Convex polyhedral tilings hidden in crystals and quasicrystals

Abstract

A lattice in a Euclidean space gives rise to facet-to-facet and space-filling convex polyhedral tilings called the Voronoi tiling and its dual Delaunay tiling of the Euclidean space. Given a subspace of the Euclidean space, we develop a systematic way of constructing facet-to-facet and space-filling convex polyhedral tilings of the subspace called the Namikawa tilings, which are generalization of the Delaunay tiling. Our results here amplify the author’s previous work with Seshadri that was obtained as a by-product of Geometric Invariant Theory in algebraic geometry. In the orthonormal setting where the lattice is spanned by an orthonormal basis of the ambient Euclidean space, the method of Namikawa tilings turns out to be the “cut and project” method relevant not only to periodic convex polyhedral tilings hidden in crystals but also to aperiodic convex polyhedral tilings hidden in quasicrystals. In another paper, the author showed that a Voronoi tiling is hidden in the “standard realization”, in the sense of Kotani and Sunada, of crystals in the case of maximal abelian covering of finite graphs. In this connection, we give an affirmative answer to a question raised by Kotani to the effect that the crystal lies in the 1-skeleton of a nondegenerate convex polyhedral tiling that is a subdivision of the Voronoi tiling appearing in the other paper of the author.