@article{hoffmann_sageman-furnas_steinmeier_2024, title={Skew parallelogram nets and universal factorization}, url={https://arxiv.org/abs/2401.08467}, DOI={10.48550/ARXIV.2401.08467}, abstractNote={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.}, publisher={arXiv}, author={Hoffmann, Tim and Sageman-Furnas, Andrew O. and Steinmeier, Jannik}, year={2024} }
 @article{bobenko_hoffmann_sageman-furnas_2023, title={Isothermic tori with one family of planar curvature lines and area constrained hyperbolic elastica}, url={https://arxiv.org/abs/2312.14956}, DOI={10.48550/ARXIV.2312.14956}, abstractNote={In 1883, Darboux gave a local classification of isothermic surfaces with one family of planar curvature lines using complex analytic methods. His choice of real reduction cannot contain tori. We classify isothermic tori with one family of planar curvature lines. They are found in the second real reduction of Darboux's description. We give explicit theta function formulas for the family of plane curves. These curves are particular area constrained hyperbolic elastica. With a Euclidean gauge, the Euler--Lagrange equation is lower order than expected. In our companion paper (arXiv:2110.06335) we use such isothermic tori to construct the first examples of compact Bonnet pairs: two isometric tori related by a mean curvature preserving isometry. They are also the first pair of isometric compact immersions that are analytic. Additionally, we study the finite dimensional moduli space characterizing when the second family of curvature lines is spherical. Isothermic tori with planar and spherical curvature lines are natural generalizations of Wente constant mean curvature tori, discovered in 1986. Wente tori are recovered in a limit case of our formulas.}, publisher={arXiv}, author={Bobenko, Alexander I. and Hoffmann, Tim and Sageman-Furnas, Andrew O.}, year={2023} }
 @article{schamberger_roschger_ziege_anselme_amar_bykowski_castro_cipitria_coles_dimova_et al._2022, title={Curvature in Biological Systems: Its quantification, Emergence and Implications Across the Scales}, volume={12}, url={http://dx.doi.org/10.1002/adma.202206110}, DOI={10.1002/adma.202206110}, abstractNote={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.}, journal={Advanced Materials}, publisher={Wiley}, author={Schamberger, Barbara and Roschger, Andreas and Ziege, Ricardo and Anselme, Karine and Amar, Martine Ben and Bykowski, Michał and Castro, André P. G. and Cipitria, Amaia and Coles, Rhoslyn and Dimova, Rumiana and et al.}, year={2022}, month={Dec}, pages={2206110} }
 @article{bobenko_hoffmann_sageman-furnas_2023, title={Compact Bonnet Pairs: isometric tori with the same curvatures}, url={https://arxiv.org/abs/2110.06335}, DOI={10.48550/ARXIV.2110.06335}, abstractNote={We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These Bonnet pair tori are the first examples of compact Bonnet pairs. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique smooth compact immersion. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. Our construction uses the relationship between Bonnet pairs and isothermic surfaces. The Bonnet pair tori arise as conformal transformations of an isothermic torus with one family of planar curvature lines. We classify such isothermic tori in our companion paper (arXiv:2312.14956). The above approach stems from computational investigations of a 5x7 quad decomposition of a torus using a discrete differential geometric analog of isothermic surfaces and Bonnet pairs.}, publisher={arXiv}, author={Bobenko, Alexander I. and Hoffmann, Tim and Sageman-Furnas, Andrew O.}, year={2023} }
 @article{sageman-furnas_chern_ben-chen_vaxman_2019, title={Chebyshev nets from commuting PolyVector fields}, volume={38}, url={http://dx.doi.org/10.1145/3355089.3356564}, DOI={10.1145/3355089.3356564}, abstractNote={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.}, number={6}, journal={ACM Transactions on Graphics}, publisher={Association for Computing Machinery (ACM)}, author={Sageman-Furnas, Andrew O. and Chern, Albert and Ben-Chen, Mirela and Vaxman, Amir}, year={2019}, month={Nov}, pages={1–16} }
