Two-dimensional Tiling Research Articles

Hairygami: Analysis of DNA Nanostructures' Conformational Change Driven by Functionalizable Overhangs.

DNA origami is a widely used method to construct nanostructures by self-assembling designed DNA strands. These structures are often used as "pegboards" for templated assembly of proteins, gold nanoparticles, aptamers, and other molecules, with applications ranging from therapeutics and diagnostics to plasmonics and photonics. Imaging these structures using atomic force microscopy (AFM) or transmission electron microscope (TEM) does not capture their full conformation ensemble as they only show their shape flattened on a surface. However, certain conformations of the nanostructure can position guest molecules into distances unaccounted for in their intended design, thus leading to spurious interactions between guest molecules that are designed to be separated. Here, we use molecular dynamics simulations to capture a conformational ensemble of two-dimensional (2D) DNA origami tiles and show that introducing single-stranded overhangs, which are typically used for functionalization of the origami with guest molecules, induces a curvature of the tile structure in the bulk. We show that the shape deformation is of entropic origin, with implications for the design of robust DNA origami breadboards as well as a potential approach to modulate structure shape by introducing overhangs. We then verify experimentally that the DNA overhangs introduce curvature into the DNA origami tiles under divalent as well as monovalent salt buffer conditions. We further experimentally verify that DNA origami functionalized with attached proteins also experiences such induced curvature. We provide the developed simulation code implementing the enhanced sampling to characterize the conformational space of DNA origami as open source software.

Multiplication cubes and multiplication automata

Abstract We extend previously known two-dimensional multiplication tiling systems that simulate multiplication by two natural numbers p and q in base $pq$ to higher dimensional multiplication tessellation systems. We develop the theory of these systems and link different multiplication tessellation systems with each other via macrotile operations that glue cubes in one tessellation system into larger cubes of another tessellation system. The macrotile operations yield topological conjugacies and factor maps between cellular automata performing multiplication by positive numbers in various bases.

Symmetry-Guided Inverse Design of Self-Assembling Multiscale DNA Origami Tilings.

Recent advances enable the creation of nanoscale building blocks with complex geometries and interaction specificities for self-assembly. This nearly boundless design space necessitates design principles for defining the mutual interactions between multiple particle species to target a user-specified complex structure or pattern. In this article, we develop a symmetry-based method to generate the interaction matrices that specify the assembly of two-dimensional tilings, which we illustrate using equilateral triangles. By exploiting the allowed 2D symmetries, we develop an algorithmic approach by which any periodic 2D tiling can be generated from an arbitrarily large number of subunit species, notably addressing an unmet challenge of engineering 2D crystals with periodicities that can be arbitrarily larger than the subunit size. To demonstrate the utility of our design approach, we encode specific interactions between triangular subunits synthesized by DNA origami and show that we can guide their self-assembly into tilings with a wide variety of symmetries, using up to 12 unique species of triangles. By conjugating specific triangles with gold nanoparticles, we fabricate gold-nanoparticle supracrystals whose lattice parameter spans up to 300 nm. Finally, to generate economical design rules, we compare the design economy of various tilings. In particular, we show that (1) higher symmetries allow assembly of larger unit cells with fewer subunits and (2) linear supracrystals can be designed more economically using linear primitive unit cells. This work provides a simple algorithmic approach to designing periodic assemblies, aiding in the multiscale assembly of supracrystals of nanostructured "meta-atoms" with engineered plasmonic functions.

Nearly Ideal Random Tiling with Dodecagonal Symmetry from Pentablock Quarterpolymers of the AB1CB2D Type.

