# Towards Infinite Tilings with Symmetric Boundaries

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## Abstract

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## 1. Introduction

## 2. Results

## 3. Discussion

## 4. Materials and Methods

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Typical structure within the large-time regime, visualized as $\varphi =1$ in $\Omega $, $\varphi =0.5$ at $z=0.01$, 0.13, 0.25, 0.37, 0.49 and $\varphi =0.5$ at $z=0.01$, 0.13, 0.25, 0.37, 0.49 projected to $z=0$, from left to right. The boundary conditions are $\mathbf{n}\xb7\nabla \varphi =\mathbf{n}\xb7\nabla \mu =0$ on $\partial \Omega $. The corresponding videos of the coarsening process are provided in SI as Videos S5–S7.

**Figure 2.**Four samples with identical boundaries but distinct inner structure. The samples are translational invariant in x- and y-direction.

**Figure 3.**Screenshots of the visualization software, here in addition color-coded according to the individual tiles used. The dark magenta lines indicate the user-interaction. Moving the mouse horizontally evolves time, moving it vertically zooms in and out.

**Figure 5.**Development of the interface area of the 3D-structure and the interface-length of a slice through the center of the domain over time, averaged for all tiles.

**Figure 6.**Middle slice of the domain in an early time instance (

**a**), a later point in time (

**b**) and the later time-point zoomed out until the interface-length equals that of the early one (

**c**). In the last image the unzoomed region is framed to indicate the level of zoom applied.

**Figure 7.**

**Left**: density-histograms of the distances between interfaces for three selected timesteps (all tiles combined).

**Right**: a square subregion of timestep 130 is considered which, after zooming to the size of a full tile, has the same interface area as timestep 1900. The histograms match almost exactly, which computationally indicates the statistically self-similar structure.

**Figure 8.**Typical structure, highlighting one of the two phases and the adaptively refined mesh along the diffuse interface.

**Figure 10.**(

**a**) Layout of $3\times 3$ tiles: diagonal recurrence cannot be avoided. (

**b**) Layout of $4\times 4$ tiles: the bottom two rows are duplicates of the top two; all other arrangements would yield diagonal recurrence. (

**c**) Layout of $5\times 5$ tiles: recurrences in non-obvious pattern.

**Figure 11.**Initial five tiles and their connecting boundary sides. a) Tile ${A}_{0}$ is simulated on ${\Gamma}^{0}$; b,d) ${B}_{tmp}$ and ${C}_{tmp}$ are intermediate steps in preparation of ${B}_{0}$ and ${C}_{0}$, respectively; c,e,f,g) tiles ${B}_{0}$ through ${E}_{0}$ are simulated on domains that extend ${\Gamma}^{1}$ towards ${\Gamma}^{0}$ only in directions where fresh boundary data is required, the other boundaries are fixed.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Stenger, F.; Voigt, A.
Towards Infinite Tilings with Symmetric Boundaries. *Symmetry* **2019**, *11*, 444.
https://doi.org/10.3390/sym11040444

**AMA Style**

Stenger F, Voigt A.
Towards Infinite Tilings with Symmetric Boundaries. *Symmetry*. 2019; 11(4):444.
https://doi.org/10.3390/sym11040444

**Chicago/Turabian Style**

Stenger, Florian, and Axel Voigt.
2019. "Towards Infinite Tilings with Symmetric Boundaries" *Symmetry* 11, no. 4: 444.
https://doi.org/10.3390/sym11040444