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An aperiodic hexagonal tile
Abstract
We show that a single prototile can fill space uniformly but not admit a
periodic tiling. A two-dimensional, hexagonal prototile with markings that
enforce local matching rules is proven to be aperiodic by two independent
methods. The space--filling tiling that can be built from copies of the
prototile has the structure of a union of honeycombs with lattice constants of
, where a sets the scale of the most dense lattice and n takes all
positive integer values. There are two local isomorphism classes consistent
with the matching rules and there is a nontrivial relation between these
tilings and a previous construction by Penrose. Alternative forms of the
prototile enforce the local matching rules by shape alone, one using a
prototile that is not a connected region and the other using a
three--dimensional prototile.
... It has been postulated, and subsequently answered to a lesser degree than expected in [56], the question "Does there exist a single-prototile that tiles the plane aperiodically?" The Taylor-Socolar tile detailed in [56] achieves this, but by the use of a tile that is defined with gaps between its various pieces -though tilings of the plane utilising this tile cover every point. ...
... It has been postulated, and subsequently answered to a lesser degree than expected in [56], the question "Does there exist a single-prototile that tiles the plane aperiodically?" The Taylor-Socolar tile detailed in [56] achieves this, but by the use of a tile that is defined with gaps between its various pieces -though tilings of the plane utilising this tile cover every point. ...
... Indeed, the work of Socolar et al. in [56] is a very interesting way of determining the dynamics of this aperiodic tiling system. We will consider more the dynamics of tilings in Chapter 6 -but it is worth noting that it is an open problem as to whether the tile-by-tile tilings of the plane due to the method in [56] does indeed lead to planar tilings. ...
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.
... Analogous constructions in higher dimensions were obtained for E = G = Z 3 (or more precisely R 3 ) in [7,8,31] and for E = G = Z n (or more precisely R n ), n ≥ 3, in [12]. Lauchli [35] aperiodic (W) 36 Robinson [30] completion-undecidable (W) 32 Robinson [14] aperiodic (W) 24 Grünbaum-Shephard [14] aperiodic (W) 24 Robinson [14] aperiodic (W) This list is primarily adapted from [16], and incorporates from that reference some corrections to the values of J in several lines of this table. Constructions labeled (W) arise from a Wang tile construction. ...
... Similarly for the "Ammann A2" construction in [1] (with J = 2 · 4 = 8). The aperiodic tiling of R 2 (or the hexagonal lattice) construction in [32] involves a class of twelve tiles that are all isometric to a single tile (twelve being the order of the symmetry group of the hexagon). ...
We construct an example of a group G = Z 2 × G 0 for a finite abelian group G 0 , a subset E of G 0 , and two finite subsets F 1 , F 2 of G , such that it is undecidable in ZFC whether Z 2 × E can be tiled by translations of F 1 , F 2 . In particular, this implies that this tiling problem is aperiodic , in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles F 1 , F 2 , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in Z 2 ). A similar construction also applies for G = Z d for sufficiently large d . If one allows the group G 0 to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F . The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.
... ('ein Stein' is German and means 'one tile'.) About a decade ago, a first solution to this problem was proposed (Socolar & Taylor, 2011). The tile is mainly hexagonal but with non-connected borders to enforce the aperiodicity. ...
... See the figure here below. The motif created are Sierpinski triangles like in previously discovered tilings, as in (Socolar & Taylor, 2011), (Mampusti & Whittaker, 2020) and (Gradit & Van Dongen, 2022). ...
Can the entire plane be paved with a single tile that forces aperiodicity? This is known as the ein Stein problem (in German, ein Stein means one tile). This paper presents an aperiodic monotile for the tiler. It is based on the monotile developed by Taylor and Socolar (whose aperiodicity is forced by means of a non-connected tile that is mainly hexagonal) and motif-based hexagonal tilings that followed this major discovery. The proposed monotile consists of two layers. No motif is needed to make the monotile aperiodic. Additional motifs can be added to the monotile to provide some insights. The proof of aperiodicity is presented with the use of such motifs.
... In 2011 studies on the "einstein problem" (in German "ein Stein" = "one stone") came to a remarkable result, when Socolar and Taylor [11] presented their decorated single prototile forcing nonperiodicity of any generated tiling by two matching rules. Other attempts were made also with one tile and two matching rules [7]. ...
... Rule type 1. Matching rules for adjacent tiles (no colors required) [9] Rule type 2. Colors required for matching adjacent tiles [10] Rule type 3. Non-adjacent but pairwise matching within a neighborhood [11] Rule type 4. Configurations of permitted adjacent tiles from an atlas [2] Rule type 5. Checks with adjacent tiles following a sequence or tree ( [7] or above cases) Rule type 6. A given number of translation classes must be found in the tiling (see below). ...
The so-called "einstein problem" (a pun playing with the famous scientist's name and the German term "ein Stein" for "one stone") asks for a simply connected prototile only allowing nonperiodic tilings without need of any matching rule. So far, researchers come only close to this demand by defining decorated prototiles forcing nonperiodicity of any generated tiling using matching rules. In this paper a class of spiral tilings (and one non-spiral example) is linked to a weaker form of the einstein problem where one or several seed tiles are used. Furthermore, the classical types of matching rules are listed and some new types are discussed. Mathematics Subject Classification (2010). 52Cxx.
... (1) We denote the collection of standard R1-tilings by Ω a because these tilings are easily seen to be mutually local derivable (MLD; see [2]) to the arrowed hex tilings [1]. Alternatively, these tilings are precisely those whose R1-decorations come from the Socolar-Taylor tilings [21]. ...
... Triangles t of T n which appear at the centers of the next level always correspond to R1-triangles, with turns directing counterclockwise about t, so the map from Ω a to S 2 2 is 1-to-1 everywhere except over points having an infinite-level vertex (that is, a point which is a vertex of every T n ). These points correspond to CHT tilings [21], where the map is 3-to-1. ...