Two-dimensional tiling manners as cross-sectional views of cylindrical domain assembly formed by pentablock quarterpolymers of the AB1CB2D type in bulk were investigated. Several binary and ternary blends from three mother polymers having different ϕB1/ϕB2 ratios (ϕB1 and ϕB2 are the volume fractions of the B1 and B2 blocks, respectively) represent nonperiodic but ordered triangle/square tilings, where the N3/N4 ratios (N3 and N4 are the numbers of triangles and squares in the observed area, respectively) are all close enough to the theoretical value of 4/√3 ≑ 2.31 for the dodecagonal quasicrystalline (DDQC) state, irrespective of the total number of polygons. The TEM images, having almost the same N3/N4 ratios, were proved to show 4- and 6-fold symmetries in terms of the angular appearance of equilateral polygon sides via image analyses. Among them, a ternary blend showed a nearly ideal random tiling pattern that is almost equivalent to the theoretically predicted tiling by SCFT. Moreover, the magnitude of phason strain estimated for a TEM image from the ternary blend was proved to be quite small when the observing area is narrow, while it deviates from the ideal quasicrystalline tiling with an increasing number of vertices in the observing area.

Hyperuniformity classes of quasiperiodic tilings via diffusion spreadability.

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α>0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wave number k≡|k| in the vicinity of the origin, e.g., S(k)∼|k|^{α} in cases where S(k) varies continuously with k as k→0. In this paper, we show that the spreadability is an effective method for determining α for quasiperiodic systems where S(k) is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Phys. Rev. E 104, 054102 (2021)10.1103/PhysRevE.104.054102] that, for media with finite α, the long-time behavior of the excess spreadability S(∞)-S(t) can be fit to a power law of the form t^{-(d-α)/2}, where d is the space dimension, to accurately extract α for the continuous case. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density χ[over ̃]_{_{V}}(k) of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract α for the one-dimensional (1D) limit-periodic period-doubling chain (α=1) and the 1D quasicrystalline Fibonacci chain (α=3) to within 0.02% of the analytically known exact results. Moreover, we obtain a value of α=5.97±0.06 for the two-dimensional Penrose tiling and present plausible theoretical arguments strongly suggesting that α is exactly equal to six. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-k region of the scattering information used to compute the spreadability and obtain an accurate value of α, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests that one can obtain a good estimate of α for an infinite self-similar quasicrystal from a modestly sized finite sample. The methods described here offer a simple and general procedure to characterize accurately the large-scale translational order present in quasicrystalline and limit-periodic media in any space dimension that are self-similar. Moreover, the scattering information extracted from these two-phase media encoded in χ[over ̃]_{_{V}}(k), can be used to estimate their physical properties, such as their effective dynamic dielectric constants, effective dynamic elastic constants, and fluid permeabilities.

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.

Droplet tilings in precessive fields: hysteresis, elastic defects, and annealing.

Two-component Marangoni contracted droplets can be arranged into arbitrary two-dimensional tiling patterns where they display rich dynamics due to vapor-mediated long-range interactions. Recent work has characterized the centered hexagonal honeycomb lattice, showing it to be a highly frustrated system with many metastable states and relaxation occurring over multiple timescales [Molina et al., Proc. Natl. Acad. Sci. U. S. A., 2021, 118, e2020014118]. Here, we study this system under the influence of a rotating gravitational field. High amplitudes are able to completely disrupt droplet-droplet interactions, making it possible to identify a transition between field-dominated and interaction-dominated regimes. The system displays complex hysteresis behavior, the details of which are connected to the emergence of linear mesoscale structures. These mesoscale features display an elasticity that is governed by the balance between gravity and long-range vapor-mediated attractions. We find that disorder plays an important role in determining the dynamics of these features. Finally, we demonstrate annealing the system by progressively reducing the field amplitude, a process that reduces configurational energy compared to a rapid quench. The ability to manipulate vapor-mediated interactions in deliberately designed droplet tilings provides a novel platform for table-top explorations of multi-body interactions.

