Regular Tessellations Research Articles

Clockwise vs anti-clockwise chiral metamaterials: Influence of base tessellation symmetry and rotational direction of chiralisation on mechanical properties

Chiral honeycombs are a class of auxetic metamaterials which are characterised by a lack of plane symmetry. Besides the traditional chiral honeycombs based on regular monohedral tessellations such as the hexachiral and tetrachiral honeycombs, a vast range of auxetic chiral metamaterials may be produced through the ‘chiralisation’ of Euclidean tessellations made from polyhedral tilings and/or irregular monohedral polygons. In this work, we show for the first time, how the direction of the geometric chiral transformation, i.e. clockwise vs anti-clockwise chiralisation, has an influence on the mechanical properties and deformation modes of the resultant chiral metamaterial systems. This influence, which is a result of the intrinsic absence of axial symmetry in the original base tessellation, can lead to the production of two unique and distinct chiral metamaterial configurations with completely different mechanical properties originating from a single base tessellation. In this work we demonstrate and quantify, through a wide range of numerical simulations and experimental tests on additively-manufactured prototypes, the effect of the direction of chiralisation on two tessellations: a Florent pentagonal system with hexagonal rotational symmetry and a hexagonal monohedral tessellation with trigonal rotational symmetry. The results obtained show that the clockwise and anti-clockwise chiral structures exhibit significantly different mechanical properties and deformation modes, highlighting the increased versatility of this relatively novel class of chiral metamaterials.

Integrable symplectic maps with a polygon tessellation

Identifying integrable dynamics remains a formidable challenge, and despite centuries of research, only a handful of examples are known to date. In this article, we explore a distinct form of area-preserving (symplectic) mappings derived from the stroboscopic Poincaré cross section of a kicked rotator—an oscillator subjected to an external force periodically switched on in short pulses. The significance of this class of problems extends to various applications in physics and mathematics, including particle accelerators, crystallography, and studies of chaos. Notably, Suris's theorem constrains the integrability within this category of mappings, outlining potential scenarios with analytic invariants of motion. In this paper, we challenge the assumption of the analyticity of the invariant by exploring piecewise linear transformations on a torus (T2) and associated systems on the plane (R2), incorporating arithmetic quasiperiodicity and discontinuities. Introducing a new automated technique, we discovered previously unknown scenarios featuring polygonal invariants that form perfect tessellations and, moreover, fibrations of the plane or torus. This work reveals a novel category of planar tilings characterized by discrete symmetries that emerge from the invertibility of transformations and are intrinsically linked to the presence of integrability. Our algorithm relies on the analysis of the Poincaré rotation number and its piecewise monotonic nature for integrable cases, contrasting with the noisy behavior in the case of chaos, thereby allowing for clear separation. Some of the newly discovered systems exhibit the peculiar behavior of “integrable diffusion,” characterized by infinite and quasirandom hopping between tiles while being confined to a set of invariant segments. Finally, through the implementation of a smoothening procedure, all mappings can be generalized to quasi-integrable scenarios with suppressed volume occupied by chaotic trajectories, thereby opening doors to potential practical applications. Published by the American Physical Society 2024

When rational functions meet virtual elements: the lightning virtual element method

We propose a lightning Virtual Element Method that eliminates the stabilisation term by actually computing the virtual component of the local VEM basis functions using a lightning approximation. In particular, the lightning VEM approximates the virtual part of the basis functions using rational functions with poles clustered exponentially close to the corners of each element of the polygonal tessellation. This results in two great advantages. First, the mathematical analysis of a priori error estimates is much easier and essentially identical to the one for any other non-conforming Galerkin discretisation. Second, the fact that the lightning VEM truly computes the basis functions allows the user to access the point-wise value of the numerical solution without needing any reconstruction techniques. The cost of the local construction of the VEM basis is the implementation price that one has to pay for the advantages of the lightning VEM method, but the embarrassingly parallelizable nature of this operation will ultimately result in a cost-efficient scheme almost comparable to standard VEM and FEM.

Three-Point-Star Deoxyribonucleic Acid Tiles with the Core Arm Length at Three Half-Turns for Two-Dimensional Archimedean Tilings and Beyond.