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.
... In 2011 studies on the "einstein problem" (in German "ein Stein" = "one stone") came to a remarkable result, when Socolar and Taylor [11] presented their decorated single prototile forcing nonperiodicity of any generated tiling by two matching rules. Other attempts were made also with one tile and two matching rules [7]. ...
... Rule type 1. Matching rules for adjacent tiles (no colors required) [9] Rule type 2. Colors required for matching adjacent tiles [10] Rule type 3. Non-adjacent but pairwise matching within a neighborhood [11] Rule type 4. Configurations of permitted adjacent tiles from an atlas [2] Rule type 5. Checks with adjacent tiles following a sequence or tree ( [7] or above cases) Rule type 6. A given number of translation classes must be found in the tiling (see below). ...
(Accepted for publication 2021 in European Journal of Combinatorics)
Abstract:
The so-called "einstein problem" (a pun playing with the famous scientist's name and the German term "ein Stein" for "one stone") asks for a simply connected prototile only allowing nonperiodic tilings without need of any matching rule. So far, researchers come only close to this demand by defining decorated prototiles forcing nonperiodicity of any generated tiling using matching rules. In this paper a class of spiral tilings is linked to a weaker form of the einstein problem where one or several seed tiles are used. Furthermore, the classical types of matching rules are listed and some new types are discussed.
... The best current solution to the monotile problem without overlapping tiles is the Socolar-Taylor tile [13,14]. In satisfying the above requirements, however, one must choose between whether one prefers simple matching rules or simple geometry; the tile has two forms. ...
... (3) It is not hard to show that tilings in Ω ST R1 exist. In fact, they are precisely those whose R1-decorations come from the Socolar-Taylor tilings [13]. ...
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.
... This approach could involve using truncated tiling patterns to create more intricate and expressive architectural forms. Additionally, incorporating aperiodic patterns, as proposed by previous researchers (Akpanya et al., 2023;Socolar & Taylor, 2011;Wagiri, Shih, Harsono, & Lin, 2023;Walter, Ligler, & Gürsoy, 2024), could unlock innovative design possibilities and expand the creative potential of the system. Moreover, when comparing the results with traditional masonry bricks, there is a notable discrepancy, as shown in Table 5. ...
The reuse of building components is a key challenge in sustainable construction. This study proposes a novel framework for designing reusable topological interlocking systems using hexagon-shaped blocks with osteomorphic interfaces and Ultra-High-Performance Concrete (UHPC). The proposed framework integrates computational design, optimization processes, and experimental fabrication to enhance material efficiency and reusability. Finite Element Analysis (FEA) was employed to optimize block geometry and structural performance, identifying a 30 mm osteomorphic profile as the optimal design. Using AHP-Express, the reusability assessment demonstrated an 84% reusability rate for the proposed system, significantly surpassing the 53% rate of traditional masonry bricks. The findings highlight the potential of this framework to bridge theoretical advancements with practical applications, paving the way for sustainable and adaptable construction practices. This study provides a foundation for future research in modular systems and circular economy principles in architecture.
... Apart from Smith Hat in [14], several other aperiodic monotiles are studied in [12,[15][16][17] under different conditions. The idea of the proofs of aperiodicity is to enforce some large structure in the resulting tilings. ...
We give a simple alternative proof that the monotile introduced by [14] is aperiodic.
... 7.2], that tile the space only in aperiodic way assuming that each two tiles are congruent using a rigid motion symmetry while reflections are not allowed. When the tiles are not required to be polytopes or to be convex, even one tile may enforce aperiodic structure considering all isometries as in the case of the Socolar-Taylor tile [55] in R 2 or the Hat or Spectre tiles by Smith, Myers, Kaplan, and Goodman-Strauss [53,54], or translations only as was recently shown by Greenfeld and Tao [25] in the space of sufficiently large dimension. We refer to [35] and references therein for a more comprehensive survey of the topic. ...
We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely, we show that if a convex five-dimensional polytope P tiles with translations, then P is an affine image of the Dirichlet-Voronoi polytope for a five-dimensional lattice. Our proof is based on an exhaustive combinatorial analysis of possible dual 3-cells and incident dual 4-cells encoding local structures around two-dimensional faces of five-dimensional parallelohedron P and their edges. The analysis is aimed to prove existence of a free direction for P and is paired with new properties established for parallelohedra (in any dimension) that have a free direction that guarantee the Voronoi conjecture for P.
... This concept was further expanded with an alternative approach using colored corners [13] and the development of recursive Wang Tiles for real-time blue noise generation [12]. A more theoretical approach demonstrated aperiodic tiling of the plane using a single prototile [24]. These classical approaches laid the foundation for efficient tiling techniques, addressing challenges in creating diverse, seamless patterns while balancing computational efficiency and visual quality. ...
Image tiling -- the seamless connection of disparate images to create a coherent visual field -- is crucial for applications such as texture creation, video game asset development, and digital art. Traditionally, tiles have been constructed manually, a method that poses significant limitations in scalability and flexibility. Recent research has attempted to automate this process using generative models. However, current approaches primarily focus on tiling textures and manipulating models for single-image generation, without inherently supporting the creation of multiple interconnected tiles across diverse domains. This paper presents Tiled Diffusion, a novel approach that extends the capabilities of diffusion models to accommodate the generation of cohesive tiling patterns across various domains of image synthesis that require tiling. Our method supports a wide range of tiling scenarios, from self-tiling to complex many-to-many connections, enabling seamless integration of multiple images. Tiled Diffusion automates the tiling process, eliminating the need for manual intervention and enhancing creative possibilities in various applications, such as seamlessly tiling of existing images, tiled texture creation, and 360{\deg} synthesis.
