might account for the absence of seven-fold symmetry in experimentally observed quasicrystals