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

TL;DR

The paper develops a unified framework for angles in higher-order planar tilings by establishing monotonicity of extreme angles across order‑$k$ Delaunay, Iglesias, and Brillouin mosaics for locally finite, coarsely dense, and generic point sets, extending Sibson’s angle-maximization concepts. It leverages orthogonal duality, weighted point constructions, and the k‑order Voronoi/Brillouin/Igles mosaics to prove that $\alpha(M_k(A))\ge\alpha(M_{k+1}(A))$ for $M\in\{\mathrm{Del},\mathrm{Igl},\mathrm Bri\}$ and that the corresponding $\omega$ inequalities hold for $M\in\{\mathrm{Vor},\mathrm Bri,\mathrm Igl\}$, with certain non-generic cases clarified. For stationary Poisson point processes, the angle distributions are independent of the order and are given by Miles’ formula for the Del/Delaunay case, with the Brillouin and Iglesias mosaics sharing the same distributions through duality; the average distribution is concave and governed by $h(t)=\tfrac{2}{3}[(\pi-2t)\cos t+2\sin t]\sin t$. The results are corroborated by computational experiments showing monotone behavior in Brillouin tilings and order-invariant angle distributions, suggesting broad relevance to stochastic geometry and materials science where higher-order Brillouin zones arise.

Abstract

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.

Figures (6)

• Figure 1: Top row: three tessellations of the integer lattice: ${\rm Vor}_{5}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}{\hbox{${\rm Vor}_{5}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}$}}$ on the left, ${\rm Vor}_{6}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}{\hbox{${\rm Vor}_{6}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}$}}$ on the right, and their overlay, ${\rm Bri}_{6}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}{\hbox{${\rm Bri}_{6}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}$}}$, in the middle. For every point in the dark blue region, the point in the center is among the five closest on the left, the sixth closest in the middle, and among the six closest on the right. Bottom row: the corresponding mosaics: ${\rm Del}_{5}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}{\hbox{${\rm Del}_{5}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}$}}$ on the left, ${\rm Del}_{6}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}{\hbox{${\rm Del}_{6}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}$}}$ on the right, and ${\rm Igl}_{6}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}{\hbox{${\rm Igl}_{6}{({{\mathbb Z}{\hbox{${\mathbb Z}$}}^2})}$}}$ in the middle. Observe that the mosaics are indeed orthogonal duals of the tessellations.
• Figure 2: For zero enclosed points, we get the triangle $abc$ in ${\rm Igl}_{1}{({A})}{\hbox{${\rm Igl}_{1}{({A})}$}}$, the hexagon constructed by equal tri-sections of the edges of $abc$ in ${\rm Igl}_{2}{({A})}{\hbox{${\rm Igl}_{2}{({A})}$}}$, and a translate of $abc$ scaled by $- \frac{1}{5}$ in ${\rm Igl}_{3}{({A})}{\hbox{${\rm Igl}_{3}{({A})}$}}$.
• Figure 3: Top: the sequences of min angles (in blue) and of max angles (in orange) in the order-$k$ Delaunay mosaics of a lattice without four cocircular points on the left and of a random periodic set on the right. We draw the vertical axes for the minima and maxima on opposite sides of each panel. Note that the min angles decrease monotonically with increasing $k$, but the max angles are not monotonic. Bottom: the sequences of min and max angles in the order-$k$ Brillouin tessellations of the lattice on the left and the random periodic set on the right. Both the min and the max angles are monotonic in $k$.
• Figure 4: The graphs of $f$, $g$, and $h$ in solid black, dotted black, and blue.
• Figure 5: The distribution of the angles in the first $57$ Brillouin zones of $0$ in ${\mathbb Z}{\hbox{${\mathbb Z}$}}^2$ on the left, and in a perturbation of ${\mathbb Z}{\hbox{${\mathbb Z}$}}^2$ on the right. The orange curves smoothe out the histogram skylines.

Theorems & Definitions (9)

• Definition 1: Orthogonal Dual
• Proposition 2: Vertical Projection
• Lemma 3: Vertex Characterization
• Definition 4: Orthogonal Dual Mosaics
• Lemma 5: Angles and Supplementary Angles
• Lemma 6: Angle Inequalities
• Theorem 7: Monotonicity of Extreme Angles
• Theorem 8: Angle Distributions
• Lemma 9: Counterexample to Monotonicity