... It came as a great surprise in the early 1980's that a family of noncrystalline (aperiodic) states of matter, called "quasicrystals" [2], also have pure point diffraction, but with a twist, namely, they densely fill reciprocal space exhibiting symmetries that would be prohibited for crystals. A less familiar phylum of point patterns obeying (1) are limit-periodic systems [3,4]. These are deterministic point patterns that consist of a union of an infinite set of distinct periodic structures, and hence are also characterized by dense Bragg peaks. ...
The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call {\it effectively limit-periodic}. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor S(k), proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor for primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. We identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes.
... Taylor [7] and Taylor and Socolar [6] introduced a planar aperiodic tiling which can be built from a single hexagonal prototile allowing translations, rotations, and reflections. The tiling is based on the familiar hexagonal tiling of the plane, but if one distinguishes the prototile in its direct and reflected forms, then the matching rules allow only aperiodic tilings to appear. ...
We study the intimate relationship between the Penrose and the Taylor-Socolar tilings, within both the context of double hexagon tiles and the algebraic context of hierarchical inverse sequences of triangular lattices. This unified approach produces both types of tilings together, clarifies their relationship, and offers straightforward proofs of their basic properties.
... Different from Penrose and many other aperiodic tiles [26]- [28], the Hat polykite is derived from a periodic tile, i.e., the Laves [3,4,6,4] tile [29] (in the following sections, we refer to the Laves [3,4,6,4] tile as a Kite). Figure 1 shows a Hat polykite on a grid of Laves [3,4,6,4] tiling. ...
The discovery of the ’Einstein’ monotile represents one of the most significant advancements in geometry in 2023, prompting research across multiple disciplines. This paper proposes a phased array with a limited beam scanning range based on the ’Einstein’ monotile (Hat polykite), characterized by low grating lobe levels and high aperture efficiency. The proposed phased array reduces implementation complexity compared to periodic subarrays and enhances engineering practicality relative to other aperiodic tiles, particularly in load-bearing lattice configurations. Two examples of Hat polykite-based phased arrays are presented in this paper. Example A presents a sparse phased array, where each element is embedded within a Hat polykite, optimized for a maximum grating lobe level of -15 dB. Example B features a subarray comprising eight antenna elements based on the Hat polykite. This configuration achieves 90% aperture efficiency while keeping the maximum grating lobe level below -14 dB within an ± 18 ° main beam scanning range.
... Grünbaum and Shephard explicitly permit reflections in their definition of monohedral tilings [GS16, Section 1.2], so that when they later ask for a tile that "only admits monohedral non-periodic tilings" [GS16,Section 10.7], we can be confident that they considered reflected tiles to be in play. Ammann's aperiodic pairs of tiles were considered congruent under reflections [AGS92], and the disconnected aperiodic Taylor-Socolar tile [ST11,ST12] also requires reflections to tile the plane. Likewise, we regard the hat as an aperiodic monotile. ...
... Coming back to geometric tilings, the tiling by Penrose kept the record for tile number for 50 years, unless one authorizes tiles to be disconnected or to communicate through their vertices, see [ST11]. In 2023, Smith, Myers, Kaplan and Goodman-Strauss [SMKGS24] described new aperiodic tilings with only one tile. ...
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In 2023, two striking, nearly simultaneous, mathematical discoveries have excited their respective communities, one by Greenfeld and Tao, the other (the Hat tile) by Smith, Myers, Kaplan and Goodman-Strauss, which can both be summed up as the following: there exists a single tile that tiles, but not periodically (sometimes dubbed the einstein problem). The two settings and the tools are quite different (as emphasized by their almost disjoint bibliographies): one in euclidean geometry, the other in group theory. Both are highly nontrivial: in the first case, one allows complex shapes; in the second one, also the space to tile may be complex.
We propose here a framework that embeds both of these problems. We illustrate our setting by transforming the Hat tile into a new aperiodic group monotile, and we describe its symmetries.
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... Motivated by these experimental systems exhibiting aperiodic long-range order with trigonal and hexagonal symmetry, we explore a two-parameter family of quasiperiodic trigonal and hexagonal tilings. We believe that the addition of such a generic family of tilings to the recent collection of ad-hoc aperiodic hexagonal tilings [20][21][22] will facilitate more quantitative analysis of the 3-fold surfaces of icosahedral quasicrystals, and of the other systems mentioned above. We also note that two of us have already used one of the hexagonal members of this family of tilings to study the Hubbard model on a nonminimal-rank quasicrystal, and discovered ferrimagnetically ordered states [23]. ...
Exploring nonminimal‐rank quasicrystals, which have symmetries that can be found in both periodic and aperiodic crystals, often provides new insight into the physical nature of aperiodic long‐range order in models that are easier to treat. Motivated by the prevalence of experimental systems exhibiting aperiodic long‐range order with hexagonal and trigonal symmetry, we introduce a generic two‐parameter family of 2‐dimensional quasiperiodic tilings with such symmetries. We focus on the special case of trigonal and hexagonal Fibonacci, or golden‐mean, tilings, analogous to the well studied square Fibonacci tiling. We first generate the tilings using a generalized version of de Bruijn's dual grid method. We then discuss their interpretation in terms of projections of a hypercubic lattice from six dimensional superspace. We conclude by concentrating on two of the hexagonal members of the family, and examining a few of their properties more closely, while providing a set of substitution rules for their generation.
... They sometimes act as a symbolic proxy for neighbour relationships that could easily be encoded geometrically, but they can also determine more complex relationships between tiles. The Taylor-Socolar tile [ST11] is a regular hexagon with matching rules in the form of markings in the interiors of tiles. The matching rules force aperiodicity, but they require non-adjacent tiles to exchange information. ...
... A good example of a similar self-assembly for tilings can be seen in the recently announced breakthrough solution to the "Ein Stein problem" 2 [3,25,26] where one polygonal tile can selfassemble itself into a tiling of R 2 but only in an aperiodic way, this property is also refered to being an aperiodic monotile. The Socolar-Taylor tile [27] was the first example of a single tile that enforces aperiodicity of the associated tiling in R 2 , but this tile is not simply connected. We refer to [4] for more background and history of the study of aperiodic order. ...
