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Pentaplexity A Class of Non-Periodic Tilings of the Plane
... Despite the recent discovery of monotile tilings based on a single prototile [1,2], aperiodic tilings usually employ two or more shapes as their prototiles. A well-known example is the pentagonal Penrose tiling (P1), which uses six polygonal shapes to tile the plane: three types of regular pentagons, along with thin rhombuses, crowns, and pentacles that fill the gaps between pentagons [3][4][5]. These six prototiles are subject to specific matching constraints, often represented by bumps and dents on their edges, which enforce aperiodicity. ...
... The number of prototiles in P1 can be reduced to two by reassembling the six tile shapes -cutting them into pieces and recomposing them into new forms. Two paired sets of prototiles were thus discovered: kites and darts (P2), and thick and thin rhombuses (P3), both incorporating matching rules that enforce aperiodicity [3,5,6]. These three Penrose tilings are renowned for providing deep insight into the structure of quasicrystals [7][8][9]. ...
... (i) Local isomorphism (LI): Two tilings are locally isomorphic if and only if every finite region in one tiling is contained somewhere in the other, and vice versa [3,5,6]. In fact, the term 'P3 tiling' does not refer to a single realization of the tiling but to all members of its LI class. ...
We present a novel variant of a planar quasiperiodic tiling with tenfold symmetry, employing the same thick and thin rhombuses as the celebrated rhombic Penrose tiling. Despite its distinct visual appearance, this new tiling shares several key features with its predecessor, including similar vertex environments, polygonal acceptance domains based on regular pentagons, and an inflation/deflation symmetry associated with the golden mean as its fundamental scaling ratio. Additional complexities arise from an increased number of prototiles and a dual grid pattern that incorporates folded lines alongside ordinary straight lines. This tiling exhibits a high density of a compact decagonal motif forming a two-tiered, five-petaled flower pattern, which spans a substantial portion of the tiling. We identify a slightly enhanced degree of hyperuniform order compared to the standard rhombic Penrose tiling.
... [2,14]. In our model, the form of the phase φ n (r) depends on the choice of a vector field forming quasicrystals, which are non-periodic tilings of the plane that nevertheless exhibit long-range order [26]. These structures feature symmetries associated with non-crystallographic point groups, such as the pentagonal or octagonal point groups, which are incompatible with translational symmetries. ...
... (2) over a square grid of side length 28λ SPP in MATLAB [31]. These quasiperiodic tilings are of a well-known form [21,26] and are rotationally symmetric about the origin of the polygon. In particular, the decagonal tiling we have recovered shares its symmetries with the first Penrose tiling [26], and selecting a polygon of N = 10 produces the same interference pattern as the one shown here for N = 5. ...
... These quasiperiodic tilings are of a well-known form [21,26] and are rotationally symmetric about the origin of the polygon. In particular, the decagonal tiling we have recovered shares its symmetries with the first Penrose tiling [26], and selecting a polygon of N = 10 produces the same interference pattern as the one shown here for N = 5. Note that modifying the wavelength and/or surface permittivity serves only to spatially rescale the field pattern, and as such the quasicrystalline patterns in Fig. 2 are plotted in units of λ SPP . ...
Photonic skyrmion and meron lattices are structured light fields with topologically protected textures, analogous to magnetic skyrmions and merons. Here, we report the theoretical existence of mixed skyrmion and meron quasicrystals in an evanescent optical field. Topological quasiperiodic tilings of even and odd point group symmetries are demonstrated in both the electric field and spin angular momentum. These quasicrystals contain both skyrmions and merons of Néel-type topology, marking the first demonstration of quasiperiodic tilings of such quasiparticles in a three-dimensional vector field. These results are in agreement with the observations of quasiperiodic arrangements of carbon nanoparticles in water driven by ultrasound and pave the way toward engineering hybrid topological states of light, which may have potential applications in optical manipulation, metrology, and information processing.
... Recently, the topological properties [36][37][38] , and cold atom simulation [39][40][41] of 2D quasicrystals have also received extensive attention. As the classical paradigm and pioneer of quasicrystals, the static electron properties of 2D Penrose tiling 42 have been widely studied in the density of states 43,44 and the localization of wave functions 45 . To generate Penrose tiling, various methods have been developed, including matching rules 42,46 , self-similarity transformations 45 , the generalized dual method (GDM) 1,2 , and superspace projection [47][48][49][50][51][52] . ...
... As the classical paradigm and pioneer of quasicrystals, the static electron properties of 2D Penrose tiling 42 have been widely studied in the density of states 43,44 and the localization of wave functions 45 . To generate Penrose tiling, various methods have been developed, including matching rules 42,46 , self-similarity transformations 45 , the generalized dual method (GDM) 1,2 , and superspace projection [47][48][49][50][51][52] . Notably, including Penrose tilings, GDM and five-dimensional superspace projection can generate various quasiperiodic tilings, which are collectively termed generalized Penrose tilings [49][50][51][52] . ...
... Hence, the parameter γ, the fractional part of P 4 i = 0 γ i , dictates the LIC of configurations, and its absolute value can be consistently mapped to the interval [0, 0.5]. In particular, when γ = 0, the tiling belongs to the Penrose LIC and can be obtained by the "matching rule" 42,46 . Varying γ i È É and enlarging the rhombuses side length l to 8 a.u., we obtain a series of tilings, exhibited in Fig. 1. ...
Electron diffraction spectroscopy is a fundamental tool for investigating quasicrystal structures, which unveils the quasiperiodic long-range order. Nevertheless, it falls short in effectively distinguishing separate local isomorphism classes. This is a long outstanding problem. Here, we study the high-order harmonic generation in two-dimensional generalized Penrose quasicrystals to optically resolve different local isomorphism classes. The results reveal that: (i) harmonic spectra from different parts of a quasicrystal are identical, even though their atomic arrangements vary significantly. (ii) The harmonic yields of diverse local isomorphism classes exhibit variations, providing a way to distinguish local isomorphism classes. (iii) The rotational symmetry of harmonic yield can serve as a characteristic of quasicrystal harmonics and is consistent with the orientation order. Our results not only pave the way for confirming the experimental reproducibility of quasicrystal harmonics and identifying quasicrystal local isomorphism classes, but also shed light on comprehending electron dynamics influenced by the vertex environments.