 @article{heidemann_sageman-furnas_sharma_rehfeldt_schmidt_wardetzky_2018, title={Topology Counts: Force Distributions in Circular Spring Networks}, volume={120}, url={http://dx.doi.org/10.1103/physrevlett.120.068001}, DOI={10.1103/physrevlett.120.068001}, abstractNote={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.}, number={6}, journal={Physical Review Letters}, publisher={American Physical Society (APS)}, author={Heidemann, Knut M. and Sageman-Furnas, Andrew O. and Sharma, Abhinav and Rehfeldt, Florian and Schmidt, Christoph F. and Wardetzky, Max}, year={2018}, month={Feb} }
 @article{heidemann_sageman-furnas_sharma_rehfeldt_schmidt_wardetzky_2018, title={Topology determines force distributions in one-dimensional random spring networks}, volume={97}, url={http://dx.doi.org/10.1103/physreve.97.022306}, DOI={10.1103/physreve.97.022306}, abstractNote={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.}, number={2}, journal={Physical Review E}, publisher={American Physical Society (APS)}, author={Heidemann, Knut M. and Sageman-Furnas, Andrew O. and Sharma, Abhinav and Rehfeldt, Florian and Schmidt, Christoph F. and Wardetzky, Max}, year={2018}, month={Feb} }
 @article{baek_sageman-furnas_jawed_reis_2018, title={Form finding in elastic gridshells}, volume={115}, url={http://dx.doi.org/10.1073/pnas.1713841115}, DOI={10.1073/pnas.1713841115}, abstractNote={Significance Elastic gridshells arise from the buckling of an initially planar grid of rods. The interaction of elasticity and geometric constraints makes their actuated shapes difficult to predict using classical methods. However, recent progress in extreme mechanics reveals the benefits of structures that buckle by design, when exploiting underlying geometry. Here, we demonstrate the geometry-driven nature of elastic gridshells. We use a geometric model, originally for woven fabric, to rationalize their actuated shapes and describe their nonlocal response to loading. Validation is provided with precision experiments and rod-based simulations. The prominence of geometry in elastic gridshells that we identify should allow for our results to transfer across length scales from architectural structures to micro/nano–1-df mechanical actuators and self-assembly systems.}, number={1}, journal={Proceedings of the National Academy of Sciences}, publisher={Proceedings of the National Academy of Sciences}, author={Baek, Changyeob and Sageman-Furnas, Andrew O. and Jawed, Mohammad K. and Reis, Pedro M.}, year={2018}, month={Jan}, pages={75–80} }
 @article{hoffmann_sageman-furnas_2016, title={A 2x2 Lax Representation, Associated Family, and Bäcklund Transformation for Circular K-Nets}, volume={56}, url={http://dx.doi.org/10.1007/s00454-016-9802-6}, DOI={10.1007/s00454-016-9802-6}, number={2}, journal={Discrete & Computational Geometry}, publisher={Springer Science and Business Media LLC}, author={Hoffmann, Tim and Sageman-Furnas, Andrew O.}, year={2016}, month={Sep}, pages={472–501} }
 @article{hoffmann_sageman-furnas_wardetzky_2016, title={A Discrete Parametrized Surface Theory in R^3}, volume={7}, url={http://dx.doi.org/10.1093/imrn/rnw015}, DOI={10.1093/imrn/rnw015}, abstractNote={We propose a discrete surface theory in ℝ3 that unites the most prevalent versions of discrete special parametrizations. Our theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. Our theory is not restricted to integrable geometries, but extends to a general surface theory. A quad net is a map from a strongly regular polytopal cell decomposition of a surface with all faces being quadrilaterals into ℝ3 with nonvanishing straight edges. A polytopal cell decomposition is strongly regular if each edge connects distinct vertices and meets at most two faces. Notice, in particular, that nonplanar faces are admissible. In discrete differential geometry, quad nets are understood as discretizations of parametrized surfaces [9, 10, 13]. In this agenda many classes of special surfaces have been discretized using algebro-geometric