Delone sets are discrete point sets X in R d characterized by parameters ( r , R ), where (usually) 2 r is the smallest inter-point distance of X , and R is the radius of a largest “empty ball” that can be inserted into the interstices of X . The regularity radius ρ ^ d is defined as the smallest positive number ρ such that each Delone set with congruent clusters of radius ρ is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our “Weak Conjecture” states that ρ ^ d = O ( d 2 log 2 d ) R as d → ∞ , independent of r . This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius 2 r and those with full-dimensional sets of d -reachable points. We also offer support for the plausibility of a “Strong Conjecture”, stating that ρ ^ d = O ( d log 2 d ) R as d → ∞ , independent of r .
... Focus has been applied to such tilings which exhibit rotational symmetries which are incommensurate with periodicity or translational symmetry: 5-, or greater than 6-fold. However, this is not a required condition -aperiodic tilings can, of course, be 2-, 3-, 4-, or 6-fold [1][2][3][4][5][6][7][8][9][10][11]. ...
Symmetry sharing facilitates coherent interfaces which can transition from periodic to aperiodic structures. Motivated by the design and construction of such systems, we present hexagonal aperiodic tilings with a single edge-length which can be considered as decorations of a periodic lattice.
We introduce these tilings by modifying an existing family of golden-mean trigonal and hexagonal tilings, and discuss their properties in terms of this wider family. Then, we show how the vertices of these new systems can be considered as decorations or sublattice sets of a periodic triangular lattice, before introducing methods to designing coherent aperiodic to periodic interfaces.
... On the other hand, to implement a phased array, the imperative for a genuinely complete plane is non-negotiable. There are also some aperiodic tiles consisting of a limited number of units, while tiling a complete plane from these units often requires adherence to additional specific conditions [38]. For example, a Taylor-Socolar tile comprises only one unit, yet the edges of this unit are uniquely symbolized. ...
... Though his set contained more than 20,000 different tiles, this was subsequently reduced all the way to 2 by Penrose [10]. In three dimensions, the problem was solved by Schmitt [12] for so-called weak aperiodicity, and then by Socolar and Taylor [16] for a more stringent version of aperiodicity (though still not so-called strong aperiodicity). The last-mentioned article also solves the problem when tiles are allowed to be disconnected shapes, as well as when edges of tiles are colored, which imposes some restrictions on how two tiles can be joined. ...
... Our preliminary experiments show that it seems to be the case and therefore this Golden Hex substitutive structure may be automatically enforced for all tilings by Tile ( √ 3 : 1). Apart from Smith Hat in [11], several other aperiodic monotiles are studied in [13,9,12] under different conditions. The idea of the proofs of aperiodicty is to enforce some large structure in the resulting tilings. ...
We give an alternative simple proof that the monotile introduced by Smith, Myers, Kaplan and Goodman-Strass is aperiodic.
... Grünbaum and Shephard explicitly permit reflections in their definition of monohedral tilings [GS16, Section 1.2], so that when they later ask for a tile that "only admits monohedral non-periodic tilings" [GS16,Section 10.7], we can be confident that they considered reflected tiles to be in play. Ammann's aperiodic pairs of tiles were considered congruent under reflections [AGS92], and the disconnected aperiodic Taylor-Socolar tile [ST11,ST12] also requires reflections to tile the plane. Likewise, we regard the hat as an aperiodic monotile. ...
The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.
... This result extended earlier results of Taylor [21] and of Socolar and Taylor [18]; see also [3] for further details. Here, a functional monotile is derived from an inflation rule with decorated hexagons that defines a unique local isometry (LI) class of tilings with perfect local rules. ...
The recently discovered Hat tiling admits a 4-dimensional family of shape deformations, including the 1-parameter family already known to yield alternate monotiles. The continuous hulls resulting from these tilings are all topologically conjugate dynamical systems, and hence have the same dynamics and topology. We construct and analyze a self-similar element of this family, and use it to derive properties of the entire family. This self-similar tiling has pure-point dynamical spectrum, which we compute explicitly, and comes from a natural cut-and-project scheme with 2-dimensional internal space. All other members of the Hat family, in particular the original version constructed from 30-60-90 right triangles, are obtained via small modifications of the projection from this cut-and-project scheme.
... The Taylor-Socolar tile [ST11] comes within a hair's breadth of being aperiodic. Their hexagonal prototile tiles aperiodically, but only when additional matching conditions are enforced, preventing certain adjacencies that would otherwise be permitted. ...
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.
... Consider the Taylor-Socolar tile [10,11]. It is an example of an einstein, with a slight caveat of being non-connected. ...
In this paper we prove a new result: there exist non-periodic colourings of monohedral periodic tilings, forced by finite patches.
To appear in "The Mathematical Intelligencer".
... 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.
... 5 for example can be seen as a local rule. Local rules can also be seen as a way to impose conditions on the validity of a tiling without using tiles of complex shapes for example the cut-and-notch Penrose rhombus tiles that where used to define the Penrose rhombus tiling [Pen74] and their coloured rhombus equivalent shown in Figure 2.29 page 48, or even to simplify tiles that are not connex such as in the case of the Socolar-Taylor tile that can either be realized as a non-connex tile or a labelled hexagon tile [ST11] There are three ways to define local rules either by forbidden pattern, by an atlas of allowed local configurations or by a colouring or labelling of the tiles. We will first quickly present the forbidden patterns definition and the atlas definition without going in much details, and then give a more detailed presentation of colouring of tiles. ...
