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Penrose tilings as coverings of congruent decagons
Abstract
The open problem of tiling theory whether there is a single aperiodic two-dimensional prototile with corresponding matching rules, is answered for coverings instead of tilings. We introduce admissible overlaps for the regular decagon determining only nonperiodic coverings of the Euclidean plane which are equivalent to tilings by Robinson triangles. Our work is motivated by structural properties of quasicrystals.
... -Elser & Henley [11] and Audier & Guyot [1] have obtained models for icosahedral quasicrystals by decorating the Ammann rhombohedra occuring in a tiling of the 3D space defined by projection [10,18]. -In his quasi-unit cell picture Steinhardt [34] has shown (following an idea of Petra Gummelt [13]) that the atomic structure can be described entirely by using a single repeating cluster which overlaps (shares atoms with) neighbour clusters. The model is determined by the overlap rules and the atom decoration of the unit cell. ...
... The cluster C can be regarded as a covering cluster, and Q as a quasiperiodic set which can be covered by partially occupied copies of a single cluster. This kind of covering is different from the covering of Penrose tiling by a decorated decagon [13] proposed by Gummelt in 1996 or the coverings of discrete quasiperiodic sets presented by Kramer, Gummelt, Gähler et. al. in [21]. ...
... In our case the generating cluster C is a finite set of points and the quasiperiodic pattern is obtained by projection. In [13,21] the covering clusters are congruent overlapping polytopes (with an asymetric decoration) and the structure is generated by imposing certain overlap rules which restrict the possible relative positions and orientations of neighbouring clusters. When the theory from [13,21] is applied to quasicrystals, atomic positions are assigned to the covering clusters. ...
Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagram of a quasicrystal admits as symmetry group a finite group G, and there is a G-cluster C (union of orbits of G) such that the quasicrystal can be regarded as a quasiperiodic packing of interpenetrating copies of C. We present an algorithm which leads from any G-cluster C directly to a multi-component model set Q such that the arithmetic neighbours of any point x belonging to Q are distributed on the sites of the translated copy x+C of C. Our mathematical algorithm may be useful in quasicrystal physics.
... Five copies of the P1 Gummelt motif can then be formed into a new pentagonal ring (c). The black-and-white portion of (a) is the P3 rhombus Gummelt motif (Gummelt, 1996) . This decagonal motif (c) is also used to create the other figures of (d), except for the star region in the center made from five obtuse rhombuses. ...
... The pentagonal motif with the red arrow has two decagons rotated by 108°, and the pentagonal motif with the green arrow has four decagons also rotated by 108°, with CW or CCW indicated by the curved blue arrows. Although the Gummelt motif (Gummelt, 1996) is not the decagon of Figure 2.3, it still works in a similar way (b). For simplicity, we will not be using the pentagonal motifs in (b) and will instead use those in (d), since they are still covering structures, minus the oblate rhombus star at the center of the pentagonal motifs-it has been added in any way for the sake of simplicity, and it will make it easier to understand the construction method. ...
... This is to say, it fills space and it is allowed to overlap itself. Gummelt later proved that these decorated decagonal motifs ( Figure 1.3) do indeed cover space in 2D (Gummelt, 1996), and so they now bear her name. Gummelt's prescription for overlaps, however, is actually quite complex-and it is not a recipe for constructing the decagonal motifs into a space-filling tiling. ...
In this thesis, I will show a fundamentally new way to understand and construct quasicrystal tilings based on a new formulation of the quasi-unit cell called layering. There are many methods for generating quasitilings including Ammann/pentagrid methods, cut-and-project higher-dimensional projections, deflation/inflation substitutions, and various approaches using matching rules and non-local forced moves. The only other method which has a real-space quasi-unit cell comes from covering theory. However, covering approaches, much like matching-rule approaches, do not provide a recipe for error-free construction of the quasitiling and lack a description of the phason flips—a new type of local particle movement only observed in quasicrystals. In this thesis, I will showcase that layering does provide a way to construct perfect quasitiling from a quasi-unit cell and naturally gives rise to a real-space description of the phason mode of particle movement. This new method was applied to the three Penrose pentagonal tilings, the Ammann-Beenker octagonal tiling, the Tübingen triangle decagonal tiling, the Niizeki-Gähler dodecagonal tiling, and finally the Ammann-Kramer-Neri icosahedral tiling. In addition, simulations were performed using a patchy hard-particle set of Penrose rhombuses as a demonstration of the power of the analysis method. The last chapter of this thesis reports the formation of a binary crystal of hard polyhedra due solely to entropic forces. Although the alternating arrangement of octahedra and tetrahedra is a known space-tessellation from Maurolyctus in 1529 (Lagarias, 2015), it had not previously been observed in self-assembly simulations. Both known one-component phases—the dodecagonal quasicrystal of tetrahedra and the densest-packing of octahedra in the Minkowski lattice—are found to coexist with the binary phase. Apart from an alternative, monoclinic packing of octahedra, no additional crystalline phases were observed.
... We begin by observing that any even-valence vertex in the Penrose tiling (a 4-vertex or 6-vertex) has no even-valence neighbors and precisely two even-valence second-nearest neighbors (see Fig. 1) [80]. A proof of this statement relies on considering the local empire of the 6-vertex in Fig. 4, which is large enough to cover the entire tiling when allowing for overlaps [81]. ...
... To see that the matching generated by the dimer inflation algorithm in Fig. 8(b) is maximum, recall the following facts. First, the local empire of the 6-vertex (Fig. 11) is sufficiently large that it can cover the entire Penrose tiling, allowing for overlaps [81]. Second, monomers cannot cross monomer membranes (solid thick black edges in Fig. 11 for even-loop segments, dashed thick black edges for potential even-loop segments). ...
... This property of P2 is in contrast to P3 in which each vertex appears in both sublattices. The P2 tiling admits a covering (allowing for overlaps) by a set of tiles known as a cartwheel [54,81,90]. Some cartwheels are highlighted in Fig. 14. Figure 15 shows a maximum matching of the same region obtained using the Hopcroft-Karp algorithm. ...
We study the classical dimer model on rhombic Penrose tilings, whose edges and vertices may be identified as those of a bipartite graph. We find that Penrose tilings do not admit perfect matchings (defect-free dimer coverings). Instead, their maximum matchings have a monomer density of 81−50φ≈0.098 in the thermodynamic limit, with φ=(1+5)/2 the golden ratio. Maximum matchings divide the tiling into a fractal of nested closed regions bounded by loops that cannot be crossed by monomers. These loops connect second-nearest-neighbor even-valence vertices, each of which lies on such a loop. Assigning a charge to each monomer with a sign fixed by its bipartite sublattice, we find that each bounded region has an excess of one charge, and a corresponding set of monomers, with adjacent regions having opposite net charge. The infinite tiling is charge neutral. We devise a simple algorithm for generating maximum matchings and demonstrate that maximum matchings form a connected manifold under local monomer-dimer rearrangements. We show that dart-kite Penrose tilings feature an imbalance of charge between bipartite sublattices, leading to a minimum monomer density of (7−4φ)/5≈0.106 all of one charge.
