A two-vertex theorem for normal tilings

Domokos, G.; Horváth, Ákos G.; Regős, Krisztina

doi:10.1007/s00010-022-00888-0

Cited by 4 publications

(2 citation statements)

References 13 publications

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“…Following (Domokos et al 2022), we also introduce a concept, which is more sensitive to the actual shape of the cells: Definition 5 In a normal, convex tiling the corner degree v* of a cell is equal to the number of its vertices and the corner degree n* of a node is equal to the number of vertices overlapping at that node (Fig. 2).…”

Section: Mathematical Conceptsmentioning

confidence: 99%

A discrete time evolution model for fracture networks

Domokos

Regős

2022

Cent Eur J Oper Res

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We examine geological crack patterns using the mean field theory of convex mosaics. We assign the pair $$\left({\overline{n } }^{*},{\overline{v } }^{*}\right)$$ n ¯ ∗ , v ¯ ∗ of average corner degrees (Domokos et al. in A two-vertex theorem for normal tilings. Aequat Math https://doi.org/10.1007/s00010-022-00888-0, 2022) to each crack pattern and we define two local, random evolutionary steps R0 and R1, corresponding to secondary fracture and rearrangement of cracks, respectively. Random sequences of these steps result in trajectories on the $$\left({\overline{n } }^{*},{\overline{v } }^{*}\right)$$ n ¯ ∗ , v ¯ ∗ plane. We prove the existence of limit points for several types of trajectories. Also, we prove that celldensity$$\overline{\rho }= \frac{{\overline{v } }^{*}}{{\overline{n } }^{*}}$$ ρ ¯ = v ¯ ∗ n ¯ ∗ increases monotonically under any admissible trajectory.

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Section: Mathematical Conceptsmentioning

confidence: 99%

A discrete time evolution model for fracture networks

Domokos

Regős

2022

Cent Eur J Oper Res

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“…Two-dimensional (2D) natural patterns at all scales, ranging from molecular assemblies ( 1 , 2 ) to macroscopic entities ( 3 , 4 ), often appear as polygonal tessellations ( 5 ). The corresponding mathematical theory ( 6 , 7 ) could be harnessed to obtain a deeper understanding of the structure-forming process. Our aim here is to make this step for molecular monolayers.…”

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Polygonal tessellations as predictive models of molecular monolayers

Regős

Pawlak

Wang

et al. 2023

Proc. Natl. Acad. Sci. U.S.A.

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Molecular self-assembly plays a very important role in various aspects of technology as well as in biological systems. Governed by covalent, hydrogen or van der Waals interactions–self-assembly of alike molecules results in a large variety of complex patterns even in two dimensions (2D). Prediction of pattern formation for 2D molecular networks is extremely important, though very challenging, and so far, relied on computationally involved approaches such as density functional theory, classical molecular dynamics, Monte Carlo, or machine learning. Such methods, however, do not guarantee that all possible patterns will be considered and often rely on intuition. Here, we introduce a much simpler, though rigorous, hierarchical geometric model founded on the mean-field theory of 2D polygonal tessellations to predict extended network patterns based on molecular-level information. Based on graph theory, this approach yields pattern classification and pattern prediction within well-defined ranges. When applied to existing experimental data, our model provides a different view of self-assembled molecular patterns, leading to interesting predictions on admissible patterns and potential additional phases. While developed for hydrogen-bonded systems, an extension to covalently bonded graphene-derived materials or 3D structures such as fullerenes is possible, significantly opening the range of potential future applications.

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An Evolution Model for Polygonal Tessellations as Models for Crack Networks and Other Natural Patterns

2023

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We introduce and study a general framework for modeling the evolution of crack networks. The evolution steps are triggered by exponential clocks corresponding to local micro-events, and thus reflect the state of the pattern. In an appropriate simultaneous limit of pattern domain tending to infinity and time step tending to zero, a continuous time model, specifically a system of ODE is derived that describes the dynamics of averaged quantities. In comparison with the previous, discrete time model, studied recently by two of the present three authors, this approach has several advantages. In particular, the emergence of non-physical solutions characteristic to the discrete time model is ruled out in the relevant nonlinear version of the new model. We also comment on the possibilities of studying further types of pattern formation phenomena based on the introduced general framework.

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A two-vertex theorem for normal tilings

Cited by 4 publications

References 13 publications

A discrete time evolution model for fracture networks

A discrete time evolution model for fracture networks

Polygonal tessellations as predictive models of molecular monolayers

An Evolution Model for Polygonal Tessellations as Models for Crack Networks and Other Natural Patterns

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