... We relate constructed quasiperiodic structures to aperiodic tilings of the plane (or space). Perhaps the best-known aperiodic tiling of the plane was designed by Penrose [14,33,34]. De Bruijn [9] showed that every Penrose tiling can be obtained by projecting a 5−dimensional periodic lattice to a 2−dimensional plane. ...
... In addition to theoretical studies of quasiperiodic tilings [14,33,34,9], physical quasicrystals [39,46,18,41] have been realized and studied beginning with the discovery by Dan Shechtman of an Al-Mn alloy with icosahedral symmetry [39]. Since this discovery, many groups have synthesized quasicrystals in various ways. ...
... [46]). The N −dimensional representation allows us to relate the constructed particle patterns to quasiperiodic tilings in d dimensions (e.g., Penrose tilings [14,33,34]). ...
We develop a method to design tunable quasiperiodic structures of particles suspended in a fluid by controlling standing acoustic waves. One application of our results is to ultrasound directed self-assembly, which allows fabricating composite materials with desired microstructures. Our approach is based on identifying the minima of a functional, termed the acoustic radiation potential, determining the locations of the particle clusters. This functional can be viewed as a two- or three-dimensional slice of a similar functional in higher dimensions as in the cut-and-project method of constructing quasiperiodic patterns. The higher dimensional representation allows for translations, rotations, and reflections of the patterns. Constrained optimization theory is used to characterize the quasiperiodic designs based on local minima of the acoustic radiation potential and to understand how changes to the controls affect particle patterns. We also show how to transition smoothly between different controls, producing smooth transformations of the quasiperiodic patterns. The developed approach unlocks a route to creating tunable quasiperiodic and moir\'e structures known for their unconventional superconductivity and other extraordinary properties. Several examples of constructing quasiperiodic structures, including in two and three dimensions, are given.
... where = (( / ) 2 − 2 SPP ) 1/2 is the wavevector component normal to the interface, and are the amplitude and phase of the th propagating SPP, respectively, and SPP = 0 1 2 /( 1 + 2 ) 1/2 is the SPP wavenumber with 1 being the (positive valued) permittivity of the dielectric occupying > 0 and 2 being the (negative valued) permittivity of the metal occupying < 0. We also assume = 1 everywhere. of optical field to host quasicrystals, which are non-periodic tilings of the two-dimensional plane that nevertheless exhibit long-range order [23]. These structures feature symmetries associated with the non-crystallographic point groups, such as the pentagonal or octagonal point groups, which are incompatible with translational symmetries. ...
... [27] and SPP = 567 nm. Figure 2 depicts quasicrystals in the electric field at the air-gold interface for decagonal, octagonal and dodecagonal symmetries. These quasiperiodic tilings are of a well-known form [18,23] and are rotationally symmetric about the origin of the polygon. In particular, the decagonal tiling we have recovered shares its symmetries with the first Penrose tiling [23], and selecting a polygon of = 10 produces the same interference pattern as the one shown here for = 5. ...
... These quasiperiodic tilings are of a well-known form [18,23] and are rotationally symmetric about the origin of the polygon. In particular, the decagonal tiling we have recovered shares its symmetries with the first Penrose tiling [23], and selecting a polygon of = 10 produces the same interference pattern as the one shown here for = 5. The center of each quasiperiodic tilings can be displaced by altering the amplitudes from +1 to −1 for different (not shown here). ...
Photonic skyrmion and meron lattices are structured light fields with topologically protected textures, analogous to magnetic skyrmions and merons. Here, we report the theoretical existence of mixed skyrmion and meron quasicrystals in an evanescent optical field. Topological quasiperiodic tilings of even and odd point group symmetries are demonstrated in both the electric field and spin angular momentum. These quasicrystals contain both skyrmions and merons of N\'eel-type topology. Interestingly, the quasiperiodic tilings are in agreement with the observations of quasiperiodic arrangements of carbon nanoparticles in water driven by ultrasound, and pave the way towards engineering hybrid topological states of light which may have potential applications in optical manipulation, metrology and information processing.
... As a necessary ingredient in this story, we introduce a new characterization of crystals and (self-similar) quasicrystals which naturally extends from flat space to hyperbolic space. Traditional self-similar quasicrystals (which live in Euclidean space), like the famous Penrose tiling [52][53][54], are special objects that are of interest to both physicists [55][56][57][58][59][60] and mathematicians [61][62][63][64][65][66]. The new self-similar quasicrystals introduced here (which live in hyperbolic space) are similarly special and interesting. ...
... Let us specify to a discrete surface made only of p-gons, assumed without boundary for simplicity. Then we can use (53) to give another purely combinatorial version of the Euler characteristic. Specifically, we have pF ¼ 2E, and if we let nðvÞ ¼ fnumber of Edges ðor FacesÞ at vg, then ...
Discrete geometries in hyperbolic space are of longstanding interest in pure mathematics and have come to recent attention in holography, quantum information, and condensed matter physics. Working at a purely geometric level, we describe how any regular tessellation of ( d + 1 )-dimensional hyperbolic space naturally admits a d -dimensional boundary geometry with self-similar “quasicrystalline” properties. In particular, the boundary geometry is described by a local, invertible, self-similar substitution tiling, that discretizes conformal geometry. We greatly refine an earlier description of these local substitution rules that appear in the 1D/2D example and use the refinement to give the first extension to higher dimensional bulks; including a detailed account for all regular 3D hyperbolic tessellations. We comment on global issues, including the reconstruction of bulk geometries from boundary data, and introduce the notion of a “holographic foliation”: a foliation by a stack of self-similar quasicrystals, where the full geometry of the bulk (and of the foliation itself) is encoded in any single leaf in a local, invertible way. In the { 3 , 5 , 3 } tessellation of 3D hyperbolic space by regular icosahedra, we find a 2D boundary quasicrystal admitting points of 5-fold symmetry which is not the Penrose tiling, and record and comment on a related conjecture of William Thurston. We end with a large list of open questions for future analytic and numerical studies.