Robustness of Flat Bands on the Perturbed Kagome and the Perturbed Super-Kagome Lattice

We study spectral properties of perturbed discrete Laplacians on two-dimensional Archimedean tilings. The perturbation manifests itself in the introduction of non-trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian exhibits flat bands, namely the (3.6)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3.6)^2$$\\end{document} Kagome lattice and the (3.122)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3.12^2)$$\\end{document} “Super-Kagome” lattice. We characterize all possible choices for edge weights which lead to flat bands. Furthermore, we discuss spectral consequences such as the emergence of new band gaps. Among our main findings is that flat bands are robust under physically reasonable assumptions on the perturbation, and we completely describe the perturbation-spectrum phase diagram. The two flat bands in the Super-Kagome lattice are shown to even exhibit an “all-or-nothing” phenomenon in the sense that there is no perturbation, which can destroy only one flat band while preserving the other.

Tactile perception of randomness: Effect of varying stimulus size and participants age.

We investigated participants' ability to differentiate between random and organized two-dimensional tactile tiles with embossed dots and examined how this ability varies with size and participant age. Four experiments were conducted to evaluate the effect of these variations on participants' capacity to utilize touch in identifying which of two stimuli exhibited greater randomness. Participants were instructed to explore embossed tiles using both hands. The tiles had varying levels of randomness from organized to random sets. In Experiments 1, 2, and 4, the sets were of equal size, while in Experiment 3, they differed in size. Results revealed a significant difference between the random and organized sets, with random stimuli being more easily discernible. These findings suggest that touch can be utilized to discern random patterns on tactile maps or displays. However, older participants encountered difficulties making this distinction, indicating similarities between vision and touch in perceiving randomness.

K-theory of two-dimensional substitution tiling spaces from AF-algebras

Given a two-dimensional substitution tiling space, we show that, under some reasonable assumptions, the K-theory of the groupoid C⁎-algebra of its unstable groupoid can be explicitly reconstructed from the K-theory of the AF-algebras of the substitution rule and its analogue on the 1-skeleton. We prove this by generalizing the calculations done for the chair tiling in [18] using relative K-theory and excision, and packaging the result into an exact sequence purely in topology. From this exact sequence, it appears that one cannot use only ordinary K-theory to compute using the dimension-filtration on the unstable groupoid. Several examples are computed using Sage and the results are compiled in a table.

Static universality of the Ising and Blume–Capel models on two-dimensional Penrose tiles

We study the phase transitions and the critical behavior of the Ising and Blume–Capel models on two kinds of two-dimensional Penrose tiles employing Monte Carlo methods and detailed finite-size scaling analysis. The Wolff algorithm is used for the Ising model whereas the Wang–Landau and Metropolis algorithms are used for the Blume–Capel model to obtain critical temperatures and exponents. The phase diagram of the Blume–Capel model reveals the presence of the double transition, the reentrant behavior, and the tricriticality in the vicinity of the critical single-ion anisotropy. Our results indicate that continuous phase transitions of the Ising and Blume–Capel models on the Penrose tiles belong to the two-dimensional Ising universality class. However, the scaling functions depend on the spin magnitude, anisotropy, and graph shape. Lastly, we verify that the critical Binder cumulant is also universal but depends on the shape of the graph.

Confined states in the tight-binding model on the hexagonal golden-mean tiling

We study the tight-binding model with two distinct hoppings (tL, tS ) on the two-dimensional hexagonal golden-mean tiling and examine the confined states with E = 0, where E is the eigenenergy. Some confined states found in the case tL = tS are exact eigenstates even for the system with tL ≠ t S, where their amplitudes are smoothly changed. By contrast, the other states are no longer eigenstates of the system with tL ≠ t S. This may imply the existence of macroscopically degenerate states which are characteristic of the system with tL = t S, and that a discontinuity appears in the number of the confined states in the thermodynamic limit.