... An intriguing puzzle that has eluded researchers for three decades is identifying the mechanism that selects an ordered yet non-periodic state. One possible explanation is that quasicrystal are energy minimizing struc-tures, whose structure is forced by specific interatomic interactions [5], by maximizing the density of some favorable motif [6,7], or by creating a deep pseudogap in the electronic density of states [8][9][10]. Another possibility is that the structural ambiguity is an intrinsic characteristic of quasicrystals [11][12][13]. ...
... A typical example of these clusters as they appear in our simulations is shown in Fig. 3d, taken from the equilibrium ensemble at T=1200K, followed by relaxation. Mixed chemical occupation breaks the cluster symmetry and can serve as a stabilizing source of entropy, while, together with the cluster covering, it is potentially a means of forcing quasiperiodicity [6,7]. ...
... FIG.6. Calculated Al-Cu-Fe phase diagrams at T=0K (left) and T=600K (right). ...
Icosahedral quasicrystals spontaneously form from the melt in simulations of Al--Cu--Fe alloys. We model the interatomic interactions using oscillating pair potentials tuned to the specific alloy system based on a database of density functional theory (DFT)-derived energies and forces. Favored interatomic separations align with the geometry of icosahedral motifs that overlap to create face-centered icosahedral order on a hierarchy of length scales. Molecular dynamics simulations, supplemented with Monte Carlo steps to swap chemical species, efficiently sample the configuration space of our models, which reach up to 9846 atoms. Exchanging temperatures of independent trajectories (replica exchange) allows us to achieve thermal equilibrium at low temperatures. By optimizing structure and composition we create structures whose DFT energies reach to within 2 meV/atom of the energies of competing crystal phases. Free energies obtained by adding contributions due to harmonic and anharmonic vibrations, chemical substitution disorder, phasons, and electronic excitations, show that the quasicrystal becomes stable against competing phases at temperatures above 600K. The average structure can be described succinctly as a cut through atomic surfaces in six-dimensional space that reveal specific patterns of preferred chemical occupancy. Atomic surface regions of mixed chemical occupation demonstrate the proliferation of phason fluctuations, which can be observed in real space through the formation, dissolution and reformation of large scale icosahedral motifs -- a picture that is hidden from diffraction refinements due to averaging over the disorder and consequent loss of information concerning occupancy correlations.
... One can even go beyond tilings and consider coverings of space. 80 In the case of the Penrose tiling, Gummelt's decagon covering 58 with a single cluster (and overlap rules encoded by the shading) has proved very popular, because it allows the description of a quasicrystal structure in terms of a single fundamental building block. The three allowed (pairwise) overlaps of the marked decagons, shown in Figure 3, are characterised by matching decorations. ...
... The three allowed (pairwise) overlaps of the marked decagons, shown in Figure 3, are characterised by matching decorations. Figure 4 shows a patch of a corresponding covering, which is mutually locally derivable (MLD) with the Penrose tiling of Figure 2. 58,59 This covering also has an interpretation in terms of 'maxing rules', 49,61,72 where maximisation of one type of specified cluster leads to the Penrose rhombus tiling (up to zero density deviations). 72 Covering rules of either type have become quite fashionable in materials science. ...
Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details.
... From T 1 and T 2 we can get two new undecorated prototiles, nicknamed Snake and Dog, by shifting two outside half thin rhombs to the corresponding gaps (Fig. 4) 3 . Again the snake and dog tiles can be transformed into further and smoother shapes given by a hexagon and a pentagon, both concave and irregular. ...
... Gummelt constructed a decorated decagonal tile and showed that when two kinds of overlaps between pairs of tiles are allowed, these tiles can cover the plane only nonperiodically [3]. Socolar and Taylor presented an undecorated, but not connected aperiodic monotile that is based on a regular hexagon [5]. Figure 13 shows an approximation to an aperiodic monotile from the Penrose tiles, a shape that represents both kite and dart as well as possible. ...
We present aperiodic sets of prototiles whose shapes are based on the well-known Penrose rhomb tiling. Some decorated prototiles lead to an exact Penrose rhomb tiling without any matching rules. We also give an approximate solution to an aperiodic monotile that tessellates the plane (including five types of gaps) only in a nonperiodic way.
... From T 1 and T 2 we can get two new undecorated prototiles, nicknamed Snake and Dog, by shifting two outside half thin rhombs to the corresponding gaps (Fig. 4) 3 . Again the snake and dog tiles can be transformed into further and smoother shapes given by a hexagon and a pentagon, both concave and irregular. ...
... Gummelt constructed a decorated decagonal tile and showed that when two kinds of overlaps between pairs of tiles are allowed, these tiles can cover the plane only nonperiodically [3]. Socolar and Taylor presented an undecorated, but not connected aperiodic monotile that is based on a regular hexagon [5]. Figure 13 shows an approximation to an aperiodic monotile from the Penrose tiles, a shape that represents both kite and dart as well as possible. ...
We present aperiodic sets of prototiles whose shapes are based
on the well-known Penrose rhomb tiling. Some decorated prototiles
lead to an exact Penrose rhomb tiling without any matching
rules. We also give an approximate solution to an aperiodic
monotile that tessellates the plane (including five types of gaps)
only in a nonperiodic way.
... An intriguing puzzle that has eluded researchers for three decades is identifying the mechanism that selects an ordered yet nonperiodic state. One possible explanation is that quasicrystals are energetic minima, whose structure is forced by specific interatomic interactions [7], by maximizing the density of some favorable motif [8,9], or by creating a deep pseudogap in the electronic density of states [10][11][12]. Another possibility is that the structural ambiguity is an intrinsic characteristic of quasicrystals [13][14][15]. ...
... A typical example of these clusters as they appear in our simulations is shown in Fig. 3(d), taken from the equilibrium ensemble at T = 1200 K, followed by relaxation. Mixed chemical occupation breaks the cluster symmetry and can serve as a stabilizing source of entropy, while, together with the cluster covering, it is potentially a means of forcing quasiperiodicity [8,9]. ...
Icosahedral quasicrystals spontaneously form from the melt in simulations of Al-Cu-Fe alloys. We model the interatomic interactions using oscillating pair potentials tuned to the specific alloy system based on a database of density functional theory (DFT)-derived energies and forces. Favored interatomic separations align with the geometry of icosahedral motifs that overlap to create face-centered icosahedral order on a hierarchy of length scales. Molecular dynamics simulations, supplemented with Monte Carlo steps to swap chemical species, efficiently sample the configuration space of our models, which reach up to 9846 atoms. Exchanging temperatures of independent trajectories (replica exchange) allows us to achieve thermal equilibrium at low temperatures. By optimizing structure and composition we create structures whose DFT energies reach to within ∼2 meV/atom of the energies of competing crystal phases. Free energies obtained by adding contributions due to harmonic and anharmonic vibrations, chemical substitution disorder, phasons, and electronic excitations, show that the quasicrystal becomes stable against competing phases at temperatures above 600 K. The average structure can be described succinctly as a cut through atomic surfaces in six-dimensional space that reveal specific patterns of preferred chemical occupancy. Atomic surface regions of mixed chemical occupation demonstrate the proliferation of phason fluctuations, which can be observed in real space through the formation, dissolution and reformation of large-scale icosahedral motifs—a picture that is hidden from diffraction refinements due to averaging over the disorder and consequent loss of information concerning occupancy correlations.
