# Symmetry-Break in Voronoi Tessellations

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## Abstract

**:**

## 1. Introduction

_{1}and x

_{2}, then the set of all points with the same distance from x

_{1}and x

_{2}is a hyperplane, which has codimension 1. The hyperplane bisects perpendicularly the segment from x

_{1}and x

_{2}. In general, the set of all points closer to a point ${x}_{i}\in X$ than to any other point ${x}_{j}\ne {x}_{i}$, ${x}_{j}\in X$ is the interior of a convex (N-1)-polytope usually called the Voronoi cell for x

_{i}. The set of the (N-1)-polytopes Π

_{i}, each corresponding to - and containing - one point ${x}_{i}\in X$, is the Voronoi tessellation corresponding to X. Extensions to the case of non-Euclidean spaces have also been presented [3,4].

_{i}as vertices of regular triangles, squares, and hexagons, respectively. Hexagonal tessellation is optimal both in terms of perimeter-to-area ratio and in terms of cost [50,51]. The extremal properties of such a tessellation are clearly highlighted in [52], where it is noted that a Gibbs system of repulsive charges in 2D arranges spontaneously for low temperatures (freezes) as a regular hexagonal crystal. Moreover, the fact that Rayleigh-Bènard convective cells may generate an hexagonal tessellation [23] is related to the fact that, under specific parameters ranges, the stationary solutions of the 2D Swift-Hohemberg equation generate naturally hexagonal tessellations [53].

_{i}, with a spatially homogeneous Gaussian noise of parametrically controlled strength. The strength of considered perturbation ranges up to the point where typical displacements become larger than the lattice unit vector, which basically leads to the limiting case of the Poisson-Voronoi process.

## 2. Data and Methods

#### 2.a. Scaling Properties of Voronoi tessellations

_{i}in a generic region Γ∈P

^{n}is ${\rho}_{0}|\Gamma |$, where $|\Gamma |$ is the Lebesgue n-measure of Γ, whereas the fluctuations in the number of points are $\approx \sqrt{{\rho}_{0}|\Gamma |}$. If ${\rho}_{0}|\Gamma |>>1$, we are in the thermodynamic limit and boundary effects are negligible, so that the number of cells of the Voronoi tessellation resulting from the set of points x

_{i}and contained inside Γ is ${N}_{V}\approx {\rho}_{0}|\Gamma |$.

#### 2.b. Some Exact Results

#### 2.b.1. 2D Tessellations

_{i}with sides $l=\left|{\overrightarrow{v}}_{1}\right|=\left|{\overrightarrow{v}}_{2}\right|,{\overrightarrow{v}}_{1}\perp {\overrightarrow{v}}_{2}$, the Voronoi cell Π

_{i}corresponding to x

_{i}is given by the square centred in x

_{i}with the same side length and orientation as the x

_{i}grid, so that the grid of the vertices y

_{i}of the tessellation is translated with respect to the x

_{i}grid by l/2 in both orthogonal directions (the verse is not relevant). Therefore, the vertices of the Voronoi tessellation resulting from the points y

_{i}are nothing but the initial points x

_{i}. If $l={l}_{S}={\rho}_{0}^{-1/2}=\left|{\overrightarrow{v}}_{1}\right|=\left|{\overrightarrow{v}}_{2}\right|$, we will have ${\rho}_{0}$ points – and ${\rho}_{0}$ corresponding square Voronoi cells - in ${\Gamma}_{1}$. Similarly, a regular hexagonal honeycomb tessellation featuring ${\rho}_{0}$ points and ${\rho}_{0}$ corresponding regular Voronoi cells in ${\Gamma}_{1}=\left[0,1\right]\otimes \left[0,1\right]$ is obtained by using a grid of points x

_{i}set as regular triangles with sides ${l}_{T}=\sqrt{2/\left(\sqrt{3}{\rho}_{0}\right)}=\left|{\overrightarrow{v}}_{1}\right|=\left|{\overrightarrow{v}}_{2}\right|$, where the angle between ${\overrightarrow{v}}_{1}$ and ${\overrightarrow{v}}_{2}$ is 60°. Finally, the regular triangular tessellation featuring a density ${\rho}_{0}$ of Voronoi cells derives from a regular grid of points x

_{i}set as hexagons with sides ${l}_{H}=\sqrt{4/\left(3\sqrt{3}{\rho}_{0}\right)}=\left|{\overrightarrow{v}}_{1}\right|/\sqrt{3}=\left|{\overrightarrow{v}}_{2}\right|/\sqrt{3}$, again with an angle of 60° between the ${\overrightarrow{v}}_{1}$ and ${\overrightarrow{v}}_{2}$. Therefore, the regular hexagonal and the regular triangular tessellation are conjugate via Voronoi tessellation.

