On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distribution of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles in 1970

On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

For a locally finite set in $\mathbb{R}^2$, the order-$k$ Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in $k$. As an example, a stationary Poisson point process in $\mathbb{R}^2$ is locally finite, coarsely dense, and generic with probability one. For such a set, the distribution of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-$1$ Delaunay mosaics given by Miles in 1970

Brillouin Zones of Integer Lattices and Their Perturbations

For a locally finite set, ⊆ℝ , the th Brillouin zone of ∈ is the region of points ∈ℝ for which ‖−‖ is the th smallest among the Euclidean distances between and the points in . If is a lattice, the th Brillouin zones of the points in are translates of each other, and together they tile space. Depending on the value of , they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in ℝ2 , and the convergence of the maximum volume of a chamber to zero for the integer lattice