... Any rhombus vertex is the projection of a hole from Λ = A4. In addition we show a covering of the Penrose tiling by overlapping decagons (heavy lines) [13]. Each decagon covers 10 rhombus tiles and is centred at the projection of a lattice point (black square) of A4. ...
... A new approach to quasiperiodic structure came with the idea of covering [13]. In a covering of a tiling, every tile becomes part of a small number of covering clusters. ...
Mathematicians have been interested in non-periodic tilings of space for decades; however, it was the unexpected discovery of non-periodically ordered structures in intermetallic alloys which brought this subject into the limelight. These fascinating materials, now called quasicrystals, are characterised by the coexistence of long-range atomic order and 'forbidden' symmetries which are incompatible with periodic arrangements in three-dimensional space. In the first part of this review, we summarise the main properties of quasicrystals, and describe how their structures relate to non-periodic tilings of space. The celebrated Penrose and Ammann-Beenker tilings are introduced as illustrative examples. The second part provides a closer look at the underlying mathematics. Starting from Bohr's theory of quasiperiodic functions, a general framework for constructing non-periodic tilings of space is described, and an alternative description as quasiperiodic coverings by overlapping clusters is discussed.
... So, even if matching rules can provide a toy model for energetic stabilization, they cannot explain why quasicrystals form and how they grow. A similar problem is also present in a slightly relaxed scenario, known as "maxing rules", see [28,37]. ...
This introductory survey deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical crystallography. In particular, we focus on their interplay with various physically motivated equivalence concepts such as local indistinguishability and local equivalence. Various discrete patterns with non-crystallographic symmetries are described in detail, and some of their magic properties are introduced. This perfectly ordered world is augmented by a brief introduction to the stochastic world of random tilings.
... Decagonal clusters feature prominently many models. Overlaps of these clusters [16] create shapes such as hexagons, boats and stars, leading in general to hexagon-boat-stardecagon (HBSD) tilings, and these tilings emerge at multiple length scales [14,17]. With growing computer power we can now perform full ab-initio total energy calculations for systems of several hundred atoms or more. ...
Atomic structures of Al-Co-Cu decagonal quasicrystals (dQCs) are investigated using empirical oscillating pair potentials (EOPP) in molecular dynamic (MD) simulations that we enhance by Monte Carlo (MC) swapping of chemical species and replica exchange. Predicted structures exhibit planar decagonal tiling patterns and are periodic along the perpendicular direction. We then recalculate the energies of promising structures using first-principles density functional theory (DFT), along with energies of competing phases. We find that our τ -inflated sequence of QC approximants (QCAs) are energetically unstable at low temperature by at least 3 meV/atom. Extending our study to finite temperatures by calculating harmonic vibrational entropy, as well as anharmonic contributions that include chemical species swaps and tile flips, our results suggest that the quasicrystal phase is entropically stabilized at temperatures in the range 600-800 K and above. It decomposes into ordinary (though complex) crystal phases at low temperatures, including a partially disordered B2-type phase. We discuss the influence of density and composition on QC phase stability; we compare the structural differences between Co-rich and Cu-rich quasicrystals; and we analyze the role of entropy in stabilizing the quasicrystal, concluding with a discussion of the possible existence of “high entropy” quasicrystals.
Published by the American Physical Society 2024
... [18]), and it is challenging to extend our method. P. Gummelt [10] gave a single decagon with special markings, which covers the plane but only in non-periodic ways. This covering encodes a version of Penrose tiling by Robinson's triangle. ...
Inspired by \cite{DBZ},we generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. For such a substitution, the substitution matrix may have non-integer entries. We give the meaning of such a matrix by showing that the right Perron--Frobenius eigenvector gives the patch frequency of the resulting tiling. We also show that the expansion constant is an algebraic integer under mild conditions. In general, overlapping substitutions may yield a patch with contradictory overlaps of tiles, even if it is locally consistent. We give a sufficient condition for an overlapping substitution to be consistent, that is, no such contradiction will emerge. We also give many one-dimensional examples of overlapping substitution. We finish by mentioning a construction of overlapping substitutions from Delone multi-sets with inflation symmetry.
... Gummelt [Gum96] and Jeong and Steinhardt [SJ96,JS97] describe a single regular decagon that can cover the plane with copies that are allowed to overlap by prescribed rules, but only non-periodically, in a manner tightly coupled to the Penrose tiling. Senechal [Sen] similarly describes simple rules that allow copies of the Penrose dart to overlap and cover the plane, but never periodically. ...
... The theoretical models provide important information about the formation mechanism and physical properties of quasicrystals. [2][3][4][5][6] There are several ways to generate a quasilattice, including the inflation-deflation transformation, the general dual method and the cut-and-projection method. [7][8][9][10] However, the formation of a real quasicrystal is different from the above processes, and the growth mechanisms of a perfect quasicrystal have not been thoroughly understood. ...
Based on the substitution rule and symmetry, we propose a method to generate an octagonal quasilattice consisting of square and rhombus tiles. Local configurations and Ammann lines are used to guide the growth of the tiles in a quasiperiodic order. The structure obtained is a perfect eight-fold symmetric quasilattice, which is confirmed by the radial distribution function and the diffraction pattern.
... The classic instance of such results can be found in Penrose Tilings, specifically the presentation from [33], and the original article by Penrose in [48] -wherein Penrose shows how you can acquire aperiodic tilings of the plane from as few as two prototiles. Indeed, two distinct but related prototile sets are given: the Penrose Rhombi and Penrose Kite and Dart prototile sets. ...
In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory. We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and do not restrict ourselves to the usual problems concerning finite sets of prototiles. We first consider two domino problems: whether a given set of prototiles S has total planar tilings, which we denote TILE, or whether it has infinite connected but not necessarily total tilings, WTILE (short for `weakly tile'). We show that both , and thereby both TILE and WTILE are -complete. We also show that the opposite problems, and SNT (short for `Strongly Not Tile') are such that and so both and SNT are both -complete. Next we give some consideration to the problem of whether a given (infinite) set of prototiles is periodic or aperiodic. We study the sets PTile of periodic tilings, and ATile of aperiodic tilings. We then show that both of these sets are complete for the class of problems of the form . We also present results for finite versions of these tiling problems. We then move on to consider the Weihrauch reducibility for a general total tiling principle CT as well as weaker principles of tiling, and show that there exist Weihrauch equivalences to closed choice on Baire space, . We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to . Finally, we give a prototile set of 15 prototiles that can encode any Elementary Cellular Automaton (ECA). We make use of an unusual tile set, based on hexagons and lozenges that we have not see in the literature before, in order to achieve this.