Electronic states of a disordered two-dimensional quasiperiodic tiling: From critical states to Anderson localization

We consider critical eigenstates in a two dimensional quasicrystal and their evolution as a function of disorder. By exact diagonalization of finite size systems we show that the evolution of properties of a typical wave-function is non-monotonic. That is, disorder leads to states delocalizing, until a certain crossover disorder strength is attained, after which they start to localize. Although this non-monotonic behavior is only present in finite-size systems and vanishes in the thermodynamic limit, the crossover disorder strength decreases logarithmically slowly with system size, and is quite large even for very large approximants. The non-monotonic evolution of spatial properties of eigenstates can be observed in the anomalous dimensions of the wave-function amplitudes, in their multifractal spectra, and in their dynamical properties. We compute the two-point correlation functions of wave-function amplitudes and show that these follow power laws in distance and energy, consistent with the idea that wave-functions retain their multifractal structure on a scale which depends on disorder strength. Dynamical properties are studied as a function of disorder. We find that the diffusion exponents do not reflect the non-monotonic wave-function evolution. Instead, they are essentially independent of disorder until disorder increases beyond the crossover value, after which they decrease rapidly, until the strong localization regime is reached. The differences between our results and earlier studies on geometrically disordered ``phason-flip'' models lead us to propose that the two models are in different universality classes. We conclude by discussing some implications of our results for transport and a proposal for a Mott hopping mechanism between power law localized wave-functions, in moderately disordered quasicrystals.

Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing

The concrete realization of topological quantum computing using low-dimensional quasiparticles, known as anyons, remains one of the important challenges of quantum computing. A topological quantum computing platform promises to deliver more robust qubits with additional hardware-level protection against errors that could lead to the desired large-scale quantum computation. We propose quasicrystal materials as such a natural platform and show that they exhibit anyonic behavior that can be used for topological quantum computing. Different from anyons, quasicrystals are already implemented in laboratories. In particular, we study the correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of the one-dimensional Fibonacci chain and the two-dimensional Penrose tiling quasicrystals. A concrete encoding on these tiling spaces of topological quantum information processing is also presented by making use of inflation and deflation of such tiling spaces. While we outline the theoretical basis for such a platform, details on the physical implementation remain open.

Self-Assembly of DNA Tiles with G-Quadruplex DNAzyme Catalytic Activity for Sensing Applications.

DNA tiles form through self-assembly of a small number of DNA strands that interact through basic repeated interactions, allowing the growth of nanoscale structures seeded by molecular inputs. If an approach for catalytic signal amplification can be integrated into the resultant nanostructure, then one can anticipate biosensing or diagnostic applications mediated by DNA tile self-assembly. Here, two-dimensional DNA tiles with split quadruplexes were designed as diagnostic tools for nucleic acid sensing without the use of protein enzymes. The presence of a target sequence leads to formation of extended microscale structures with arrayed multiple G-quadruplexes across the tile plane, with catalytic activity coupled to a colorimetric reporter. Such a mechanism has potential for low-cost signal amplification using unmodified DNA without the use of protein enzymes for biosensing.

Enhanced Second-Harmonic-Generation Efficiency and Birefringence in Melillite Oxychalcogenides Sr2MGe2OS6 (M = Mn, Zn, and Cd)

Simultaneous regulation and control of non-linear second-harmonic-generation (SHG) coefficients (deff) and linear birefringence (Δn) of non-centrosymmetric (NCS) crystals is a crucial aspect of improving the non-linear optical (NLO) performance, yet it remains a huge challenge in modern laser techniques and science. Herein, a series of melilite-type oxide-derived NCS oxychalcogenides, Sr2MGe2OS6 (M = Mn, Zn, and Cd), are successfully designed and synthesized through a facile partial isovalent chemical substitution strategy. These isostructural compounds crystallize in the tetragonal NCS space group P4̅21m (no. 113) and feature unique two-dimensional Cairo pentagonal tiling layers composed of heteroligand [GeOS3] and tetrahedral [MS4] asymmetric building units (ABUs). Compared to the parent melilite-type oxides, the derived title compounds not only successfully realize the phase matchability transformation [i.e., from non-phase-matching (NPM) to phase-matching (PM)] but also greatly improve the IR-NLO performance (especially, simultaneously boosting deff and Δn). Sr2CdGe2OS6 exhibits the best comprehensive performance among oxychalcogenides, including a wide PM cutoff edge (>525 nm), a strong deff (0.8 × AgGaS2), a giant laser-induced damage threshold (19.2 × AgGaS2), a large band gap (3.62 eV), as well as a broad transmission cutoff region (0.28–12.0 μm). Furthermore, highly distorted [GeOS3] ABU in this melilite-type structure is proved to be the profitable bifunctional NLO-active unit, which can contribute to the search and design of novel IR-NLO crystals with a large Δn and a strong deff based on the experimental results and theoretical calculations. This work demonstrates the first examples of PM melilite-type oxychalcogenides and offers new perspectives on the phase matchability transformation through the partial isovalent chemical substitution approach, which will play a constructive role in the future design of high-performance IR-NLO heteroanionic materials.