2D Archimedean tiling and complex tessellation patterns assembled from soft materials including modular DNA tiles have attracted great interest because of their specific structures and potential applications in nanofabrication, nanoelectronics, nanophotonics, biomedical sensing, drug delivery, therapeutics, etc. Traditional three- and four-point-star DNA tiles with the core arm length at two half-turns (specified as three- and four-point-star-E previously and abbreviated as 3PSE and 4PSE tiles here) have been applied to assemble intricate tessellations through tuning the size of inserted nT (n = 1-7, T is thymine) loops on helper strands at the tile center. Following our recent findings using a new type of four-point-star tiles with the core arm length at three half-turns (specified as four-point-star-O previously and abbreviated as 4PSO tiles here) to assemble DNA tubes and flat 2D arrays, we report here the cross-hybridization weaving architectures at the tile center to construct three new 3PSO tiles with circular DNA oligonucleotides of 96-nt (nucleotides) serving as the scaffolds, further the monotonous and combinatory E- and O-tilings on one type of 3PSO tiles to create 2D Archimedean tiling patterns (6.6.6) and (4.8.8), and finally, the combination of 3PSO with 4PSO as well as 2PSO tiles to tile into complex tessellation patterns. The easy realization of regular and intricate DNA tessellations with 2-4PSO tiles not only richens the fundamental DNA modules and complex DNA nanostructures in types but also broadens the potential application scopes of DNA nanostructures in nanofabrication, DNA computing, biomedicine, etc.

Pearcey universality at cusps of polygonal lozenge tilings

AbstractWe study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved by Aggarwal‐Huang and Aggarwal‐Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with nonintersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formulas.

Prism[2]dihydrophenazines: Synthesis, Configurational Analysis, and Supramolecular Tessellation through Exo-Wall Interactions.

Macrocyclic arenes have gained considerable attention for their structural diversity and widespread applications. In this research, a new kind of macrocyclic arenes, namely prism[2]dihydrophenazines (anti-P2P20, syn-P2P20, and P2P22), composed of two dihydrophenazine derivatives subunits bridged by methylene groups, were conveniently synthesized by AlCl3-catalyzed one-pot condensation in 1,2-dichloroethane. Both anti-P2P20 and its isomer syn-P2P20 exhibited flexible and convertible conformation with narrow cavity, while P2P22 possessed rigid and rhombic-like skeleton due to the more steric hindrance on subunits. In addition, the selection of electron-deficient guest was found to influence the outside binding behavior of syn-P2P20. Fantastic regular supramolecular tessellation was fabricated by tiling of syn-P2P20 with tetrafluoro-1,4-benzoquinone (TFB) through the exo-wall interactions. Using 1,5-difluoro-2,4-dinitrobenzene (DFN) as a linker, only the regular 2D network superstructure with periodic units in a plane was obtained through cocrystallization. This work not only reports the construction of supramolecular tessellations by using prism[2]dihydrophenazines as building blocks, but also provides a new perspective for the design of macrocyclic arenes and fabrication of 2D supramolecular materials.

Anomalous and Chern topological waves in hyperbolic networks

Hyperbolic lattices are a new type of synthetic materials based on regular tessellations in non-Euclidean spaces with constant negative curvature. While so far, there has been several theoretical investigations of hyperbolic topological media, experimental work has been limited to time-reversal invariant systems made of coupled discrete resonances, leaving the more interesting case of robust, unidirectional edge wave transport completely unobserved. Here, we report a non-reciprocal hyperbolic network that exhibits both Chern and anomalous chiral edge modes, and implement it on a planar microwave platform. We experimentally evidence the unidirectional character of the topological edge modes by direct field mapping. We demonstrate the topological origin of these hyperbolic chiral edge modes by an explicit topological invariant measurement, performed from external probes. Our work extends the reach of topological wave physics by allowing for backscattering-immune transport in materials with synthetic non-Euclidean behavior.

Mechanical properties and failure modes of additively-manufactured chiral metamaterials based on Euclidean tessellations: an experimental and finite element study

