The impact of aperiodic order on mathematics

Mathematics has been strongly influenced by problems arising from physics. The existence of quasicrystals as strongly ordered structures which cannot be periodic has raised various mathematical questions that have stimulated developments in the areas of discrete geometry, harmonic analysis, group theory and ergodic theory. It seems that extra “internal dimensions” are useful in describing certain features of quasicrystal structure and their diffraction spectra. In particular N-dimensional crystalline symmetries can appear in the diffraction spectra of model sets. This paper describes recent work in discrete geometry suggested by the modeling of atomic positions in quasicrystals by Delone sets with restrictions on interpoint distances. It suggests one mechanism for the appearance and usefulness of “internal dimensions” in describing ordered aperiodic structures.