A tiling is a covering of the plane by tiles which do not overlap. We are mostly interested in edge-to-edge rhombus tilings, this means that the tiles are unit rhombuses and any two tiles either do not intersect at all, intersect on a single common vertex or along a full common edge. Substitutions are applications that to each tile associate a patch of tiles (which usually has the same shape as the original tile but bigger), a substi- tution can be extended to tilings by applying it to each tile and gluing the obtained patches together. Substitutions are a way to grow and define tilings with a strong hierarchical structure. Discrete planes are edge-to-edge rhombus tilings with finitely many edge directions that can be lifted in Rn and which approximate a plane in Rn, such a tiling is also called planar. Note that discrete planes are a relaxed version of cut-and-project tilings. In this thesis we mostly study edge-to-edge substitution rhombus tilings lifted in Rn. We prove that the Sub Rosa tilings are not discrete planes, the Sub Rosa tilings are edge-to-edge substitution rhombus tilings with n-fold rotational symmetry that were defined by Jarkko Kari and Markus Rissa- nen [KR16] and which were good candidates for being discrete planes. We define a new family of tilings which we call the Planar Rosa tilings which are subsitution discrete planes with n-fold rotational symmetry.We also study the multigrid method which is a construction for cut- and-project tiling and we give an explicit construction for cut-and-project rhombus tilings with global n-fold rotational symmetry.
... [4][5][6][7][8]), and the initial intrigue of the exhibition of perfect long-range order in structures with 'forbidden' rotational symmetry, or, orders of symmetry not associated with periodicity. However, neither quasicrystals nor aperiodic tilings are restricted to displaying non-periodic rotational symmetries [9], and indeed, there are examples of aperiodic tilings which share symmetries with periodic systems [10][11][12][13][14]. Further definitions of similar tilings represents an exciting arena of research: both the exploration of the physical properties of these stand-alone tilings, and the possibility of investigating interfacial periodic-aperiodic systems which share rotational symmetries. ...
We present a multi-edge-length aperiodic tiling which exhibits 6--fold rotational symmetry. The edge lengths of the tiling are proportional to 1:, where is the golden mean . We show how the tiling can be generated using simple substitution rules for its three constituent tiles, which we then use to demonstrate the bipartite nature of the tiling vertices. As such, we show that there is a relatively large sublattice imbalance of . Similarly, we define allowed vertex configurations before analysing the tiling structure in 4-dimensional hyperspace.
... (Note in the brackets that it is non-trivial question if there exists a tiling set that is not regularly tiling. An example of such set was constructed [16] in 2011 only.) ...
In 1954, G. Polya conjectured that the counting function of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set is lesser (resp. greater) than . Here is the spectral parameter, and is the volume of the unit ball. We prove this conjecture for both Dirichlet and Neumann boundary problems for any circular sector, and for the Dirichlet problem for a ball of arbitrary dimension. We heavily use the ideas from \cite{LPS}.
... Aperiodic tilings have proved to be an essential tool in understanding, modelling, and representing quasicrystalline structures. However, there are limited examples which explore aperiodic tilings exhibiting rotational symmetries commensurate with periodic real-space structures i.e. 2-, 3-, 4-, and 6-fold [1][2][3][4]. This is despite the fact quasicrystals are not restricted to exhibiting 'forbidden' symmetries only [5], and the obvious benefits of studying the transition between periodic and aperiodic structures (facilitated by a sharing of rotational symmetries). ...
Exploring nonminimal-rank quasicrystals, which have symmetries that can be found in both periodic and aperiodic crystals, often provides new insight into the physical nature of aperiodic long-range order in models that are easier to treat. Motivated by the prevalence of experimental systems exhibiting aperiodic long-range order with hexagonal and trigonal symmetry, we introduce a generic two-parameter family of 2-dimensional quasiperiodic tilings with such symmetries. We focus on the special case of trigonal and hexagonal Fibonacci, or golden-mean, tilings, analogous to the well studied square Fibonacci tiling. We first generate the tilings using a generalized version of de Bruijn's dual grid method. We then discuss their interpretation in terms of projections of a hypercubic lattice from six dimensional superspace. We conclude by concentrating on two of the hexagonal members of the family, and examining a few of their properties more closely, while providing a set of substitution rules for their generation.
... 5 for example can be seen as a local rule. Local rules can also be seen as a way to impose conditions on the validity of a tiling without using tiles of complex shapes for example the cut-and-notch Penrose rhombus tiles that where used to define the Penrose rhombus tiling [Pen74] and their coloured rhombus equivalent shown in Figure 2.29 page 48, or even to simplify tiles that are not connex such as in the case of the Socolar-Taylor tile that can either be realized as a non-connex tile or a labelled hexagon tile [ST11] There are three ways to define local rules either by forbidden pattern, by an atlas of allowed local configurations or by a colouring or labelling of the tiles. We will first quickly present the forbidden patterns definition and the atlas definition without going in much details, and then give a more detailed presentation of colouring of tiles. ...
A tiling is a covering of the plane by tiles which do not overlap. We are mostly interested in edge-to-edge rhombus tilings, this means that the tilesare unit rhombuses and any two tiles either do not intersect at all, intersect on a single common vertex or along a full common edge. Substitutions are applications that to each tile associate a patch of tiles (which usually has the same shape as the original tile but bigger), a substitution can be extended to tilings by applying it to each tile and gluing the obtained patches together. Substitutions are a way to grow and define tilings with a strong hierarchical structure. Discrete planes are edge-to-edge rhombus tilings with finitely many edge directions that can be lifted in RR^n and which approximate a plane in RR^n, such a tiling is also called planar. Note that discrete planes are a relaxed version of cut-and-project tilings.In this thesis we mostly study edge-to-edge substitution rhombus tilings lifted in RR^n. We prove that the Sub Rosa tilings are not discrete planes, the Sub Rosa tilings are edge-to-edge substitution rhombus tilings with n-fold rotational symmetry that were defined by Jarkko Kari and Markus Rissanen [KR16] and which were good candidates for being discrete planes. We define a new family of tilings which we call the Planar Rosa tilings which are subsitution discrete planes with n-fold rotational symmetry. We also study the multigrid method which is a construction for cutand-project tiling and we give an explicit construction for cut-and-project rhombus tilings with global n-fold rotational symmetry
... 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.
