Skew parallelogram nets and universal factorization

We obtain many objects of discrete differential geometry as reductions of skew parallelogram nets, a system of lattice equations that may be formulated for any unit associative algebra. The Lax representation is linear in the spectral parameter, and paths in the lattice give rise to polynomial dependencies. We prove that generic polynomials in complex two by two matrices factorize, implying that skew parallelogram nets encompass all systems with such a polynomial representation. We demonstrate factorization in the context of discrete curves by constructing pairs of B\"acklund transformations that induce Euclidean motions on discrete elastic rods. More generally, we define a hierarchy of discrete curves by requiring such an invariance after an integer number of B\"acklund transformations. Moreover, we provide the factorization explicitly for discrete constant curvature surfaces and reveal that they are slices in certain 4D cross-ratio systems. Encompassing the discrete DPW method, this interpretation constructs such surfaces from given discrete holomorphic maps.Comment: 32 pages, 11 figure

Topology counts: force distributions in circular spring networks

Filamentous polymer networks govern the mechanical properties of many biological materials. Force distributions within these networks are typically highly inhomogeneous and, although the importance of force distributions for structural properties is well recognized, they are far from being understood quantitatively. Using a combination of probabilistic and graph-theoretical techniques we derive force distributions in a model system consisting of ensembles of random linear spring networks on a circle. We show that characteristic quantities, such as mean and variance of the force supported by individual springs, can be derived explicitly in terms of only two parameters: (i) average connectivity and (ii) number of nodes. Our analysis shows that a classical mean-field approach fails to capture these characteristic quantities correctly. In contrast, we demonstrate that network topology is a crucial determinant of force distributions in an elastic spring network.Comment: 5 pages, 4 figures. Missing labels in Fig. 4 added. Reference fixe

Wire mesh design

We present a computational approach for designing wire meshes, i.e., freeform surfaces composed of woven wires arranged in a regular grid. To facilitate shape exploration, we map material properties of wire meshes to the geometric model of Chebyshev nets. This abstraction is exploited to build an efficient optimization scheme. While the theory of Chebyshev nets suggests a highly constrained design space, we show that allowing controlled deviations from the underlying surface provides a rich shape space for design exploration. Our algorithm balances globally coupled material constraints with aesthetic and geometric design objectives that can be specified by the user in an interactive design session. In addition to sculptural art, wire meshes represent an innovative medium for industrial applications including composite materials and architectural façades. We demonstrate the effectiveness of our approach using a variety of digital and physical prototypes with a level of shape complexity unobtainable using previous methods

Topology Counts: Force Distributions in Circular Spring Networks

Filamentous polymer networks govern the mechanical properties of many biological materials. Force distributions within these networks are typically highly inhomogeneous, and, although the importance of force distributions for structural properties is well recognized, they are far from being understood quantitatively. Using a combination of probabilistic and graph-theoretical techniques, we derive force distributions in a model system consisting of ensembles of random linear spring networks on a circle. We show that characteristic quantities, such as the mean and variance of the force supported by individual springs, can be derived explicitly in terms of only two parameters: (i) average connectivity and (ii) number of nodes. Our analysis shows that a classical mean-field approach fails to capture these characteristic quantities correctly. In contrast, we demonstrate that network topology is a crucial determinant of force distributions in an elastic spring network. Our results for 1D linear spring networks readily generalize to arbitrary dimensions

Topology determines force distributions in one-dimensional random spring networks

Networks of elastic fibers are ubiquitous in biological systems and often provide mechanical stability to cells and tissues. Fiber-reinforced materials are also common in technology. An important characteristic of such materials is their resistance to failure under load. Rupture occurs when fibers break under excessive force and when that failure propagates. Therefore, it is crucial to understand force distributions. Force distributions within such networks are typically highly inhomogeneous and are not well understood. Here we construct a simple one-dimensional model system with periodic boundary conditions by randomly placing linear springs on a circle. We consider ensembles of such networks that consist of N nodes and have an average degree of connectivity z but vary in topology. Using a graph-theoretical approach that accounts for the full topology of each network in the ensemble, we show that, surprisingly, the force distributions can be fully characterized in terms of the parameters (N,z). Despite the universal properties of such (N,z) ensembles, our analysis further reveals that a classical mean-field approach fails to capture force distributions correctly. We demonstrate that network topology is a crucial determinant of force distributions in elastic spring networks