PurposeThe “chiralisation” of Euclidean polygonal tessellations is a novel, recent method which has been used to design new auxetic metamaterials with complex topologies and improved geometric versatility over traditional chiral honeycombs. This paper aims to design and manufacture chiral honeycombs representative of four distinct classes of 2D Euclidean tessellations with hexagonal rotational symmetry using fused-deposition additive manufacturing and experimentally analysed the mechanical properties and failure modes of these metamaterials.Design/methodology/approachFinite Element simulations were also used to study the high-strain compressive performance of these systems under both periodic boundary conditions and realistic, finite conditions. Experimental uniaxial compressive loading tests were applied to additively manufactured prototypes and digital image correlation was used to measure the Poisson’s ratio and analyse the deformation behaviour of these systems.FindingsThe results obtained demonstrate that these systems have the ability to exhibit a wide range of Poisson’s ratios (positive, quasi-zero and negative values) and stiffnesses as well as unusual failure modes characterised by a sequential layer-by-layer collapse of specific, non-adjacent ligaments. These findings provide useful insights on the mechanical properties and deformation behaviours of this new class of metamaterials and indicate that these chiral honeycombs could potentially possess anomalous characteristics which are not commonly found in traditional chiral metamaterials based on regular monohedral tilings.Originality/valueTo the best of the authors’ knowledge, the authors have analysed for the first time the high strain behaviour and failure modes of chiral metamaterials based on Euclidean multi-polygonal tessellations.

Hyperbolic photonic topological insulators

Topological photonics provides a new degree of freedom to robustly control electromagnetic fields. To date, most of established topological states in photonics have been employed in Euclidean space. Motivated by unique properties of hyperbolic lattices, which are regular tessellations in non-Euclidean space with a constant negative curvature, the boundary-dominated hyperbolic topological states have been proposed. However, limited by highly crowded boundary resonators and complicated site couplings, the hyperbolic topological insulator has only been experimentally constructed in electric circuits. How to achieve hyperbolic photonic topological insulators is still an open question. Here, we report the experimental realization of hyperbolic photonic topological insulators using coupled ring resonators on silicon chips. Boundary-dominated one-way edge states with pseudospin-dependent propagation directions have been observed. Furthermore, the robustness of edge states in hyperbolic photonic topological insulators is also verified. Our findings have potential applications in the field of designing high-efficient topological photonic devices with enhanced boundary responses.

Self‐avoiding walks and polygons on hyperbolic graphs

AbstractWe prove that for the ‐regular tessellations of the hyperbolic plane by ‐gons, there are exponentially more self‐avoiding walks of length than there are self‐avoiding polygons of length . We then prove that this property implies that the self‐avoiding walk is ballistic, even on an arbitrary vertex‐transitive graph. Moreover, for every fixed , we show that the connective constant for self‐avoiding walks satisfies the asymptotic expansion as ; on the other hand, the connective constant for self‐avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self‐avoiding walks of length is comparable to the th power of their connective constant. Some of these results were previously obtained by Madras and Wu for all but finitely many regular tessellations of the hyperbolic plane.

Regular Tessellations of Maximally Symmetric Hyperbolic Manifolds

We first briefly summarize several well-known properties of regular tessellations of the three two-dimensional maximally symmetric manifolds, E2, S2, and H2, by bounded regular tiles. For instance, there exist infinitely many regular tessellations of the hyperbolic plane H2 by curved hyperbolic equilateral triangles whose vertex angles are 2π/d for d=7,8,9,… On the other hand, we prove that there is no curved hyperbolic regular tetrahedron which tessellates the three-dimensional hyperbolic space H3. We also show that a regular tessellation of H3 can only consist of the hyperbolic cubes, hyperbolic regular icosahedra, or two types of hyperbolic regular dodecahedra. There exist only two regular hyperbolic space-fillers of H4. If n>4, then there exists no regular tessellation of Hn.

Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations.

Tessellations of the hyperbolic spaces by regular polygons support discrete quantum and classical models with unique spectral and topological characteristics. Resolving the true bulk spectra and the thermodynamic response functions of these models requires converging periodic boundary conditions and our Letter delivers a practical and rigorous solution for this open problem on generic {p,q}-tessellations. This enables us to identify the true spectral gaps of bulk Hamiltonians and construct all but one topological models that deliver the topological gaps predicted by the K theory of the lattices. We demonstrate the emergence of the expected topological spectral flows whenever two such bulk models are deformed into each other and prove the emergence of topological channels whenever a soft physical interface is created between different topological classes of Hamiltonians.