Ferrimagnetically ordered states in the Hubbard model on the hexagonal golden-mean tiling

We study magnetic properties of the half-filled Hubbard model on the two-dimensional hexagonal golden-mean tiling. We find that the vertex model of the tiling is bipartite, with a sublattice imbalance of $\sqrt{5}/(6\tau^3)$ (where $\tau$ is the golden mean), and that the non-interacting tight-binding model gives macroscopically degenerate states at $E=0$. We clarify that each sublattice has specific types of confined states, which in turn leads to an interesting spatial pattern in the local magnetizations in the weak coupling regime. Furthermore, this allows us to analytically obtain the lower bound on the fraction of the confined states as $(\tau+9)/(6\tau^6)\sim 0.0986$, which is conjectured to be the exact fraction. These results imply that a ferrimagnetically ordered state is realized even in the weak coupling limit. The introduction of the Coulomb interaction lifts the macroscopic degeneracy at the Fermi level, and induces finite staggered magnetization as well as uniform magnetization. Likewise, the spatial distribution of the magnetizations continuously changes with increasing interaction strength. The crossover behavior in the magnetically ordered states is also addressed in terms of the perpendicular space analysis.

Higher-order topological Anderson insulators in quasicrystals

The disorder effects on higher-order topological phases in periodic systems have attracted much attention. However, in aperiodic systems, such as quasicrystalline systems, the interplay between disorder and higher-order topology is still unclear. In this paper, we investigate the effects of disorder on two types of second-order topological insulators, including a quasicrystalline quadrupole insulator and a modified quantum spin Hall insulator, in a two-dimensional Amman-Beenker tiling quasicrystalline lattice. We demonstrate that the higher-order topological insulators are robust against weak disorder in both models. More striking, the disorder-induced higher-order topological insulators called higher-order topological Anderson insulators are found at a certain region of disorder strength in both models. Our paper extends the study of the interplay between disorder and higher-order topology to quasicrystalline systems.

Disorder-induced chiral and helical Majorana edge modes in a two-dimensional Ammann-Beenker quasicrystal

Recent research on disorder effects in topological phases in quasicrystalline systems has received much attention. In this work, by numerically computing the (spin) Bott index and the thermal conductance, we reveal the effects of disorder on a class D chiral topological superconductor and a class DIII time-reversal-invariant topological superconductor in a two-dimensional Ammann-Beenker tiling quasicrystalline lattice. We demonstrate that both the topologically protected chiral and helical Majorana edge modes are robust against weak disorder in the quasicrystalline lattice. More fascinating is the discovery of disorder-induced topologically nontrivial phases exhibiting chiral and helical Majorana edge modes in class D and DIII topological superconductor systems, respectively. Our findings open the door for the research on disorder-induced Majorana edge modes in quasicrystalline systems.

Tegula – exploring a galaxy of two-dimensional periodic tilings

Periodic tilings play a role in the decorative arts, in construction and in crystal structures. Combinatorial tiling theory allows the systematic generation, visualization and exploration of such tilings of the plane, sphere and hyperbolic plane, using advanced algorithms and software. Here we present a “galaxy” of tilings that consists of the set of all 2.4 billion different types of periodic tilings that have Dress complexity up to 24. We make these available in a database and provide a new program called Tegula that can be used to search and visualize such tilings.