Curvature in Biological Systems: Its Quantification, Emergence, and Implications across the Scales

© 2023 The Authors. Advanced Materials published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.Surface curvature both emerges from, and influences the behavior of, living objects at length scales ranging from cell membranes to single cells to tissues and organs. The relevance of surface curvature in biology is supported by numerous experimental and theoretical investigations in recent years. In this review, first, a brief introduction to the key ideas of surface curvature in the context of biological systems is given and the challenges that arise when measuring surface curvature are discussed. Giving an overview of the emergence of curvature in biological systems, its significance at different length scales becomes apparent. On the other hand, summarizing current findings also shows that both single cells and entire cell sheets, tissues or organisms respond to curvature by modulating their shape and their migration behavior. Finally, the interplay between the distribution of morphogens or micro-organisms and the emergence of curvature across length scales is addressed with examples demonstrating these key mechanistic principles of morphogenesis. Overall, this review highlights that curved interfaces are not merely a passive by-product of the chemical, biological, and mechanical processes but that curvature acts also as a signal that co-determines these processes.A.P.G.C. and P.R.F. acknowledge the funding from Fundação para a Ciência e Tecnologia (Portugal), through IDMEC, under LAETA project UIDB/50022/2020. T.H.V.P. acknowledges the funding from Fundação para a Ciência e Tecnologia (Portugal), through Ph.D. Grant 2020.04417.BD. A.S. acknowledges that this work was partially supported by the ATTRACT Investigator Grant (no. A17/MS/11572821/MBRACE, to A.S.) from the Luxembourg National Research Fund. The author thanks Gerardo Ceada for his help in the graphical representations. N.A.K. acknowledges support from the European Research Council (grant 851960) and the Gravitation Program “Materials Driven Regeneration,” funded by the Netherlands Organization for Scientific Research (024.003.013). M.B.A. acknowledges support from the French National Research Agency (grant ANR-201-8-CE1-3-0008 for the project “Epimorph”). G.E.S.T. acknowledges funding by the Australian Research Council through project DP200102593. A.C. acknowledges the funding from the Deutsche Forschungsgemeinschaft (DFG) Emmy Noether Grant CI 203/-2 1, the Spanish Ministry of Science and Innovation (PID2021-123013O-BI00) and the IKERBASQUE Basque Foundation for Science.Peer reviewe

Chebyshev nets from commuting PolyVector fields

We propose a method for computing global Chebyshev nets on triangular meshes. We formulate the corresponding global parameterization problem in terms of commuting PolyVector fields, and design an efficient optimization method to solve it. We compute, for the first time, Chebyshev nets with automatically-placed singularities, and demonstrate the realizability of our approach using real material

The Sphereprint: An Approach to Quantifying the Conformability of Flexible Materials

The Sphereprint is introduced as a means to characterize hemispherical conformability, even when buckling occurs, in a variety of flexible materials such as papers, textiles, nonwovens, films, membranes, and biological tissues. Conformability is defined here as the ability to fit a doubly curved surface without folding. Applications of conformability range from the fit of a wound dressing, artificial skin, or wearable electronics around a protuberance such as a knee or elbow to geosynthetics used as reinforcements. Conformability of flexible materials is quantified by two dimensionless quantities derived from the Sphereprint. The Sphereprint ratio summarizes how much of the specimen conforms to a hemisphere under symmetric radial loading. The coefficient of expansion approximates the average stretching of the specimen during deformation, accounting for hysteresis. Both quantities are reproducible and robust, even though a given material folds differently each time it conforms. For demonstration purposes, an implementation of the Sphereprint test methodology was performed on a collection of cellulosic fibrous assemblies. For this example, the Sphereprint ratio ranked the fabrics according to intuition from least to most conformable in the sequence: paper towel, plain weave, satin weave, and single knit jersey. The coefficient of expansion distinguished the single knit jersey from the bark weave fabric, despite them having similar Sphereprint ratios and, as expected, the bark weave stretched less than the single knit jersey did during conformance. This work lays the foundation for engineers to quickly and quantitatively compare the conformance of existing and new flexible materials, no matter their construction