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.