... This is highlighted in grey. Kites and darts consisting of two triangles are called Robinson triangles, after observations by Robinson in 1975(Gummelt, 1996. The kites and darts are shown in detail in Figure 9b. ...
Islamic floral patterns warrant further research and analysis as they are an important aspect of the cultural heritage of Islamic patterns. These floral patterns are aesthetically inspired by flowers, leaves, vines, and stems and feature characteristics such as symmetry, interlacing, and pattern repetition. This study analysed a five-pointed rose pattern (peony flower) and its elements, such as the curved lines that make up the leaves and flowers. A new floral pattern featuring a botanical motif and curved lines was designed and distributed using kite and dart tiling. The floral pattern was designed using the pentagram reflection of the Penrose tiling method to suit modern design requirements of looking like a Shamsah. The results of the floral ornament and newly designed patterns were then reviewed in order to facilitate the generation of new patterns accurately and quickly through computer design software. Thus, the problem of time and effort in designing Islamic floral patterns was solved. This study also provides suggestions for future studies on Islamic floral patterns.
... Therefore, it is sufficient to work with the decagon in figure 3(a). Tiling with overlapping decagons of figure 3(a) was first proposed by Gummelt [22]. ...
The present work offers a different perspective for the 5-fold symmetric quasicrystallography by employing affine H 2 as a subgroup of affine A 4 . It is shown that the projection of the Voronoi cell of the root lattice A 4 can be dissociated as identical five decagons up to a rotation tiled by thick and thin rhombuses. Projection of the Voronoi cell of the weight lattice onto the Coxeter plane tessellates the plane with four different tiles: thick and thin rhombuses with different edge lengths and two types of hexagons. Structure of the local dihedral symmetry H 2 fixing a particular point on the Coxeter plane is determined.
... Gummelt [Gum96] and Jeong and Steinhardt [SJ96,JS97] describe a single regular decagon that can cover the plane with copies that are allowed to overlap by prescribed rules, but only non-periodically, in a manner tightly coupled to the Penrose tiling. Senechal [Sen] similarly describes simple rules that allow copies of the Penrose dart to overlap and cover the plane, but never periodically. ...
A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.
... We have defined the affine extension of H 2 as the subgroup of affine A 4 . The projection of the Voronoi cell V(0) defines a decagon tiled by the projected rhombohedral facets in terms of thick and thin rhombuses as shown in Fig. 4. The same decagonal patch has been obtained earlier by different techniques (Gummelt, 1996;Baake et al., 1988). Here we have studied it with an analysis of the group-theoretical applications based on the affine A 4 and its affine subgroup H 2 . ...
The projections of lattices may be used as models of quasicrystals, and the particular affine extension of the H2 symmetry as a subgroup of A4, discussed in this work, presents a different perspective on fivefold symmetric quasicrystallography. Affine H2 is obtained as the subgroup of affine A4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of A4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine A4 symmetry. Facets of the Voronoi cells of the root and weight lattices are identified. Four adjacent rhombohedral facets of the Voronoi cell V(0) of A4 project into the decagonal orbit of H2 as thick and thin rhombuses where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of H2. It is shown that the thick and thin rhombuses constitute the finite fragments of the tiles of the Coxeter plane with the action of the affine H2 symmetry. Projection of the Voronoi cell of the weight lattice onto the Coxeter plane tessellates the plane with four different tiles: thick and thin rhombuses with different edge lengths obtained from the projection of the square faces and two types of hexagons obtained from the projection of the hexagonal faces of the Voronoi cell. The structure of the local dihedral symmetry H2 fixing a particular point on the Coxeter plane is determined.
... Since it was first shown that a particular patch of unit tiles (Gummelt decagon, GD) can cover the Penrose tiling (PT) with well defined overlaps and without gaps (covering) (Gummelt, 1996), this structural subunit has been commonly used as the prototype for a quasi-unit cell (Steinhardt et al., 1998;Abe et al., 2000). A quasi-unit cell is just a structural repeat unit analogous to the unit cells of periodic crystal structures. ...
Specific structural repeat units can be used as quasi-unit cells of decagonal quasicrystals. So far, the most famous and almost exclusively employed one has been the Gummelt decagon. However, in an increasing number of cases Lück decagons have been found to be more appropriate without going into depth. The diversities and commonalities of these two basic decagonal clusters and of some more general ones are discussed. The importance of the type of underlying tiling for the correct classification of a quasi-unit cell is demonstrated.
... It allows a more physical description of a quasicrystal structure than the structurally fully equivalent tiling-based models do [6][7][8] . In the case of decagonal quasicrystals (DQCs), the so-far most frequently used quasi-unit-cell is based on the Gummelt decagon [9][10][11][12] , if the quasi-unit cell approach is employed at all. ...
A high-angle annular dark field scanning transmission electron microscopy study of the intermetallic compound Al74Cr15Fe11 reveals a quasiperiodic structure significantly differing from the ones known so far. In contrast to the common quasi-unit-cells based on Gummelt decagons, the present structure is related to a covering formed by Lück decagons, which can also be described by a Hexagon-Bow-Tie tiling. The description of quasicrystal structures by only one repeating unit is desirable. Here the authors experimentally identify a quasi-unit-cell corresponding to a Lück decagon to describe the randomly ordered structure of a decagonal quasicrystalline phase.
... While tilings are based on two or more unit-tiles, coverings can cover the plane (space) by partially overlapping copies of a single structural repeat-unit (quasi-unit-cell [2][3][4][5][6] ). In the case of decagonal quasicrystals, the so-far most abundant quasi-unit-cell is based on the Gummelt decagon [7][8][9][10] . Such a quasi-unit-cell, when decorated with atoms (atomic cluster), is the counterpart to a unit cell of a periodic structure. ...
A high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) study of the intermetallic compound Al74Cr15Fe11 reveals a novel kind of aperiodic order. In contrast to the common quasi-unit-cells based on Gummelt decagons, the present structure is related to a covering formed by Andritz decagons, which can also be described by a Hexagon-Bowtie (HB) tiling. This is the first observation of a decagonal quasicrystal with a structure significantly differing from the ones known so far.
... Another possibility, as previously mentioned above, is that perhaps the geometry of the space graph is a generalized higher-dimensional version of variant tiling (see Figure 18), Penrose tiling 25 (see Figure 17) [34], Gummelt covering, [35] or a quasi-crystal type of structure. [36] Our understanding of such structures is evolving but much still remains to be learned about them. ...