Published by the American Physical Society 2025
... We show that, among the whole Hat family of tilings, a parameter region around a member of the family, known as the Specter tiling (see Fig. 1), provides ideal properties for unaliased array analysis. For similar inter-station distance, the Specter outperforms all other members of the Hat tiling, as well as other aperiodic tilings such as the famous Penrose tiling [ Fig. 1c, 39,40] or the Square-Triangle tiling [Fig. 1d,41]. ...
... Unlike other aperiodic tilings, the Hat family of tiles are examples of monotiles that can tile the plane only aperiodically. They improve on the previous record holder, the Penrose tilings [ Fig. 1c; 39,40], that required a minimum of two tiles to tile the plane aperiodically. ...
Finding optimal wave sampling methods has far-reaching implications in wave physics, such as seismology, acoustics, and telecommunications. A key challenge is surpassing the Whittaker-Nyquist-Shannon (WNS) aliasing limit, establishing a frequency below which the signal cannot be faithfully reconstructed. However, the WNS limit applies only to periodic sampling, opening the door to bypass aliasing by aperiodic sampling. In this work, we investigate the efficiency of a recently discovered family of aperiodic monotile tilings, the Hat family, in overcoming the aliasing limit when spatially sampling a wavefield. By analyzing their spectral properties, we show that monotile aperiodic seismic (MAS) arrays, based on a subset of the Hat tiling family, are efficient in surpassing the WNS sampling limit. Our investigation leads us to propose MAS arrays as a novel design principle for seismic arrays. We show that MAS arrays can outperform regular and other aperiodic arrays in realistic beamforming scenarios using single and distributed sources, including station-position noise. While current seismic arrays optimize beamforming or imaging applications using spiral or regular arrays, MAS arrays can accommodate both, as they share properties with both periodic and aperiodic arrays. More generally, our work suggests that aperiodic monotiles can be an efficient design principle in various fields requiring wave sampling.
... These designs are based on the regular pentagon [Pen79]. ...
... We start with either a kite or a dart, and then iteratively replace a dart with two darts and one kite, and iteratively replace a kite with two darts and two kites. [Pen79] The dart tile is replaced by two darts and a kite (all three scaled by 1/Φ, ...
We consider graphs derived from aperiodically ordered tilings of the plane, by treating each corner of each tile as a vertex and each side of each tile as an edge. We calculate the average degree of these graphs. For the Ammann A2 tiling, we present a closed-form formula for the average degree. For the Kite and Dart Penrose tiling, the Rhomb Penrose Tiling, and the Ammann-Beenker tiling we present numerical calculations for the average degree.
... In addition to periodic lattices built on crystalline symmetries, quasiperiodic lattices may also be constructed that have no translational symmetry but higher-order rotational symmetries, such as the pentagonal Penrose tiling [28] and the octagonal Ammann-Beenker tiling [3]. Quasicrystals (quasiperiodic crystals) based on these lattices are the appropriate analogue of periodic crystals [21,31]. ...
... Array 10f is formed from the vertices of a Penrose P3 lattice [28], which is perhaps the most well-known quasiperiodic geometry. The lattice is composed of two types of rhombi, with internal angles 36 • (thin) and 72 • (fat), arranged according to matching rules such as to give the tiling an overall 10-fold symmetry, although it is often described as pentagonal.. ...
Inspired by transformation optics and photonic crystals, this paper presents a computational investigation into the interaction between water surface waves and array waveguides of cylinders with multiple previously unexplored lattice geometries, including, for the first time, quasiperiodic geometries. Extending beyond conventional square and hexagonal periodic arrays, transformation optics has opened up entirely new opportunities to investigate water wave propagation through arrays based on quasiperiodic lattices, and quasiperiodically arranged vacancy defects. Using the linear potential flow open-source code Capytaine, missing element and -scaled Fibonacci square lattices, the Penrose lattice, hexagonal lattice and Amman-Beenker lattice are investigated. The existence of band gaps for all arrays is observed. An hexagonal lattice with vacancy defects transmits the least energy. Bragg diffraction consistent with rotational symmetry is observed from all arrays. Waves will distort significantly to achieve resonance with arrays, supporting transformation-based waveguides. The possible uses include adaptation to more versatile waveguides with applications such as offshore renewable energy and coastal defence.
... Each intersection (crossing) of two multigrid lines corresponds to a tile. See Fig. 2. Penrose rhombus tilings (Penrose 1974(Penrose , 1979 are dual of regular multigrids (see Remark 2). ...
The growth pattern of an invasive cell-to-cell propagation (called the successive coronas) on the square grid is a tilted square. On the triangular and hexagonal grids, it is an hexagon. It is remarkable that, on the aperiodic structure of Penrose tilings, this cell-to-cell diffusion process tends to a regular decagon (at the limit). On any multigrid dual tiling, it tends to a polygon which we call characteristic polygon. In this article we provide a complete and self-contained proof of this result. Exploiting this elegant duality allows to fully understand why such surprising phenomena, of seeing highly regular polygonal shapes emerge from aperiodic underlying structures, happen.
... Remarkably, tilings in finite dimensions tend to be periodic and the construction of aperiodic tilings has been a long-standing open problem, especially due to its connection to the modelling of quasicristals, [6,45,62,67,78]. Nowadays, several constructions of aperiodic tilings are available, by rotations and translations of a large finite set of tiles, [8,63,79], by rotations and translations of two tiles (Penrose' 'kites' and 'darts'), [15,36,59,60], by translations and rotations of a single tile, [73,74], and, finally, only by translations of a single tile, [37]. ...