Asymptotics of noncolliding q-exchangeable random walks

We consider a process of noncolliding q-exchangeable random walks on making steps 0 (‘straight’) and −1 (‘down’). A single random walk is called q-exchangeable if under an elementary transposition of the neighboring steps the probability of the trajectory is multiplied by a parameter . Our process of m noncolliding q-exchangeable random walks is obtained from the independent q-exchangeable walks via the Doob’s h-transform for a nonnegative eigenfunction h (expressed via the q-Vandermonde product) with the eigenvalue less than 1. The system of m walks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is the q-analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive half-line and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncolliding q-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel of q-distributed random lozenge tilings of sawtooth polygons. In the limit as , with γ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel.

Methodology for Designing Kinetic Planar Surface through Tessellation

Abstract: This paper illustrates the creation of kinetic planar surface by tessellation method. Tessellation is a traditional technique to usually cover a plane without gaps and create a decorative pattern. As we talk about kinetic design, it is an attractive strategy to solve the adaptive and responsive environmental architectural problems. Mathematical tessellation are the techniques to be used to design a surface. The purpose of this study is to develop a methodology for designing the kinetic planar surface with the help of regular tessellation technique in the area of architecture, mechanical and mathematical interdisciplinary approach. Her the method comprises of repetition of interconnected geometry, whose design derive from the symmetry that is taken from the original reference tessellation.

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

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.

On face-magic labelings of regular tesselations

On face-magic labelings of regular tesselations

Finetuning Discrete Architectural Surfaces by use of Circle Packing

ABSTRACT This paper presents an algorithmic approach for the conceptual design of architectural surfaces represented by triangulated meshes. Specifically, we propose a method to optimise a surface according to user-specified geometric properties including the distribution of the Gaussian curvature and preferable boundary location. Designing a surface manually with specific Gaussian curvatures can be a time-consuming task, and the proposed method automates this task. Also, in the proposed approach, the resulting mesh could be encouraged to form a regular tessellation or kept close to those of the initial one. Our method relies on the idea in computational conformal geometry called circle packing and the discrete Ricch energy, which have been used for surface modelling. We develop a least-squares-based optimisation scheme by introducing a variant of the Ricci energy to accommodate flexibility in specifying design constraints such as boundary locations and convexity of the spanned surface, which are essential to architectural applications. We provide an open-source implementation of our method in Python. 1 1 Our codes are publicly available at https://github.com/shizuo-kaji/ricci_flow

The Activities Framework on Project-Based Learning: The Use of Autodesk Sketchbook to Improve Students' Metacognition Thinking Skills in Solving Polygon Tessellation Problems

The Activities Framework on Project-Based Learning: The Use of Autodesk Sketchbook to Improve Students' Metacognition Thinking Skills in Solving Polygon Tessellation Problems

Mesh quality agglomeration algorithm for the virtual element method applied to discrete fracture networks

We propose a quality-based optimization strategy to reduce the total number of degrees of freedom associated with a discrete problem defined over a polygonal tessellation with the Virtual Element Method. The presented Quality Agglomeration algorithm relies only on the geometrical properties of the problem polygonal mesh, agglomerating groups of neighboring elements. We test this approach in the context of fractured porous media, in which the generation of a global conforming mesh on a Discrete Fracture Network leads to a considerable number of unknowns, due to the presence of highly complex geometries (e.g. thin triangles, large angles, small edges) and the significant size of the computational domains. We show the efficiency and the robustness of our approach, applied independently on each fracture for different network configurations, exploiting the flexibility of the Virtual Element Method in handling general polygonal elements.

Engineering boundary-dominated topological states in defective hyperbolic lattices

Topological matters with lattice disclinations have been widely investigated in Euclidean space. In recent years, exotic topological phases in hyperbolic lattices, which are regular tessellations in the curved space with a constant negative curvature, have been theoretically proposed and experimentally observed, while the investigation of topologically defective hyperbolic lattices is still lacking. Here, we study topological states in the hyperbolic lattice with polygonal defects. It is shown that the topologically one-way boundary state can still exist in the defective hyperbolic Haldane model. Interestingly, we find that only a single bulk site can induce the formation of boundary-dominated one-way propagations in defective hyperbolic lattices. In experiments, we fabricate the defective hyperbolic circuit with a single bulk site to detect the boundary-dominated topological state. Frequency-dependent impedance responses clearly illustrate the existence of nontrivial band gaps and midgap topological boundary states. Furthermore, the backscattering-immune propagation protected by a single bulk site is observed by measuring the dynamics of voltage packet. Our work suggests a useful platform to study topological phases in defective hyperbolic lattices, and may have potential applications in designing high-efficient topological devices with an extremely small bulk region.