This article proposes a unified theory framework encompassing a discrete topological interpretation of physical forces, wave functions, and the nature of space and time. It provides novel explanations for the collapse of wave functions, quantum entanglement, and offers insights into the origins of quantum probabilities. This article also explains the nature of mass, Higgs field, and suggests a path for unifying quantum mechanics and gravity. Elementary particles are represented as defects in discrete topological spaces. Entangled particles are directly connected to each other through a puncture in discrete space, separated by a distance of one Planck length. Wave functions are explained as mechanical stress waves within elastic discrete space. The results of the double-slit experiment are interpreted as wave functions maximizing the probability of rupture in high-stress areas of discrete space with obvious analogies to solid state mechanics. Wave-particle duality is explained as discrete topological defects causing extended distributed stress within space lattice.
... In fact, by covering the inflated rhombohedra with RTH clusters, we can assign every single atom to a RTH cluster forming a kind of quasiperiodic covering. A covering is usually discussed for decagonal QCs, e.g. the Gummelt cluster has such a property (Gummelt, 1996), but it was never elaborated nor implied for an iQC. That feat is not possible for the 12-fold sphere packing based model of iQCs. ...
In this study, the atomic structure of the ternary icosahedral ZnMgTm quasicrystal (QC) is investigated by means of single-crystal X-ray diffraction. The structure is found to be a member of the Bergman QC family, frequently found in Zn–Mg–rare-earth systems. The ab initio structure solution was obtained by the use of the Superflip software. The infinite structure model was founded on the atomic decoration of two golden rhombohedra, with an edge length of 21.7 Å, constituting the Ammann–Kramer–Neri tiling. The refined structure converged well with the experimental diffraction diagram, with the crystallographic R factor equal to 9.8%. The Bergman clusters were found to be bonded by four possible linkages. Only two linkages, b and c, are detected in approximant crystals and are employed to model the icosahedral QCs in the cluster approach known for the CdYb Tsai-type QC. Additional short b and a linkages are found in this study. Short interatomic distances are not generated by those linkages due to the systematic absence of atoms and the formation of split atomic positions. The presence of four linkages allows the structure to be pictured as a complete covering by rhombic triacontahedral clusters and consequently there is no need to define the interstitial part of the structure (i.e. that outside the cluster). The 6D embedding of the solved structure is discussed for the final verification of the model.
... It is possible to get a structure corresponding to Penrose tiling using one element (decagon). However, such a construction is not a tiling, since both touching and partial intersection of the elements is possible [8]. Similar decagons were proposed to describe the structure of decagonal quasicrystals [9]. ...
The method for constructing 3D fractals of stars with icosahedral symmetry is proposed. A great stellated dodecahedron (I) and a small stellated dodecahedron (D) were selected as building elements. The faces of the polyhedra are equal in size. The initial polyhedron (I or D) is replicated and the copies are placed so that its centres coincide with the vertices of the star (I or D), which is not displayed. This star is called by the author the generalized star, its size is τ^N times the size of the initial star, where τ = (1 + 5^0.5) / 2 ≈ 1.618 is the golden mean, N is a non-negative integer. At each next step of the construction, the previous prefractal is replicated and the copies are placed so that its centres coincide with the vertices of the generalized star. The series of integers N forms a non-decreasing sequence. The types of generalized stars (I or D) can vary at every step. Coinciding points are counted once. Initial polyhedra can touch each other by vertices, overlap each other or stand separately. Like in the Penrose 3D tiling, all the vertices of these fractals belong to the symmetric projection of a simple 6D cubic lattice onto a 3D space.
... It is natural, for example, to ask that the tile does not have too wild a shape: it should be the closure of its interior, but one might also demand that it is a polytope, just a topological disc or perhaps merely that it is connected. And finally by 'tiling the plane' one usually means that the tiles cover the plane but that distinct tiles overlap on at most their boundaries (however, we note here Gummelt's aperiodic tile which tiles the plane with overlaps [6]). One should also specify what rules are permitted in how tiles can be placed next to each other -should these rules be forced by geometry alone, are colour matchings permitted, or can more complicated local rules be specified? ...
We present a single, connected tile which can tile the plane but only non-periodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar-Taylor tiling. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity.
... is sufficiently large that it can cover the entire Penrose tiling, allowing for overlaps 58 . Second, monomers cannot cross monomer membranes (solid thick black edges in Fig. 10 for even-loop segments, dashed thick black edges for potential even-loop segments). ...
We study the classical dimer model on rhombic Penrose tilings, whose edges and vertices may be identified with those of a bipartite graph. We find that Penrose tilings do not admit perfect matchings (defect-free dimer coverings). Instead, their maximum matchings have a monomer density of in the thermodynamic limit, with the golden ratio. Maximum matchings divide the tiling into a fractal of nested closed regions bounded by loops that cannot be crossed by monomers. These loops connect second-nearest neighbour even-valence vertices, each of which lies on such a loop. Assigning a charge to each monomer with a sign fixed by its bipartite sublattice, we find that each bounded region has an excess of one charge, and a corresponding set of monomers, with adjacent regions having opposite net charge. The infinite tiling is charge neutral. We devise a simple algorithm for generating maximum matchings, and demonstrate that maximum matchings form a connected manifold under local monomer-dimer rearrangements.
... However, we have to start this project with the most constrained definition: one cover of rectangular shape. 3. The famous Penrose tilings [14] can be described in terms of a single tile that overlaps itself [9]. In one dimension, the family of standard Sturmian words (which are right-infinite words) can be characterized in terms of covers [5]. ...
A coloration w of Z^2 is said to be coverable if there exists a rectangular block q such that w is covered with occurrences of q, possibly overlapping. In this case, q is a cover of w. A subshift is said to have the cover q if each of its points has the cover q. In a previous article, we characterized the covers that force subshifts to be finite (in particular, all configurations are periodic). We also noticed that some covers force subshifts to have zero topological entropy while not forcing them to be finite. In the current paper we work towards characterizing precisely covers which force a subshift to have zero entropy, but not necessarily periodicity. We give a necessary condition and a sufficient condition which are close, but not quite identical.
... A second view of super-tiles in quasicrystals was introduced by Gummelt[95], which deals with the construction of quasicrystals through coverings (rather than tilings). Here a quasi-patch is repeated but allowed to overlap according to some overlap rules. ...