... It concerns aperiodic tilings in the plane. See [38] for more information. • Hammer X-ray problem (3). ...
This article contains a short and entertaining list of unsolved problems in Plane Geometry. Their statement may seem naive and can be understood at an elementary level. But their solutions have refused to appear for forty years in the best case.
... This systems continues to attract attention, see for instance [44,21,25], but the question whether it opens up a completely new approach to aperiodic structures remains to be explored. Closely related is the search for planar monotiles of hexagonal shape [40,45]. • Constant-size substitutions and characteristic-p S-unit equations are also connected to the question of mixing in algebraic dynamics, as described recently in work by Derkson and Masser [17,18]. ...
Aperiodic order is a relatively young area of mathematics with connections to many other fields, including discrete geometry, harmonic analysis, dynamical systems, algebra, combinatorics and, above all, number theory. In fact, numbertheoretic methods and results are present in practically all of these connections. It was one aim of this workshop to review, strengthen and foster these connections.
... Finally in 2011, Socolar and Taylor solved this problem and constructed an example of a tiling set such that any its tiling of the plane is not periodic [17]. Thus, a tiling set that is not regularly tiling does exist. ...
In 1954, G. P\'olya conjectured that the counting function of the eigenvalues of the Laplace operator of Dirichlet (resp. Neumann) boundary value problem in a bounded set is lesser (resp. greater) than . Here is the spectral parameter, and is the constant in the Weyl asymptotics. In 1961, P\'olya proved this conjecture for tiling sets in the Dirichlet case, and for tiling sets under some additional restrictions for the Neumann case. We prove the P\'olya conjecture in the Neumann case for all tiling sets.
... In a crowning achievement of tiling theory, the existence of an aperiodic monotile was resolved almost a decade ago by Joshua Socolar and Joan Taylor [12,15]. Several candidates had been put forth prior to their monotile, but the experts immediately recognised the importance of Socolar and Taylor's discovery [1,2,3,7,8]. ...
We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar--Taylor monotile, but can be realised by shape alone. The second is a local growth rule; a direct isometry of our monotile can be added to any patch of tiles provided that a tree on the monotile connects continuously with a tree on one of its neighbouring tiles. This condition forces tilings to grow along dendrites, which ultimately results in nonperiodic tilings. Our local growth rule initiates a new method to produce tilings of the plane.
This study investigates the development of the Hat monotile, a singular tile designed by David Smith and his team, capable of covering surfaces with unique, non-repeating, aperiodic patterns. We enhance this tiling concept by integrating diverse motifs into the Hat monotile, improving its attractiveness and adaptability. Utilizing computational tools, we experiment with these tiles to assess their potential in architecture, specifically focusing on their ability to innovate facade designs through simplified fabrication processes, cost reduction, and greater design flexibility. Our research demonstrates the practical applications of these distinctive patterns, bridging the gap between mathematical theories and architectural implementation. The objective is to encourage further exploration and innovation in applying aperiodic patterns, underlining their potential to discover the beauty in asymmetrical forms.
The theory of aperiodic order expanded and developed significantly since the discovery of quasicrystals, and continues to bring many mathematical disciplines together. The focus of this workshop was on harmonic analysis and spectral theory, dynamical systems and group actions, Schrödinger operators, and their roles in aperiodic order – with links into a full range of problems from number theory to operator theory.
In the last 30 years, the mathematical theory of aperiodic order has developed enormously. Many new tilings and properties have been discovered, few of which are covered or anticipated by the early papers and books. Here, we start from the well‐known Fibonacci chain to explain some of them, with pointers to various generalisations as well as to higher‐dimensional phenomena and results. This should give some entry points to the modern literature on the subject.
Strong correlations in quasiperiodic systems have attracted much interest since the recent observation of quantum critical behavior in the Tsai-type quasicrystal compound Au51Al34Yb15. Possible long-range electronic orders in such systems have also been widely debated. This chapter reviews our theoretical investigations on correlation effects in the quasiperiodic system, such as the Mott transition, valence transition, and superconductivity. In the metallic state close to the Mott transition point, the quasiparticle weight strongly depends on the site and its geometry while the Mott transition occurs simultaneously and is not suppressed by the quasiperiodicity. On the other hand, the valence transition is suppressed by the quasiperiodicity, where the Kondo and valence-fluctuating states appear depending on the site. For superconductivity, we found unconventional weak-coupling superconductivity formed by the Cooper pairs deviating from those of Bardeen-Cooper-Schrieffer superconductivity in periodic systems. This deviation can be seen in the real-space distribution of the superconducting order parameter, jump of specific heat, and current-voltage characteristic curve. These results indicate that superconductivity in quasiperiodic systems is qualitatively different from that in periodic and random systems. In particular, the results are consistent with the superconductivity recently discovered in an Al-Mg-Zn quasicrystal and provide a clue to understanding its mechanism and property.
The problem of channel planning is typical during the design of the wireless access networks. It can have a geometric interpretation from the point of view of the plane tessellation problem. Since the wireless coverage of the target area by access points coverage zones is very similar to the plane tessellation and is studied in detail in crystallography, we may use appropriate methods of description and research. It makes possible to assess the mutual influence between access points in a distributed network and, thus, to make a conclusion about the applicability of the selected channel plan. Since the number of channels in each specific design task may differ depending on the form and structure of lattice unit cells, the structure of the plane tessellation will be different.
In this paper, channel planning in IEEE 802.11 networks is considered from the plane tessellation problem point of view, taking into account the specifics of spectrum use in these networks. Also we will consider typical structures of a plane tessellation and lattices that correspond to the translational symmetry of such structures, and a method is proposed that considers negative influence between access points of the entire plane.