We construct a bounded and symmetric convex body in (for certain cardinals ) whose translates yield a tiling of . This answers a question due to Fonf and Lindenstrauss. As a consequence, we obtain the first example of an infinite-dimensional reflexive Banach space that admits a tiling with balls (of radius 1). Further, our tiling has the property of being point-countable and lattice (in the sense that the set of translates forms a group). The same construction performed in yields a point-2-finite lattice tiling by balls of radius 1 for , which compares to a celebrated construction due to Klee. We also prove that lattice tilings by balls are never disjoint and, more generally, each tile intersects as many tiles as the cardinality of the tiling. Finally, we prove some results concerning discrete subgroups of normed spaces. By a simplification of the proof of our main result, we prove that every infinite-dimensional normed space contains a subgroup that is 1-separated and -dense, for every ; further, the subgroup admits a set of generators of norm at most . This solves a problem due to Swanepoel and yields a simpler proof of a result of Dilworth, Odell, Schlumprecht, and Zs\'ak. We also give an alternative elementary proof of Stepr\={a}ns' result that discrete subgroups of normed spaces are free.
... varying according to the salient feature in each class. By way of example, in Fig. 2, the case of a regular square lattice pattern is compared with that of a Penrose tiling as an example of a quasiperiodic pattern [41,42] and a homogeneous PPP. In all cases, the SFF exhibits a decay from unit value until a dip and a subsequent steep increase toward a plateau. ...
Point processes have broad applications in science and engineering. In physics, their use ranges from quantum chaos to statistical mechanics of many-particle systems. We introduce a spatial form factor (SFF) for the characterization of spatial patterns associated with point processes. Specifically, the SFF is defined in terms of the averaged even Fourier transform of the distance between any pair of points. We focus on homogeneous Poisson point processes and derive the explicit expression for the SFF in d -spatial dimensions. The SFF can then be found in terms of the even Fourier transform of the probability distribution for the distance between two independent and uniformly distributed random points on a d -dimensional ball, arising in the ball line picking problem. The relation between the SFF and the set of n -order spacing distributions is further established. The SFF is analyzed in detail for d = 1 , 2 , 3 and in the infinite-dimensional case, as well as for the d -dimensional Coulomb gas, as an interacting point process. As a physical application, we describe the spontaneous vortex formation during Bose-Einstein condensation in finite time recently studied in ultracold atom experiments and use the SFF to reveal the stochastic geometry of the resulting vortex patterns. In closing, we also introduce a generalization of the SFF applicable to arbitrary sets in a metric space.
Published by the American Physical Society 2025
... In physics, the lack of mathematical precision typically poses no problem when considering the precise properties of genuine quasicrystals. About ten years before the discovery of quasicrystal line materials in physics, Penrose tilings of the plane, also known as quasicrystals, first appeared in mathematics [17]. De Bruijn then built their algebraic theory [4]. ...
A quasicrystal is a mathematical term for an infinite point or tiling space. It possesses several intersecting features, including Delone, relative discreteness, and self-similarities. The model set is the basic form in which the physical quasicrystals are represented. Pisot numbers are used to identify the one-dimensional model sets' adjacent points. The quadratic irrational numbers are related to all one-dimensional model sets that have been experimentally discovered the paper offers mathematical models of quasicrystals with particular attention given to cut and projection sets for the eight folded symmetry and discuss about one dimensional cut and projection set.
... If no other edges or all other edges have rivers, we do not add a river on 3 . If one other edge ( 1 or 2 ) has a river, we add a river on 3 with an altitude between To reduce the axis-aligned artefacts, we can use Penrose tiles [5,6,10], which form aperiodic tilings. We use the P2-tiling that uses "kites and darts", both formed by joining two Robinson triangles: isoscleles triangles where the ratio of the long and short sides is the golden ratio = √ 5+1 2 . ...
This paper presents a method for generating maps with rivers and fjords. The method is based on recursive subdivision of triangles and allows unlimited zoom on details without requiring generation of a full map at high resolution.
... In this Letter, we report on unusual linear and nonlinear light localization properties at the edge of a quasiperiodic structure. We use a laser-fabricated array of waveguides, positioned at the vertices of a Penrose tiling [39]. This array is then truncated using circular masks, leaving only the waveguides with centers inside the circle of radius r. ...
Quasicrystals are unique systems that, unlike periodic structures, lack translational symmetry but exhibit long-range order dramatically enriching the system properties. While evolution of light in the bulk of photonic quasicrystals is well studied, experimental evidences of light localization near the edge of truncated photonic quasicrystal structures are practically absent. In this Letter, we observe both linear and nonlinear localization of light at the edges of radially cropped quasicrystal waveguide arrays, forming an aperiodic Penrose tiling. Our theoretical analysis reveals that for certain truncation radii, the system exhibits linear eigenstates localized at the edge of the truncated array, whereas for other radii, this localization does not occur, highlighting the significant influence of truncation on edge light localization. Using single-waveguide excitations, we experimentally confirm the presence of localized states in Penrose arrays inscribed by a femtosecond laser and investigate the effects of nonlinearity on these states. Our theoretical findings identify a family of edge solitons, and experimentally, we observe a transition from linear localized states to edge solitons as the power of the input pulse increases. Our results represent the first experimental demonstration of localization phenomena induced by the selective truncation of quasiperiodic photonic systems.
... In this Letter, we report on unusual linear and nonlinear light localization properties at the edge of a quasiperiodic structure. We use a laser-fabricated array of waveguides, positioned at the vertices of a Penrose tiling [39]. This array is then truncated using circular masks, leaving only the waveguides with centers inside the circle of radius r. ...