In this thesis we investigate designer disordered complex media for photonics and phononics applications. Initially we focus on the photonic properties and we analyse hyperuniform disordered structures (HUDS) using numerical simulations. Photonic HUDS are a new class of photonic solids, which display large, isotropic photonic band gaps (PBG)comparable in size to the ones found in photonic crystals (PC). We review their complex interference properties, including the origin of PBGs and potential applications. HUDS combine advantages of both isotropy due to disorder (absence of long-range order) and controlled scattering properties from uniform local topology due to hyperuniformity (constrained disorder). The existence of large band gaps in HUDS contradicts the longstanding intuition that Bragg scattering and long-range translational order is required in PBG formation, and demonstrates that interactions between Mie-like local resonances and multiple scattering can induce on their own PBGs. The discussion is extended to finite height effects of planar architectures such as pseudo-band-gaps in photonic slabs as well as the vertical confinement in the presence of disorder. The particular case of a silicon-on-insulator compatible hyperuniform disordered network structure is considered for TE polarised light. We address technologically realisable designs of HUDS including localisation of light in point-defect-like optical cavities and the guiding of light in freeform PC waveguide analogues. Using finite-difference time domain and band structure computer simulations, we show that it is possible to construct optical cavities in planar
hyperuniform disordered solids with isotropic band gaps that efficiently confine TE polarised radiation. We thus demonstrate that HUDS are a promising general-purpose design platform for integrated optical micro-circuitry. After analysing HUDS for photonic applications we investigate them in the context of elastic waves towards phononics applications. We demonstrate the first phononic band gaps (PnBG) for HUDS. We find that PnBGs in phononic HUDS can confine and guide elastic waves similar to photonic HUDS for EM radiation.
... It is interesting to consider the motifs and superstructures of motifs that we have identified through configurations of P2 tiles at different scales in light of previous work on coverings and quasi-unit cells. Overlapping decorated decagonal tiles has been shown 47 to be a useful method of generating pentagonal quasi- crystalline tilings according to prescribed geometrical rules that govern the overlap. These overlapping decagonal systems were referred to as 'coverings' , to distinguish them from 'tilings' , which typically do not have overlap. ...
A lithographic patterning and release method is used to create a dense, fluctuating, Brownian system of mobile colloidal kite- and dart-shaped Penrose tiles over large areas that retains quasi-crystalline order.
... Summary of all the joints and triple junctions formed by DB-1s and/or DB-2s in Mg 21 Zn 25 .Triple Junctions Three DB-1sThree DB-2sTwo DB-2s þ one DB-1 Zn atoms within the Zn-rich layer in front of the growing Mg 21 Zn 25 domains, which could consequently lower the local composition quickly to a level facilitating the nucleation and growth of other Mg 21 Zn 25 domains based on those overgrown Laves-type columns. This produced DBs with slightly overgrown Laves-type columns which could be modeled by the overlap of two such C14 Laves-type columns within Mg 21 Zn 25[41,42], as shown in Figs. 5e7. ...
... A quasicrystal may also be made up from congruent parts [3,[24][25][26][27]. But it may have no translational symmetry at all. ...
The problem of finding the toroidal unit cell of a quasicrystal is formulated in terms of a matrix (graph) divisor. Such a unit cell is an analog of the crystal’s unit cell with cyclic boundary conditions, but is not a crystal approximant of the latter. The proposed approach may be used for calculating the energy-band structure of quasicrystals.
... Any tiling belonging to the Penrose local isomorphic class can be created using only one cluster type with specific overlapping rules. It is a mirror-symmetrical decagon introduced by Gummelt [137]. Certain overlap rules imposed on the structure units (decagons or other) guarantee an ideal tiling formation. ...
From the discovery of quasicrystals, exhibiting rotational symmetries forbidden by the classical crystallography, the idea of a quasicrystalline geometry spreads out to a great variety of scientific fields, including soft matter physics, optics, or nanotechnology. However, the atomic structure of quasicrystals is still unclear and many questions are unanswered, including the most peculiar one asked by Per Bak: ‘Where are the atoms?’ [Icosahedral crystals: where are the atoms? Phys Rev Lett. 1986;56:861–864]. To answer those questions, more detailed structural analysis needs to be done. Two approaches to the structural description and refinement are used for aperiodic crystals: the higher dimensional method, well established in the community, and the statistical method. We introduce the essence of the crystallography of quasicrystals, giving the historical background and fundamentals of the higher dimensional modelling, but focus is put on the details and application of the statistical approach. We present also a case study of the decagonal Al–Cu–{Co,Ir,Rh} phase, which is an example of the crystal structure solved using the statistical method. In the last chapter, a brief discussion of the recently developed approach of a phasonic corrective factor, alongside the effect of phonons on the atomic distribution function, is given.
... Having determined the low energy states of the vertices, we joined them into the full Penrose tiling using a two step process in which nearest-neighbour interactions were considered. First the vertices were joined to form two different types of decagons, which were then fused to form the overall pattern 48 . This process is described in detail in the Supplementary Information. ...
We have created and studied artificial magnetic quasicrystals based on Penrose tiling patterns of interacting nanomagnets that lack the translational symmetry of spatially periodic artificial spin ices. Vertex-level degeneracy and frustration induced by the network topology of the Penrose pattern leads to a low energy configuration that we propose as a ground state. It has two parts, a quasi-one-dimensional rigid "skeleton" that spans the entire pattern and is capable of long-range order, and clusters of macrospins within it that are degenerate in a nearest neighbour model, and so are "flippable". These lead to macroscopic degeneracy for the array as a whole. Magnetic force microscopy imaging of Penrose tiling arrays revealed superdomains that are larger for more strongly coupled arrays. The superdomain size is larger after AC-demagnetisation and especially after annealing the array above its blocking temperature.
We study the universal groups of inverse semigroups associated with point sets and with tilings. We focus our attention on two classes of examples. The first class consists of point sets which are obtained by a cut and projection scheme (so-called model sets). Here we introduce another inverse semigroup which is given in terms of the defining data of the projection scheme and related to the model set by the empire congruence. The second class is given by one-dimensional tilings.
We have created and studied artificial magnetic quasicrystals based on Penrose tiling patterns of interacting nanomagnets that lack the translational symmetry of spatially periodic artificial spin ices. Vertex-level degeneracy and frustration induced by the network topology of the Penrose pattern leads to a low energy configuration that we propose as a ground state. Topologically induced emergent frustration means that this ground state cannot be constructed from vertices in their ground states. It has two parts, a quasi-one-dimensional rigid "skeleton" that spans the entire pattern and is capable of long-range order, and "flippable" clusters of macrospins within it. These lead to macroscopic degeneracy for the array as a whole. Magnetic force microscopy imaging of Penrose tiling arrays revealed superdomains that are larger for more strongly coupled arrays. The superdomain size is larger after AC-demagnetisation and especially after annealing the array above its blocking temperature.
Mechanical metamaterials, also known as architected materials, are rationally designed composites, aiming at elastic behaviors and effective mechanical properties beyond (“meta”) those of their individual ingredients – qualitatively and/or quantitatively. Due to advances in computational science and manufacturing, this field has progressed considerably throughout the last decade. Here, we review its mathematical basis in the spirit of a tutorial, and summarize the conceptual as well as experimental state-of-the-art. This summary comprises disordered, periodic, quasi-periodic, and graded anisotropic functional architectures, in one, two, and three dimensions, covering length scales ranging from below one micrometer to tens of meters. Examples include extreme ordinary linear elastic behavior from artificial crystals, e.g., auxetics and pentamodes, “negative” effective properties, behavior beyond classical linear elasticity, e.g., arising from local resonances, chirality, beyond-nearest-neighbor interactions, quasi-crystalline mechanical metamaterials, topological band gaps, cloaking based on coordinate transformations and on scattering cancellation, seismic protection, nonlinear and programmable metamaterials, as well as space-time-periodic architectures.