approaches for integrable geometry—originally using discrete analogues of soliton theory techniques (e.g., discrete Lax pairs and finite-gap integration [6]) to construct nets, but more recently using the notion of 3D consistency (reviewed in [10]). As in the smooth setting, these approaches have been successfully applied to space forms (see, e.g., [15–17, 22]). As an example consider the case of K-surfaces, i.e., surfaces of constant negative Gauß curvature. The integrability equations of classical surface theory are equivalent to the famous sine-Gordon equation [1, 20]. In an integrable discretization, the sine-Gordon equation becomes a finite difference equation for which integrability is encoded by a certain closing condition around a 3D cube. Both in the smooth and discrete settings, integrability is bound to specific choices of parameterizations, such as asymptotic line parametrizations for K-surfaces. In this way different classes of surfaces, such as minimal surfaces or surfaces of constant mean curvature (CMC), lead to different partial differential equations and give rise to different parametrizations. In the discrete case, this is reflected by developments that treat different special surfaces by disparate approaches. These integrable discretizations maintain characteristic properties of their smooth counterparts (e.g., the transformation theory of Darboux, Bäcklund, Bianchi, etc.). What has been lacking, however, is a unified discrete theory that lifts the restriction of special surface parametrizations. Indeed, different from the case of classical smooth surface theory, existing literature does not provide a general discrete theory for quad nets.}, journal={International Mathematics Research Notices}, publisher={Oxford University Press (OUP)}, author={Hoffmann, Tim and Sageman-Furnas, Andrew O. and Wardetzky, Max}, year={2016}, month={Jul}, pages={rnw015} }
 @inproceedings{sageman-furnas_umetani_schmidt_2015, title={Meltables: fabrication of complex 3D curves by melting}, url={http://dx.doi.org/10.1145/2820903.2820915}, DOI={10.1145/2820903.2820915}, abstractNote={We propose a novel approach to fabricating complex 3D shapes via physical deformation of simpler shapes. Our focus is on objects composed of a set of planar beams and joints, where the joints are thin parts of the object which temporarily become living hinges when heated, close to a fixed angle defined by the local shape, and then become rigid when cooled. We call this class of objects Meltables. We present a novel algorithm that computes an optimal joint sequence which approximates a 3D spline curve while satisfying fabrication constraints. This technique is used in an interactive Meltables design tool. We demonstrate a variety of Meltables, fabricated with both 3D-printing and standard PVC piping.}, booktitle={SIGGRAPH Asia 2015 Technical Briefs}, publisher={ACM}, author={Sageman-Furnas, Andrew O. and Umetani, Nobuyuki and Schmidt, Ryan}, year={2015}, month={Nov} }
 @article{garg_sageman-furnas_deng_yue_grinspun_pauly_wardetzky_2014, title={Wire mesh design}, volume={33}, url={http://dx.doi.org/10.1145/2601097.2601106}, DOI={10.1145/2601097.2601106}, abstractNote={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.}, number={4}, journal={ACM Transactions on Graphics}, publisher={Association for Computing Machinery (ACM)}, author={Garg, Akash and Sageman-Furnas, Andrew O. and Deng, Bailin and Yue, Yonghao and Grinspun, Eitan and Pauly, Mark and Wardetzky, Max}, year={2014}, month={Jul}, pages={1–12} }
 @article{sageman-furnas_goswami_menon_russell_2014, title={The Sphereprint: An approach to quantifying the conformability of flexible materials}, volume={84}, url={http://dx.doi.org/10.1177/0040517513512402}, DOI={10.1177/0040517513512402}, abstractNote={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.}, number={8}, journal={Textile Research Journal}, publisher={SAGE Publications}, author={Sageman-Furnas, Andrew O and Goswami, Parikshit and Menon, Govind and Russell, Stephen J}, year={2014}, month={May}, pages={793–807} }