Quasicrystals are unique systems that, unlike periodic structures, lack translational symmetry but exhibit long-range order dramatically enriching the system properties. While evolution of light in the bulk of photonic quasicrystals is well studied, experimental evidences of light localization near the edge of truncated photonic quasicrystal structures are practically absent. In this Letter, we observe both linear and nonlinear localization of light at the edges of radially cropped quasicrystal waveguide arrays, forming an aperiodic Penrose tiling. Our theoretical analysis reveals that for certain truncation radii, the system exhibits linear eigenstates localized at the edge of the truncated array, whereas for other radii, this localization does not occur, highlighting the significant influence of truncation on edge light localization. Using single-waveguide excitations, we experimentally confirm the presence of localized states in Penrose arrays inscribed by a femtosecond laser and investigate the effects of nonlinearity on these states. Our theoretical findings identify a family of edge solitons, and experimentally, we observe a transition from linear localized states to edge solitons as the power of the input pulse increases. Our results represent the first experimental demonstration of localization phenomena induced by the selective truncation of quasiperiodic photonic systems.
... The crystallization of chiral molecules has played a pivotal role in comprehending molecular structures [11][12][13] and remains a cornerstone method for achieving enantiopure pharmaceuticals and Check for updates 1 Empa, Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland. 2 Univ Angers, CNRS, MOLTECH-Anjou, SFR MATRIX, F-49000 Angers, France. 3 Laboratory for X-ray Nanoscience and Technologies, Paul-Scherrer-Institut (PSI), CH-5232 Villigen PSI, Switzerland. 4 Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) Station 3, CH-1015 Lausanne, Switzerland. ...
Studying the self-assembly of chiral molecules in two dimensions offers insights into the fundamentals of crystallization. Using scanning tunneling microscopy, we examine an uncommon aggregation of polyaromatic chiral molecules on a silver surface. Dense packing is achieved through a chiral triangular tiling of triads, with N and N ± 1 molecules at the edges. The triangles feature a random distribution of mirror-isomers, with a significant excess of one isomer. Chirality at the domain boundaries causes a lateral shift, producing three distinct topological defects where six triangles converge. These defects partially contribute to the formation of supramolecular spirals. The observation of different equal-density arrangements suggests that entropy maximization must play a crucial role. Despite the potential for regular patterns, all observed tiling is aperiodic. Differences from previously reported aperiodic molecular assemblies, such as Penrose tiling, are discussed. Our findings demonstrate that two-dimensional molecular self-assembly can be governed by topological constraints, leading to aperiodic tiling induced by intermolecular forces.
... We mention two remarkable recent results on aperiodic tilings in two different settings here. For tiling of the plane where rotation is allowed, Penrose first found an aperiodic set of two tiles [17]. That was the record holder for the smallest aperiodic sets in the setting of general tiling (i.e., rotation is allowed) of the plane for more than 50 years until a family of aperiodic monotiles was discovered by Smith, Myers, Kaplan, and Goodman-Strauss [21,22] in 2023. ...
Recently, two extraordinary results on aperiodic monotiles have been obtained in two different settings. One is a family of aperiodic monotiles in the plane discovered by Smith, Myers, Kaplan and Goodman-Strauss in 2023, where rotation is allowed, breaking the 50-year-old record (aperiodic sets of two tiles found by Roger Penrose in the 1970s) on the minimum size of aperiodic sets in the plane. The other is the existence of an aperiodic monotile in the translational tiling of for some huge dimension n proved by Greenfeld and Tao. This disproves the long-standing periodic tiling conjecture. However, it is known that there is no aperiodic monotile for translational tiling of the plane. The smallest size of known aperiodic sets for translational tilings of the plane is 8, which was discovered more than 30 years ago by Ammann. In this paper, we prove that translational tiling of the plane with a set of 7 polyominoes is undecidable. As a consequence of the undecidability, we have constructed a family of aperiodic sets of size 7 for the translational tiling of the plane. This breaks the 30-year-old record of Ammann.
... They can be obtained by iteration of a substitution rule over tiles. A famous class of such tilings is found in [7]. We give an example of such a tiling in Figure 1 before properly defining this notion. ...
The goal of this paper is to study the action of the group of translations over self-similar tilings in the euclidian space . It investigates the behaviour near zero for spectral measures for such dynamical systems. Namely the paper gives a H\"older asymptotic expansion near zero for these spectral measures. It is a generalization to higher dimension of a result by Bufetov and Solomyak who studied self similar-suspension flows for substitutions. The study of such asymptotics mostly involves the understanding of the deviations of some ergodic averages.
... The size of the liquid droplet is smaller than the capillary length, ca l g (γ, ρ and g are the liquid surface tension, density, and gravitational acceleration, respectively), such that the gravity is negligible. Figure 1b shows the surface, on which micropillars are arranged in the pattern of a Penrose tiling (24). This pattern has 5-fold symmetry and is self-similar (the same patterns appear at different scales) with many local symmetrical centers ( Fig. 1c-d). ...
Quasi-periodic structures of quasicrystals yield novel effects in diverse systems. However, there is little investigation on employing quasi-periodic structures in the morphology control. Here, we show the use of quasi-periodic surface structures in controlling the transition of liquid droplets. Although surface structures seem random-like, we find that on these surfaces, droplets spread to well-defined 5-fold symmetric shapes and the symmetry of droplet shapes spontaneously restore during spreading, hitherto unreported in the morphology control of droplets. To obtain physical insights into these symmetry transitions, we conduct energy analysis and perform systematic experiments by varying properties of both liquid droplet and patterned surface. The results show the dominant factors in determining droplet shapes to be surface topography and the self-similarity of the surface structure. Our findings significantly advance the control capability of the droplet morphology. Such a quasi-periodic patterning strategy can offer a new method to achieve complex patterns.