We present a single, connected tile which can tile the plane but only nonperiodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar–Taylor tilings. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity. We show that the hull of all tilings satisfying our rules is uniquely ergodic and that almost all tilings in the hull belong to a minimal core of tilings generated by substitution. Identifying tilings which are charge-flips of each other, these tilings are shown to have pure point dynamical spectrum and a regular model set structure.
Electron microscopy images of decagonal quasicrystals obtained recently have been shown to be related to cluster coverings with a Hexagon–Bow–Tie decagon as single structural unit. Most decagonal phases show more complex structural orderings than models based on deterministic tilings like the Penrose tiling. We analyze different types of decagonal random tilings and their coverings by a Hexagon–Bow–Tie decagon.
A set of X-ray data collected on a fragment of decagonite, Al71Ni24Fe5, the only known natural decagonal quasicrystal found in a meteorite formed at the beginning of the Solar System, allowed us to determine the first structural model for a natural quasicrystal. It is a two-layer structure with decagonal columnar clusters arranged according to the pentagonal Penrose tiling. The structural model showed peculiarities and slight differences with respect to those obtained for other synthetic decagonal quasicrystals. Interestingly, decagonite is found to exhibit low linear phason strain and a high degree of perfection despite the fact it was formed under conditions very far from those used in the laboratory.
Though quasicrystals were discovered more than a decade ago, the fundamental questions of why and how quasicrystals form are still unanswered. Knowledge of atomic positions, for all species present, is the most important factor in trying to answer this question, and we have therefore applied Z-contrast imaging to this problem. The quasicrystalline structure, a state with long-range order, but no periodic translation symmetry, has been commonly described by the Penrose tiling model, in which two types of tile are connected according to strict matching rules. However, such a mathematical description does not provide any physical insight into how and why quasicrystals arise. An alternative description of the ideal quasicrystal has been given by Gummelt, who showed that the ideal quasicrystalline arrangement can be obtained with a single type of tile which is allowed to overlap so as to cover the surface. The equivalence of these two descriptions was proved by Steinhardt and Jeong.
The structure of a decagonal quasicrystal in the Zn58Mg40Y2 (at.%) alloy was studied using electron diffraction and atomic resolution Z-contrast imaging techniques. This stable Frank–Kasper Zn–Mg–Y decagonal quasicrystal has an atomic structure which can be modeled with a rhombic/hexagonal tiling decorated with icosahedral units at each vertex. No perfect decagonal clusters were observed in the Zn–Mg–Y decagonal quasicrystal, which differs from the Zn–Mg–Dy decagonal crystal with the same space group P10/mmm. Y atoms occupy the center of ‘dented decagon’ motifs consisting of three fat rhombic and two flattened hexagonal tiles. About 75% of fat rhombic tiles are arranged in groups of five forming star motifs, while the others connect with each other in a ‘zigzag’ configuration. This decagonal quasicrystal has a composition of Zn68.3Mg29.1Y2.6 (at.%) with a valence electron concentration (e/a) of about 2.03, which is in accord with the Hume–Rothery criterion for the formation of the Zn-based quasicrystal phase (e/a = 2.0–2.15).
In this study, the atomic arrangement of the decagonal phase in a spark-plasma-sintered Al–Cr–Fe alloy has been studied by Cs-corrected high angle annular dark field scanning transmission electron microscopy (HAADF STEM) for the first time. We have found that three types of atomic clusters exist as basic structural units: star, flattened hexagon and decagon with an edge-length of 0.65 nm. These three types of units are linked to each other by sharing edges or inter-penetrating, with adjacent star clusters overlapping each other. The respective structural models of these structural units are proposed from the HAADF STEM images.
The hexagonal ZrNiAl-type (space group: P-62m) and the tetragonal Mo2FeB2-type (space group: P4/mbm) structures, which are frequently formed in the same Yb-based alloys and exhibit physical properties related to valence-fluctuation, can be regarded as approximants of a hypothetical dodecagonal quasicrystal. Using Pd-Sn-Yb system as an example, a model of quasicrystal structure has been constructed, of which 5-dimensional crystal (space group: P12/mmm, aDD=5.66 {\AA} and c=3.72 {\AA}) consists of four types of acceptance regions located at the following crystallographic sites; Yb [00000], Pd[1/3 0 1/3 0 1/2], Pd[1/3 1/3 1/3 1/3 0] and Sn[1/2 00 1/2 1/2]. In the 3-dimensional space, the quasicrystal is composed of three types of columns, of which c-projections correspond to a square, an equilateral triangle and a 3-fold hexagon. They are fragments of two known crystals, the hexagonal {\alpha}-YbPdSn and the tetragonal Yb2Pd2Sn structures. The model of the hypothetical quasicrystal may be applicable as a platform to treat in a unified manner the heavy fermion properties in the two types of Yb-based crystals.
How does a quasicrystal grow? Despite the decades of research that have been dedicated to this area of study, it remains one of the fundamental puzzles in the field of crystal growth. Although there has been no lack of theoretical studies on quasicrystal growth, there have been very few experimental investigations with which to test their various hypotheses. In particular, evidence of the in situ and three-dimensional (3D) growth of a quasicrystal from a parent liquid phase is lacking. To fill-in-the-gaps in our understanding of the solidification and melting pathways of quasicrystals, we performed synchrotron-based X-ray imaging experiments on a decagonal phase with composition of Al-15at%Ni-15at%Co. High-flux X-ray tomography enabled us to observe both growth and melting morphologies of the 3D quasicrystal at temperature. We determined that there is no time-reversal symmetry upon growth and melting of the decagonal quasicrystal. While quasicrystal growth is predominantly dominated by the attachment kinetics of atomic clusters in the liquid phase, melting is instead barrier-less and limited by buoyancy-driven convection. These experimental results provide the much-needed benchmark data that can be used to validate simulations of phase transformations involving this unique phase of matter.
The way to find the optimum packing of quasicrystal-constituting clusters is discussed based on the projected-cell approach. We illustrate why the quasiperiodic arrangement of partially overlapping clusters with decagonal or icosahedral symmetry is the most efficient one, by relating it to the packing of unit cells in hypercubic lattices. © Springer International Publishing Switzerland 2015. All rights reserved.
A shield-like tile (SLT) consisting of one star tile and two hexagon tiles was used as one structural block to describe the tilings of quasicrystal approximants in our previous work [J. All. Compds. 647 (2015) 797–801.] because of its merits compared with the hexagon-boat-star model. In this paper, we aim to elucidate the rules of connection and the orientations of SLTs. Two examples of high-angle annular dark-field scanning transmission electron microscopy images are provided to support these rules. The SLT orientation mapping of the quasicrystal-related structures along the pseudo-tenfold axis was achieved, and the structural characterizations were revealed intuitively. Furthermore, some new quasicrystal approximants were discussed based on the changeable rows of SLTs.