- for square tessellation, we have $\mu \left(n\right)=4$, $\mu \left(P\right)=4{\rho}_{0}^{-1/2}$;for honeycomb tessellation, we have $\mu \left(n\right)=6$ and $\mu \left(P\right)=\sqrt{24/\sqrt{3}}{\rho}_{0}^{-1/2}$,
- for triangular tessellation, we have $\mu \left(n\right)=3$ and $\mu \left(P\right)=\sqrt{36/\sqrt{3}}{\rho}_{0}^{-1/2}$.

#### 2.b.2. 3D Tessellations

- the average number of vertices is $\langle \mu \left(v\right)\rangle =96{\pi}^{2}/35\approx 27.0709$ and its standard deviation is $\langle \sigma \left(v\right)\rangle \approx 6.6708$; exploiting the Euler-Poincare relation plus the genericity property, we obtain $\langle \mu \left(e\right)\rangle =3/2\langle \mu \left(v\right)\rangle $, $\langle \mu \left(f\right)\rangle =1/2\langle \mu \left(v\right)\rangle +2$, $\langle \sigma \left(e\right)\rangle =3/2\langle \sigma \left(v\right)\rangle $, and $\langle \sigma \left(f\right)\rangle =1/2\langle \sigma \left(v\right)\rangle $;
- the average surface area is $\langle \mu \left(A\right)\rangle ={\left(256\pi /3\right)}^{1/3}\Gamma \left(5/3\right){\rho}_{0}^{-2/3}\approx 5.8209{\rho}_{0}^{-2/3}$ (with $\Gamma (\u2022)$ here indicating the usual Gamma function), and its standard deviation is $\langle \sigma \left(A\right)\rangle \approx 0.4804{\rho}_{0}^{-2/3}$;
- the average volume is, by definition, $\langle \mu \left(V\right)\rangle ={\rho}_{0}^{-1}$, whereas its standard deviation is $\langle \sigma \left(V\right)\rangle \approx 0.4231{\rho}_{0}^{-1}$.

#### 2.c. Simulations

_{i}about their deterministic positions with a spatial variance $\left|{\epsilon}^{2}\right|$. We define $\left|{\epsilon}^{2}\right|={\alpha}^{2}{l}_{S}^{2}$, thus expressing the mean squared displacement as a fraction ${\alpha}^{2}$ of the natural squared length scale, where ${l}_{S}^{2}={\rho}_{0}^{-1}$ in the 2D case and ${l}_{S}^{2}={\rho}_{0}^{-2/3}$ in the 3D case. When ensembles are considered, the probability distribution of the points x

_{i}is still periodic. The parameter ${\alpha}^{2}$ can be loosely interpreted as a normalized temperature of the lattice. By definition, if $\alpha =0$ we are in the deterministic case. We study how the statistical properties of the resulting Voronoi cells change with α, covering the whole range going from the symmetry breaking, occurring when α becomes positive, up to the progressively more and more uniform distribution of x

_{i}, obtained when $\alpha >>1$, so that the distributions of nearby points x

_{i}overlap more and more significantly. The actual simulations are performed by applying, within a customised routine, the MATLAB7.0

^{®}functions

`voronoin.m`and

`convhulln.m`, which implement the algorithm introduced in [31], to a set of points x

_{i}generated according to the considered random process. The function

`voronoin.m`associates to each point the vertices of the corresponding Voronoi cell and its volume, whereas the function

`convhulln.m`is used to generate the convex hull of the cell.

_{i}(${\rho}_{0}=10000$) belonging to the square $\left[-0.2,1.2\right]\otimes \left[-0.2,1.2\right]\supset {\Gamma}_{1}=\left[0,1\right]\otimes \left[0,1\right]$, but only the cells belonging to ${\Gamma}_{1}$ have been considered for evaluating the statistical properties. Since the external shell having a thickness of 0.2 comprises about 20 layers of cells, boundary effects due to one-step Brownian diffusion of the points nearby the boundaries, which, in the case of large values of α, cause ${\rho}_{0}$ depletion, become negligible. Another set of simulations is performed by computing an ensemble of 1,000 Poisson-Voronoi tessellations generated starting from a set of uniformly randomly distributed ${\rho}_{0}$ points per unit volume.

_{0}= 1,000,000 generated for all values of $\alpha $ ranging from 0 to 2 with step 0.01, plus additional values aimed at checking the weak- and high-noise limits, and the Poisson-Voronoi case. Similarly to the 2D case, tessellation has been performed starting from points x

_{i}belonging to the square $\left[-0.1,1.1\right]\otimes \left[-0.1,1.1\right]\otimes \left[-0.1,1.1\right]\supset {\Gamma}_{1}=\left[0,1\right]\otimes \left[0,1\right]\otimes \left[0,1\right]$, but only the cells belonging to ${\Gamma}_{1}$ have been considered.