... Only in the latter case can the Hamiltonian be cast in block off-diagonal form by dividing the Hilbert space into subspaces of even-and odd-indexed lattice sites. Such lattices provide a setting for studying disorder-immune chiral symmetry, which is also realized in two-dimensional (2D) lattices such as square, hexagonal [11,12], and even-sited ring lattices and certain Penrose tilings [15] under conditions of off-diagonal disorder. A Hamiltonian endowed with chiral symmetry features eigenvalues and eigenvectors that occur in skew-symmetric pairs in every realization of the disorder. ...
The formation of gaps -- forbidden ranges in the values of a physical parameter -- is a ubiquitous feature of a variety of physical systems: from energy bandgaps of electrons in periodic lattices and their analogs in photonic, phononic, and plasmonic systems to pseudo energy gaps in aperiodic quasicrystals. Here, we report on a `thermalization' gap for light propagating in finite disordered structures characterized by disorder-immune chiral symmetry -- the appearance of the eigenvalues and eigenvectors in skew-symmetric pairs. In this class of systems, the span of sub- thermal photon statistics is inaccessible to input coherent light, which -- once the steady state is reached -- always emerges with super-thermal statistics no matter how small the disorder level. We formulate an independent constraint that must be satisfied by the input field for the chiral symmetry to be `activated' and the gap to be observed. This unique feature enables a new form of photon-statistics interferometry: the deterministic tuning of photon statistics -- from sub-thermal to super-thermal -- in a compact device, without changing the disorder level, via controlled excitation-symmetry-breaking realized by sculpting the amplitude or phase of the input coherent field.
... Colored strong local rules. They can be traced back to [DB81], when de Bruijn proved that the Penrose tilings by rhombi -introduced with colored local rules in [Pen78] -are digitizations of an plane in R 5 based on the golden ratio (i.e., generated by vectors with entries in Q[ √ 5]). Beenker [Bee82] then tried to find colored strong local rules for another example, namely a plane in R 4 based on the silver ratio (generated by vectors with entries in Q[ √ 2]), unaware that it was already known by the "Mysterious Mr. Ammann" [Sen08] in a work that was only later published in [AGS92]. ...
A linear subspace E of has colored local rules if there exists a finite set of decorated tiles whose tilings are digitizations of E. The local rules are weak if the digitizations can slightly wander around E. We prove that a linear subspace has weak colored local rules if and only if it is computable. This goes beyond the previous results, all based on algebraic subspaces. We prove an analogous characterization for sets of linear subspaces, including the set of all the linear subspaces of .
... The dodecagonal quasicrystal with twelvefold symmetry is one of the twodimensional quasicrystals having a translational symmetry along one crystallographic direction and has so far been found in macromolecular polymers and colloids as well as in alloys such as Mn-Si-V and Cr-Ni-Si systems [1][2][3][4][5][6][7][8][9][10][11]. Its striking feature is that it has both a twelvefold rotational axis and quasiperiodicity, which can be expressed by a different tiling model from that for the decagonal quasicrystal [12][13][14][15][16]. In the case of the dodecagonal quasicrystal, the tiling model is composed of three kinds of tiles: that is, square-, triangle-, and rhombus-shaped tiles. ...
The intermetallic compound H (MnSiV) phase in the Mn-Si-V alloy system can be regarded as an approximant phase of the dodecagonal quasicrystal as one of the two-dimensional quasicrystals. To understand the features of the approximant H phase, in this study, the crystallographic features of both the H phase and the (\sigma H) reaction in Mn-Si-V alloy samples were investigated, mainly by transmission electron microscopy. It was found that, in the H phase, there were characteristic structural disorders with respect to an array of a dodecagonal structural unit consisting of 19 dodecagonal atomic columns. Concretely, penetrated structural units consisting of two dodecagonal structural units were presumed to be typical of such disorders. An interesting feature of the (\sigma H) reaction was that regions with a rectangular arrangement of penetrated structural units (RAPU) first appeared in the \sigma matrix as the initial state, and H regions were then nucleated in contact with RAPU regions. The subsequent conversion of RAPU regions into H regions eventually resulted in the formation of the approximant H state as the final state. Furthermore, atomic positions in both the H structure and the dodecagonal quasicrystal were examined using a simple plane-wave model with twelve plane waves.
... The Penrose Tilings. Let T be a tiling of Penrose's famous 'kite' and 'dart' tiles which meet along edges according to their matching rules [8]. These tilings have a hierarchical structure, based upon self-similarity with inflation constant the golden ratio. ...
A spectral sequence is defined which converges to the \v{C}ech cohomology of the Euclidean hull of a tiling of the plane with Euclidean finite local complexity. The terms of the second page are determined by the so-called ePE homology and ePE cohomology groups of the tiling, and the only potentially non-trivial boundary map has a simple combinatorial description in terms of its local patches. Using this spectral sequence, we compute the \v{C}ech cohomology of the Euclidean hull of the Penrose tilings.
... Under these more general settings, researchers find aperiodic sets of even fewer number of tiles. Notably, Penrose was the first to find an aperiodic set of two tiles [13]. More strikingly, a series of aperiodic monotile has been discovered by Smith et al. [15,16], which solves the long-standing einstein problem. ...
We show that translational tiling of the plane with a set of 8 polyominoes is undecidable, which answers a question posted by Ollinger. The techniques employed in our proof include a different orientation for simulating the Wang tiles in polyomino and a new method for encoding the colors of Wang tiles.
... The DFF can be used to characterize different classes of point patterns, with its behavior varying according to the salient feature in each class. By way of example, in Fig. 2, the case of a regular square lattice pattern is compared with that of a Penrose tiling as an example of a quasiperiodic pattern [40,41] and a homogeneous PPP. In all cases, the DFF exhibits a decay from unit value until a dip and a subsequent steep increase towards a plateau. ...