KeywordsWireless access networkIEEE 802.11Channel planningPlane tessellationLatticeUnit cell
This is an extended abstract of the paper ‘Scaling properties of the Thue–Morse measure’ by Baake, Gohlke, Kesseb¨ohmer and Schindler [1].
This book provides an interdisciplinary guide to quasicrystals, the 2011 Nobel Prize in Chemistry winning topic, by presenting an up-to-date and detailed introduction to the many fundamental aspects and applications of quasicrystals science. It reviews the most characteristic features of the peculiar geometric order underlying their structure and their reported intrinsic physical properties, along with their potential for specific applications.
The role of quasiperiodic order in science and technology is also examined by focusing on the new design capabilities provided by this novel ordering of matter. This book is specifically devoted to promoting the very notion of quasiperiodic order, and to spur its physical implications and technological capabilities. It explores the fundamental aspects of intermetallic, photonic and phononic quasicrystals, as well as soft-matter quasicrystals, including their intrinsic physical and structural properties. In addition, it thoroughly discusses experimental data and related theoretical approaches to explain them, extending the standard treatment given in most current solid state physics literature. It also outlines exciting applications in new technological devices of quasiperiodically ordered systems, including multilayered quasiperiodic systems, along with 2D and 3D designs, whilst outlining new frontiers in quasicrystals research.
This is a book about polyominoes, especially about their tilings. It covers in some detail tilings by dominoes, tilings of and by rectangles, plane tilings, and various other topics such as the tiling hierarchy, plane tilings, and colorings.
This work considers the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long-wavelength fluctuations in a broad class of one-dimensional substitution tilings. A simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit-periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between −1 and 3. Limit-periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law.
. The discrete part of the diffraction pattern of self-similar tilings, called the Bragg spectrum, is determined. Necessary and sufficient conditions for a wave vector q to be in the Bragg spectrum are derived. It is found that the Bragg spectrum can be non-trivial only if the scaling factor # of the tiling is a PV-number. In this case, the Bragg spectrum is entirely determined by the scaling factor # and the translation module of the tiling. Three types of Bragg spectra can be distinguished, belonging to quasiperiodic, limitperiodic and limit-quasiperiodic structures, respectively. 1. Introduction The selfsimilarity properties of quasicrystals have triggered considerable interest in selfsimilar tilings, which are used as simple models to describe the structure of quasicrystals. Most of the tilings used for this purpose can be obtained as a cut through a higher-dimensional periodic structure, and are thus quasiperiodic by construction. This implies, in particular, that their Fourie...
Tilings and patterns have been made and enjoyed for thousands of years. Their mathematical treatment was begun by n J. Kepleren but was then forgotten until the nineteenth-century development of crystallography. In this unique book, with its abundant illustrations, the authors explain exactly what one means by "tiling" and "pattern", restricting the treatment to two dimensions. There are many surprises; for instance, Figure 1.2.2 shows the 24 "heptiamonds", with the remark that only one of them cannot be repeated by congruent copies to fill and cover the whole plane. Chapter 2, on "Tilings by regular polygons", includes n A. J. W. Duijvestijn'sen "squared square", in which 21 different squares, with sides , are fitted together to fill a square of side 112. Any solution to the slightly simpler problem of "squaring a rectangle" can be extended to a tiling of the whole plane by infinitely many squares of different sizes. Chapter 3, on "Well-behaved tilings", tells us precisely when a tiling can be called "normal". One counterexample is the remarkable Figure 1.0.1 (repeated as a cover design) which is monohedral (all tiles congruent) but is abnormal in that some pairs of tiles share a disconnected set of boundary points. Euler's theorem is used to prove that, if every tile of a normal tiling has k vertices, where the valences are , then . Figure 3.8.6 illustrates a nice paradox: it shows a particular pentagon which is entirely surrounded by seven congruent replicas although the arrangement cannot be extended to a monohedral tiling of the whole plane. Chapter 4 describes the transition from metrical to topological tilings. Chapter 5 introduces the subject of patterns, beginning with a fascinating example (Figure 5.0.1) based on a maze. Except in some of the exercises, a discrete pattern means a planar family of mutually disjoint congruent copies of a motif with the property that for each pair of copies, say and , there is an isometry of the plane that maps the whole pattern onto itself and onto . According to this strict definition, the abnormal Figure 1.0.1 is not a pattern: every two of its infinitely many tiles are related by an isometry that maps one onto the other, but in no case is this isometry a symmetry of the whole pattern! The authors have undertaken the almost incredibly difficult task of classifying patterns so that one can say in what sense any two given patterns are of different types. Table 5.2.1 lists the 3 types of finite patterns, each type-symbol involving an integer n which is the smallest period of a rotatory symmetry; Table 5.2.2 lists the 15 types of frieze patterns; Table 5.2.3 lists the 52 types of discrete periodic patterns. The number of types becomes smaller when the arbitrary motif is replaced by a dot or other symmetrical shape. Further complications arise when the motif is allowed to be infinite, or when copies of the motif overlap. Chapter 6 combines the two topics (tilings and patterns) by making even subtler distinctions: it can happen that several tilings are "really different" even though they have the same topological type and the same pattern type. There is a historical account of the classification of tilings. Attempts by some highly respected crystallographers, such as n A. V. Shubnikoven and n V. A. Koptsiken ref[ Symmetry in science and art, English translation, Plenum, New York, 1974], "led to an almost unbelievable number of errors". A different method of classification is developed in Chapter 7. Chapter 8 describes the complications that arise when tiles are distinguished by being variously colored. Chapter 9 deals with tilings by polygons, not necessarily regular; for instance, there are 24 types of tilings by congruent pentagons. The most exciting developments are reserved for Chapters 10 and 11, on "Aperiodic tilings". Before 1966, nobody could imagine the existence of a set of prototiles which would admit infinitely many tilings of the plane although no such tiling is periodic. Obviously, even such a simple prototile as a domino admits a nonperiodic tiling; but the exciting new idea, embodied in the term "aperiodic", is a set of n prototiles which cannot possibly be arranged in a periodic fashion. n R. Bergeren ref[Mem. Amer. Math. Soc. No. 66 (1966); MR0216954 (36 #49)] discovered the first aperiodic tiling, with n=20426. Berger himself soon reduced this fantastic number to 104, n D. E. Knuthen to 92, n H. Läuchlien to 40, n R. M. Robinsonen to 35, n R. Penroseen to 34, n Robinsonen (again) to 32, and later to 24, n R. Ammannen to 16, and later to 6, then Penrose (again) to 5, and ultimately to 2! Many of the amazing ramifications of this theory, including some by n J. H. Conwayen, are published here for the first time. Chapter 11 deals with "Wang tiles": square tiles having colored edges which must match with their neighbors, only translations being allowed. These aperiodic tilings are relevant to questions of logic and computing, because it is possible to find sets of 16 Wang tiles which mimic the behavior of any Turing machine. Finally, Chapter 12 relaxes the restriction that the tiles should be topological disks, or that the tiling should cover the plane only once. The book ends appropriately with a 42-page bibliography and a 6-page index. Reviewed by H. S. M. Coxeter
The sphinx is a non-periodic tiling of the plane made of one type of tile. In order to characterise the type of order found in this tiling, the computation of its diffraction spectrum is considered. The spectrum contains a discrete component, with Bragg peaks located at the vertices of triangular lattices with lattice spacings, in reciprocal space, equal to any power of 1/2.