## 3. Results

#### 3.a. Two-dimensional case

#### 3.a.1. Number of sides of the cells

_{i}is quite different from what previously observed. Results are also shown in Figure 1a)-Figure 1b). The first feature is that an infinitesimal noise does not affect at all the tessellation, in the sense that all cells remain hexagons. Moreover, even finite-size noise basically does not distort cells in such a way that figures other than hexagons are created. We have not observed non n=6 cells for up to $\alpha \approx 0.12$ in any member of the ensemble. This has been confirmed also considering larger densities (e.g. ρ

_{0}= 1,000,000). It is more precise, though, to frame the structural stability of the hexagon tessellation in probabilistic terms: the creation of non-hexagons is very unlikely for the considered range. Since the presence of a Gaussian noise induces for each point x

_{i}a probability distribution with – an unrealistic- non-compact support, it is possible to have low-probability outliers that, at local level, can distort heavily the tessellation. Anyway, for all values of $\alpha $ we have that $\langle \mu \left(n\right)\rangle =6$ within a few permils, as imposed by the Euler’s theorem. For $\alpha >0.12$, $\langle \sigma \left(n\right)\rangle $ is positive and increases monotonically with $\alpha $; this is accompanied by a monotonic decrease with $\alpha $ of the fraction of hexagons, which are nevertheless dominant for all values of $\alpha $. For $\alpha >0.5$ the value of $\langle \sigma \left(n\right)\rangle $ is not distinguishable from what obtained for perturbed square and triangular tessellations. This implies that, from a statistical point of view, the variable n loses memory of its unperturbed state already for a rather low amount of Gaussian noise, well before becoming undistinguishable from the random Poisson case.

#### 3.a.2. Area and Perimeter of the cells

#### 3.a.3. Area and perimeter of n-sided cells

#### 3.a.4. Anomalous Scaling

#### 3.b. Three-dimensional case

#### 3.b.1. Faces, Edges, Vertices

_{i}is rather different. Results are also shown in Figure 7. Infinitesimal noise does not affect at all the tessellation, in the sense that all Voronoi cells are 14-faceted (as in the unperturbed state). Moreover, even finite-size noise basically does not distort cells in such a way that other polyhedra are created. We have not observed – also going to higher densities - any non-14 faceted polyhedron for up to $\alpha \approx 0.1$ in any member of the ensemble, so that $\langle \mu \left(f\right)\rangle =14$ and $\langle \sigma \left(f\right)\rangle =0$ in a finite range. However, since the Gaussian noise induces for each point x

_{i}a pdf with a non-compact support, as discussed in Section 3.a.1, we may have extreme fluctuations distorting heavily the tessellation. Therefore, we should interpret this result as that finding non 14-faceted cells is highly – in some sense, exponentially - unlikely.

#### 3.b.2. Area and Volume of the cells

#### 3.b.3. Shape of the cells

## 4. Summary and Conclusions

## Acknowledgements

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**Figure 1.**Ensemble mean of the mean - (a) - and of the standard deviation - (b) - of the number of sides (n) of the Voronoi cells. Note that in (a) the number of sides of all cells is 4 (3) - out of scale - for α=0 in the case of regular square (triangular) tessellation. Half-width of the error bars is twice the standard deviation computed over the ensemble. Poisson-Voronoi limit is indicated.

**Figure 2.**Ensemble mean of the mean - (a) - and of the standard deviation - (b) - of the area (A) of the Voronoi cells. Half-width of the error bars is twice the standard deviation computed over the ensemble. Poisson-Voronoi limit is indicated. In (b), linear approximation for small values of α is also shown. Values are multiplied times the appropriate power of the density in order to obtain universal functions.

**Figure 3.**Ensemble mean of the mean - (a) - and of the standard deviation - (b) - of the perimeter (P) of the Voronoi cells. Half-width of the error bars is twice the standard deviation computed over the ensemble. Poisson-Voronoi limit is indicated. In (b), linear approximation for small values of α is also shown. Values are multiplied times the appropriate power of the density in order to obtain universal functions.

**Figure 4.**Ensemble mean of A - (a) - and of P - (b) - of n-sided Voronoi cells. Half-width of the error bars is twice the ensemble standard deviation. Full ensemble mean is indicated. Linear (a) and square root (b) fits of the Poisson-Voronoi limit results as a function of n is shown. Values are multiplied times the appropriate power of the density in order to obtain universal functions.