We introduce a distance form factor (DFF) for the characterization of spatial patterns associated with point processes. Specifically, the DFF is defined in terms of the averaged even Fourier transform of the distance between any pair of points. We focus on homogeneous Poisson point processes and derive the explicit expression for the DFF in d-spatial dimensions. The DFF can then be found in terms of the even Fourier transform of the probability distribution for the distance between two independent and uniformly distributed random points on a d-dimensional ball, arising in the ball line picking problem. The relation between the DFF and the set of n-order spacing distributions is further established. The DFF is analyzed in detail for d=1,2,3 and in the infinite-dimensional case, as well as for the d-dimensional Coulomb gas, as an interacting point process.
... sentative QC, namely Penrose lattice [91]. The Holstein model [92] is a prototypical microscopic model characterizing the e-ph coupling, which has been extensively studied in the periodic-crystal systems. ...
The exotic quantum states emerging in the quasicrystal (QC) have attracted extensive interest because of various properties absent in the crystal. In this paper, we systematically study the Holstein model at half filling on a prototypical structure of QC, namely rhombic Penrose lattice, aiming at investigating the superconductivity (SC) and other intertwined ordering arising from the interplay between quasiperiodicity and electron-phonon ({\it e}-ph) interaction. Through unbiased sign-problem-free determinant quantum Monte Carlo simulations, we reveal the salient features of the ground-state phase diagram. Distinct from the results on bipartite periodic lattices at half filling, SC is dominant in a large parameter regime on the Penrose lattice. When {\it e}-ph coupling is sufficiently strong, charge-density-wave order appears and strongly suppresses the SC. The strongest SC emerges at intermediate {\it e}-ph coupling strength and pronounced pairing fluctuation exists above the SC transition temperature. The strong pairing originates from the cooperative effects of unique lattice structure and macroscopically degenerate confined states at Fermi energy which uniquely exist on the Penrose lattice. Moreover, we demonstrate the forbidden ladders substantially suppress the phase coherence of SC. Our unbiased numerical results suggest that Penrose lattice is a potential platform to realize strong SC pairing, providing a promising avenue to searching for relatively high- SC dominantly induced by {\it e}-ph coupling.
... In all that follow, we consider the case of rhombus tilings, where there are finitely many tiles up to translation, all the tiles are rhombuses and the tiling is edge-to-edge, i.e., any two tiles either intersect on a single common vertex, on a full common edge, or not at all. Throughout the article we use the famous example of Penrose rhombus tilings [Pen79], see Fig. 1. ...
2-boostrap percolation on a graph is a diffusion process where a vertex gets infected whenever it has at least 2 infected neighbours, and then stays infected forever. It has been much studied on the infinite grid for random Bernoulli initial configurations, starting from the seminal result of van Enter that establishes that the entire grid gets almost surely entirely infected for any non-trivial initial probability of infection. In this paper, we generalize this result to any adjacency graph of any rhombus tiling of the plane, including aperiodic ones like Penrose tilings. We actually show almost sure infection of the entire graph for a larger class of measure than non-trivial Bernoulli ones. Our proof strategy combines a geometry toolkit for infected clusters based on chain-convexity, and uniform probabilistic bounds on particular geometric patterns that play the role of 0-1 laws or ergodicity, which are not available in our settings due to the lack of symmetry of the graph considered.
... Seen from figure 4(b), each cyan dot has five nearest neighbour dots with the nearest neighbour distance equal to a, and each magenta dot has six nearest neighbour dots with the nearest neighbour distance of a. Considering the structural characteristics of the six-fold symmetric quasiperiodical surface (Penrose 1979) (also see figure A(b) in the supplementary material), the ratio of cyan dots c 1 is calculated to be 0.95, and the ratio of magenta dots c 2 is calculated to be 0.05. As a result, the configurational entropy of the six-fold symmetric quasiperiodical surface in figure 4(b) is calculated to be 0.20. ...
Microstructured surfaces with pillar arrays are widely used to control the wetting morphology and spreading dynamics of droplets. In both simulations and experiments, it is shown that fabricating the surface with various microstructures is a very effective method for achieving the desired symmetry of the moving contact line. However, the method for characterizing miscellaneous pillar-arrayed microstructured surfaces is still insufficient. This paper presents the configurational entropy to characterize the microstructured surfaces with pillar arrays. By calculating the configurational entropy of pillar-arrayed microstructured surfaces, the relationship between the configurational entropy and the wetting morphology of droplets is obtained. For pillar-arrayed microstructured surfaces with the configurational entropy S > 0, the droplet wetting morphology may be much more complex than those with S = 0. The relationship is found to be consistent with the previous results. Furthermore, the wetting dynamics has been analysed. This study may be useful to understand the mechanism of droplet wetting on pillar-arrayed microstructured surfaces and provide insights for the design and manufacture of microstructured surfaces.
... The Penrose tiling serves as an excellent example of how an abstract object can find its reflection in the surrounding world. The structure, proposed in late 1970s [1,2], was a response to a very interesting, if purely speculative, mathematical problem: what is the minimum set of shapes that can cover a plane only in a non-periodic manner? For almost half a century, this two-tile covering remained the best, until very recently the ultimate singletile solution was finally found [3,4]. ...
Hexagon-boat-star (HBS) pentagonal tilings often appear in the description of decagonal quasicrystals and their periodic approximants. Being related to the Penrose tiling, they differ from the latter by a significantly higher packing density of vertices, which, in turn, depends on the relative frequency of appearance of the H, B and S tiles. Since boats (also known as "ivy leaves") have the lowest packing density, reducing their number in the tiling leads to an increase in its packing density. The paper proposes an inflation rule for a chiral tiling, which in principle contains no boats and therefore has the highest possible density among HBS tilings. The relationship between the tiling and the real structures of crystal approximants of decagonal quasicrystals is discussed.