The aim of the present article is to give a review of the state-of-the-art on quasicrystal structure analysis. After a short discussion of the term “crystal” in chapter 1, the geometrical generation of quasilattices is touched in chapter 2. In the following the higher-dimensional description of 1d, 2d and 3d quasi-crystals is demonstrated in detail as well as the derivation of structure factor equations and symmetry relationships in the higher-dimensional space. Chapter 4 shows the experimental techniques and structure determination methods for the study of quasicrystals. The experimental results of structural studies performed with different tech-niques are critically reviewed in chapter 5. Some of the results of the literature research are that five years after the detection of the first quasicrystal not a single quantitative (in terms of a regular structure determination) analysis of its structure has been carried out, and that the famous Mackay-icosahedra do not play the important role as the basic structural building elements as one supposed before.
CONTENTSIntroductionChapter I. Group approach §1. Two-dimensional quasicrystallographic groups1.1. Finite generation of two-dimensional quasicrystallographic groups1.2. Examples of two-dimensional quasicrystallographic groups with infinite point group. Classification of admissible rotation angles1.3. Classification of two-dimensional quasicrystallographic groups with finite point group §2. Three-dimensional and multidimensional quasicrystallographic groups with finite point group2.1. The abstract construction of quasicrystallographic groups with finite point group2.2. Classification of admissible representations in certain special cases §3. Connection with other approaches and definitions3.1. Connection between quasiperiodic tilings and quasicrystallographic groups3.2. Connection with projections of multidimensional crystals3.3. Connection with the Mermin-Rokhsar-Wright phase multipliers3.4. Connection with the generalized Mermin-Rokhsar-Wright phase multipliers. The case of an infinite point group §4. Quasicrystallographic groups with infinite point group4.1. Complete reducibility and pseudo-orthogonality of the action of the point group on a quasilattice4.2. Classification of admissible rotation angles in the case of a quadratic quasilattice4.3. Some examples4.4. Questions connected with finite generationChapter II. The geometry of local rules §1. Definitions and notation §2. Quasiperiodicity and local rules2.1. Projection method and quasiperiodicity2.2. An example of quasiperiodic local rules2.3. The section method2.4. The construction of local rules §3. The geometry of the forbidden set3.1. The topology of complements of 2-planes §4. Multidimensional case §5. An example of local rulesAcknowledgementsAppendices1. Tilings with -symmetry2. Generalization to the case References
A new geometrical method for generating aperiodic lattices forn-fold non-crystallographic axes is described. The method is based on the self-similarity principle. It makes use of the principles of gnomons to divide the basic triangle of a regular polygon of 2n sides to appropriate isosceles triangles and to generate a minimum set of rhombi required to fill that polygon. The method is applicable to anyn-fold noncrystallographic axis.
It is first shown how these regular polygons can be obtained and how these can be used to generate aperiodic structures. In particular, the application of this method to the cases of five-fold and seven-fold axes is discussed. The present method indicates that the recursion rule used by others earlier is a restricted one and that several aperiodic lattices with five fold symmetry could be generated. It is also shown how a limited array of approximately square cells with large dimensions could be detected in a quasi lattice and these are compared with the unit cell dimensions of MnAl6 suggested by Pauling.
In addition, the recursion rule for sub-dividing the three basic rhombi of seven-fold structure was obtained and the aperiodic lattice thus generated is also shown.
Two 2D quasiperiodic tilings with generalized tenfold symmetry are derived from the lattice A4R, the reciprocal of the root lattice A4. Both tilings are built from four tiles, triangles in one case, rhombi and hexagons in the other. After a brief description of the tilings and their structures, the authors introduce the equivalence concept of mutual local derivability. They present its key properties and its application to several tenfold tilings and discuss some implications on a future classification of tilings in position space.
Soon after the announcement of their discovery in 1984 [1], quasi-crystals hit the headlines. Here was a substance—an alloy of aluminum and manganese—whose electron diffraction patterns exhibited clear and unmistakable icosahedral symmetry (a view along a five-fold axis is shown in Figure 25-1). A clear and unmistakable diffraction pattern of any sort is evidence of “long-range order”: The diffraction pattern is a picture of a Fourier transform. Long-range order is usually synonymous with periodicity, and every periodic structure has a translation lattice. But a simple argument shows that five-fold rotational symmetry is incompatible with lattices in R
2 and R
3: every lattice has a minimum distance d between its points, but if two points at this distance are centers of five-fold rotation about parallel axes, the rotations will generate an orbit with smaller distances between them (Figure 25-2). By this chain of reasoning, it appeared that the impossible had occurred.
Conventional tiling models of quasicrystals imply the existence of two or more elementary cells (tiles). A new approach is proposed that allows a quasicrystal to be thought of as a random assembly of identical interpenetrating atomic clusters. This model is shown to be equivalent to a decagonal binary tiling. On applying a random tiling hypothesis, originally postulated by Elser, to the present cluster model it is found that the free energy as a function of the alloy composition has a cusp at a point exactly corresponding to the decagonal quasicrystal. This fact helps to explain an old mystery, namely why a system is phase locked in a quasicrystalline state even thought it is incommensurate.
A structure model for the decagonal quasicrystals AlCuCo and AlNiCo is proposed. The model agrees with available experimental data, i.e., with three-dimensional and five-dimensional Patterson analysis of x-ray-diffraction data and with direct-space atomic patterns found by high-resolution electron microscopy. The model allows for a dualistic description. It can be viewed both as a set of specifically decorated overlapping decagonal clusters and as a cutting and projecting of only two simple atomic surfaces.''
We consider a ``cluster picture'' to explain the structure and formation of quasicrystals. Quasicrystal ordering is attributed to a small set of low-energy atomic clusters which determine the state of minimum free energy. The nature of the ground state as T-->0, depending upon the clusters and cluster energies, ranges from energetically stable and unique (as in Penrose tilings), to entropically stable and degenerate (as in random tilings).
TheStructureofQuasicrystals
- W Steurer
Steurer, W.: TheStructureofQuasicrystals,ZeitschriftftirKristallographie 190(1990), 179-234.
Construction of Penrose filings by a single aperiodic protoset
- P Gummelt
Gummelt, P.: Construction of Penrose filings by a single aperiodic protoset, Proc. 5th Internat. Conf. on Quasicrystals, Avignon, 1995, 84-87.
Aperiodicity and Order
- M Jari
Jari6, M.: Aperiodicity and Order, Vol. 1: Introduction to Quasicrystals, Vol. 2: Introduction to the Mathematics of Quasicrystals, San Diego, 1988 and 1989.
- B Grtinbaum
- G C Shephard
Grtinbaum, B. and Shephard, G. C.: Tilings and Patterns, Freeman, New York, 1987.