**Figure 5.**Joint distribution of the perimeter and of the area of the Voronoi cells in the Poisson-Voronoi tessellation limit in 2D. The black solid line indicates the best log-log least squares fit, with ensemble mean of the exponent $\langle \eta \rangle =2+\langle {\eta}^{\prime}\rangle =2.17$. The dashed black line reports the fit of isoperimetric quotient (see right vertical axis), which scales with the area with exponent $\langle {\eta}^{\prime}\rangle =0.17$. The effective range of applicability of the scaling law is between 1.5 and 6 in units of normalized area. Correspondingly, q ranges between 0.65 and 0.78, and ε between 0.052 and 0.062. Details in the text.

**Figure 6.**Ensemble mean of the scaling exponent fitting the power-law relation $A\propto {P}^{\eta}$ for the perturbed square, hexagonal and triangular Voronoi tessellations. The anomalous scaling ($\eta >2$) is apparent. The error bars, whose half-width is twice the ensemble standard deviation, are too small to be plotted. The Poisson-Voronoi limit (see Figure 5) is indicated. Details in the text.

**Figure 7.**Ensemble mean of the mean and of the standard deviation of the number of faces of the Voronoi cells for perturbed SC, BCC and FCC cubic crystals. The error bars, whose half-width is twice the standard deviation computed over the ensemble, are too small to be plotted. The Poisson-Voronoi limit is indicated.

**Figure 8.**Ensemble mean of the standard deviation of the volume (V) of the Voronoi cells for perturbed SC, BCC and FCC cubic crystal. The ensemble mean of the mean is set to the inverse of the density. Values are multiplied times the appropriate power of the density in order to obtain universal functions. The error bars, whose half-width is twice the standard deviation computed over the ensemble, are too small to be plotted. The Poisson-Voronoi limit is indicated.

**Figure 9.**Ensemble mean of the mean and of the standard deviation of the area (A) of the Voronoi cells for perturbed SC, BCC and FCC cubic crystals. Values are multiplied times the appropriate power of the density in order to obtain universal functions. The error bars, whose half-width is twice the standard deviation computed over the ensemble, are too small to be plotted. The Poisson-Voronoi limit is indicated.

**Figure 10.**Ensemble mean of the mean and of the standard deviation (see the different scales) of the isoperimetric quotient $Q=36\pi {V}^{2}/{A}^{3}$ of the Voronoi cells for perturbed SC, BCC and FCC cubic crystals The error bars, whose half-width is twice the standard deviation computed over the ensemble, are too small to be plotted. The Poisson-Voronoi limit is indicated. Details in the text.

**Figure 11.**Ensemble mean of the isoperimetric quotient of the Voronoi cells for perturbed SC (a), BCC (b) and FCC(c) cubic crystals, where averages are taken over cells having f faces. A white shading indicates that the corresponding ensemble is empty. More faceted cells are typically bulkier.

**Figure 12.**Joint distribution of the area and of the volume of the Voronoi cells in the Poisson-Voronoi tessellation limit. The black solid line indicates the best log-log least squares fit, with ensemble mean of the exponent $\langle \eta \rangle =3/2+\langle {\eta}^{\prime}\rangle =1.67$. The dashed black line reports the corresponding fit of isoperimetric quotient (see right vertical axis), which scales with the area with exponent $2\langle {\eta}^{\prime}\rangle =0.34$. The effective range of applicability of the scaling law is between 2 and 10 in units of normalized area. Correspondingly, Q ranges between 0.35 and 0.65, and ε between 0.056 and 0.076. Details in the text.

**Figure 13.**Ensemble mean of the scaling exponent $\eta $ fitting the power-law relation $V\propto {A}^{\eta}$ for the Voronoi cells of perturbed SC, BCC and FCC cubic crystals. The presence of an anomalous scaling ($\langle \eta \rangle >3/2$) due to the fluctuations in the shape of the cells is apparent. The error bars, whose half-width is twice the standard deviation computed over the ensemble, are too small to be plotted. The Poisson-Voronoi limit (see Figure 12) is indicated. Details in the text.

© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Lucarini, V.
Symmetry-Break in Voronoi Tessellations. *Symmetry* **2009**, *1*, 21-54.
https://doi.org/10.3390/sym1010021

**AMA Style**

Lucarini V.
Symmetry-Break in Voronoi Tessellations. *Symmetry*. 2009; 1(1):21-54.
https://doi.org/10.3390/sym1010021

**Chicago/Turabian Style**

Lucarini, Valerio.
2009. "Symmetry-Break in Voronoi Tessellations" *Symmetry* 1, no. 1: 21-54.
https://doi.org/10.3390/sym1010021