... rotation and reflection of the tiles are allowed in addition to translation). Notably, Penrose first found an aperiodic set of two tiles [18]. More strikingly, a family of aperiodic monotiles has been discovered by Smith, Myers, Kaplan, and Goodman-Strauss [21,22], which solves the long-standing Einstein problem. ...
This paper focuses on the undecidability of translational tiling of n-dimensional space with a set of k tiles. It is known that tiling with translated copies with a set of 8 tiles is undecidable. Greenfeld and Tao gave strong evidence in a series of works that for sufficiently large dimension n, the translational tiling problem for might be undecidable for just one tile. This paper shows the undecidability of translational tiling of with a set of 6 tiles.
... Example 5.3. The most well-known two-dimensional example was given by Penrose [15], represented here as Robinson triangles. We note that there are 40 prototiles ¹a i ; b i ; ra i ; rb i j i D 0; : : : ; 9º, where the subscript denotes the number of rotations by =5. ...
We show that Kellendonk’s tiling semigroup of a finite local complexity substitution tiling is self-similar, in the sense of Bartholdi, Grigorchuk and Nekrashevych. We extend the notion of the limit space of a self-similar group to the setting of self-similar semigroups, and show that it is homeomorphic to the Anderson–Putnam complex for such substitution tilings, with natural self-map induced by the substitution. Thus, the inverse limit of the limit space, given by the limit solenoid of the self-similar semigroup, is homeomorphic to the translational hull of the tiling.
... A set of prototiles is called a tileset, and oftentimes many tilings can be composed with copies of the same prototiles. The three types of Penrose tilings, denoted P1, P2 and P3 in [3], were described by Roger Penrose himself [8]. The corresponding tilesets are shown in Figure 2. Uncountably many tilings can be composed with each, but none of them is periodic: they have no translations among their symmetries. ...
... Similar studies were initiated in the framework of tilings and Delone sets (see e.g., [27]), where the existence of some point set symmetries is at the heart of the discovery of aperiodicity of quasicrystals [30]. For instance, the Penrose tiling serves as a model with ten-fold symmetry [26]. However, still few results and no systematic description of the analogous normalizer group for R d -flows exist. ...
For a Zd-topological dynamical system (X,T,Zd), an isomomorphism is a self-homeomorphism ϕ:X→X such that for some matrix M∈GL(d,Z) and any n∈Zd, ϕ∘Tn=TMn∘ϕ, where Tn denote the self-homeomorphism of X given by the action of n∈Zd. The collection of all the isomorphisms forms a group that is the normalizer of the set of transformations Tn. In the one-dimensional case, isomorphisms correspond to the notion of flip conjugacy of dynamical systems and by this fact are also called reversing symmetries. These isomorphisms are not well understood even for classical systems. We present a description of them for odometers and more precisely for constant-base Z2-odometers, which is surprisingly not simple. We deduce a complete description of the isomorphisms of some minimal Zd-substitutive subshifts. This enables us to provide the first example known of a minimal zero-entropy subshift with the largest possible normalizer group.
Architected materials achieve unique mechanical properties through precisely engineered microstructures that minimize material usage. However, a key challenge of low‐density materials is balancing high stiffness with stable deformability up to large strains. Current microstructures, which employ slender elements such as thin beams and plates arranged in periodic patterns to optimize stiffness, are largely prone to instabilities, including buckling and brittle collapse at low strains. This challenge is here addressed by introducing a new class of aperiodic architected materials inspired by quasicrystalline lattices. Beam networks derived from canonical quasicrystalline patterns, such as the Penrose tiling in two dimensions and icosahedral quasicrystals (IQCs) in three dimensions, are shown to create stiff, stretching‐dominated topologies with non‐uniform force chain distributions, effectively mitigating the global instabilities observed in periodic designs through distributed localized buckling instabilities. Numerical and experimental results confirm the effectiveness of these designs in combining stiffness and stable deformability at large strains, representing a significant advancement in the development of low‐density metamaterials for applications requiring high impact resistance and energy absorption. These results demonstrate the potential of deterministic quasi‐periodic topologies to bridge the gap between periodic and random structures, while branching toward uncharted territory in the property space of architected materials.
Symmetry, in the sense of repetitive spatial arrangements, takes many specific forms that we encounter routinely, usually recognize visually, and have some difficulty in quantifying. As there are many types of symmetry, some of them partial or imperfect, so there are many measurement approaches. Some of these consider only the outline or boundary of an object, others include the interior structure, some apply to an entire image, while others operate on individual objects. Examples of the various classes of symmetry and several methods for analysis are presented.
We show how the spectrum of normal discrete short-range infinite-volume operators can be approximated with two-sided error control using only data from finite-sized local patches. As a corollary, we prove the computability of the spectrum of such infinite-volume operators with the additional property of finite local complexity and provide an explicit algorithm. Such operators appear in many applications, e.g. as discretizations of differential operators on unbounded domains or as so-called tight-binding Hamiltonians in solid state physics. For a large class of such operators, our result allows for the first time to establish computationally also the absence of spectrum, i.e. the existence and the size of spectral gaps. We extend our results to the ε \varepsilon -pseudospectrum of non-normal operators, proving that also the pseudospectrum of such operators is computable.
In previous work a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an ergodic theorem of Nevo to provide an appropriate definition of optimal dense packings. Examples are given to illustrate various aspects of the density problem, in particular the shift in emphasis from the analysis of individual packings to spaces of packings.
The class of Cyclotomic Aperiodic Substitution Tilings (CAST) is introduced. Its vertices are supported on the 2n-th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry Dn or D2n, i.e. the tiling contains an infinite number of patches of any size with dihedral symmetry Dn or D2n only by iteration of substitution rules on a single tile.
This paper gives a detailed review on symmetry and symmetry breaking of quasicrystals. The symmetry groups of solid and soft-matter quasicrystals observed so far are summarized and the symmetry breakings of quasicrystals are systematically discussed. The present review connects several branches of mathematics and physics and related applications in science and technology are also discussed.