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.
I have exhibited several types of monotiles with matching rules that force the construction of a hexagonal parquet. The isohedral number of the resulting tiling can be made as large as desired by increasing the aspect ratio of the monotile. Aside from illustrating some elegant peculiarities of the hexagonal parquet tiling, the constructions demonstrate three points.1.
Monotiles with arbitrarily large isohedral number do exist;
2.
The additional topological possibilities afforded in 3D allow construction of a simply connected monotile with a rule enforced by shape only, which is impossible for the hexagonal parquet in 2D;
3.
The precise statement of the tiling problem matters— whether color matching rules are allowed; whether multiply connected shapes are allowed; whether spacefilling is required as opposed to just maximum density. So what about the quest for thek = ∞ monotile? Schmitt.
Single quasicrystals Al72Ni12Co16 and Al72Pd19Mn9 alloys have been grown by the floating-zone method and Zn46Mg51Ho3 alloy by the Bridgman method. In these three systems, single quasicrystals with dimensions of the order of a centimeter have been successfully grown. X-ray diffraction and neutron diffraction show that these single quasicrystals are as good as the best crystals. We therefore assert that quasicrystals are a metallic bulk matter with a degree of order equal to that of best crystals. By quenching the liquid, we observed flat solid–liquid interface perpendicular to the growth directions in Al–Ni–Co and Al–Pd–Mn alloys.
In this paper we study the three-dimensional curves generated by the iterated bending of a piece of wire, which are generalizations of the so-called “dragon curves” or “paper-folding sequences” previously studied by Davis and Knuth, Mendès France, and other writers. These “wire-bending sequences” have several surprising properties. We characterize the nth term of a wire-bending sequence in terms of the binary expansion of n. We prove that the curves traced out in R3 by many wire-bending sequences are actually bounded, although they are all aperiodic. Finally, we illustrate the close connection between wire-bending and the continued fractions for the transcendental numbers Σn ⩾ 0εng−2n, where εn = ± 1 and g ⩾ 3 is an integer.
Single quasicrystals Al72Ni12Co16 and Al72Pd19Mn9 alloys have been grown by the floating-zone method and Zn46Mg51Ho3 alloy by the Bridgman method. In these three systems, single quasicrystals with dimensions of the order of a centimeter have been successfully grown. X-ray diffraction and neutron diffraction show that these single quasicrystals are as good as the best crystals. We therefore assert that quasicrystals are a metallic bulk matter with a degree of order equal to that of best crystals. By quenching the liquid, we observed flat solid–liquid interface perpendicular to the growth directions in Al–Ni–Co and Al–Pd–Mn alloys.
Introduction The discovery of alloys with long-range orientational order and sharp diffraction images of non-crystallographic symmetry [65, 35] has initiated an intensive investigation of the possible structures and physical properties of such systems. Although there were various precursors, both theoretically and experimentally [72], it was this renewed and amplified interest that established a new branch of solid state physics, and also of discrete geometry. It is usually called the theory of quasicrystals, even though it also covers ordered structures more general than those with pure Bragg diffraction spectrum. It is now rather common to think of the regime between crystallographic and amorphous systems as an interesting area with a hierarchy of ordered states. This was not so some fifteen years ago, and it is the purpose of this contribution, and of the book as a whole, to introduce some of the ideas and methods that are needed to handle this new zoo. In particular, I wil
This paper addresses the question of whether a single tile with nearest neighbor matching rules can force a tiling in which the tiles fall into a large number of isohedral classes. A single tile is exhibited that can fill the Euclidean plane only with a tiling that contains k distinct isohedral sets of tiles, where k can be made arbitrarily large. It is shown that the construction cannot work for a simply connected 2D tile with matching rules for adjacent tiles enforced by shape alone. It is also shown that any of the following modifications allows the construction to work: (1) coloring the edges of the tiling and imposing rules on which colors can touch; (2) allowing the tile to be multiply connected; (3) requiring maximum density rather than space-filling; (4) allowing the tile to have a thickness in the third dimension.
In The mathematics of long-range aperiodic order The diffraction pattern of self–similar
- F Ahler
- R Klitzing
F. G¨ ahler and R. Klitzing. tilings, pages 141–174. In The mathematics of long-range aperiodic order (R. V. Moody, Ed., NATO ASI Series C, 489, Kluwer Academic Publishers, Dordrecht), 1997. The diffraction pattern of self–similar
Forcing nonperiodicity with a single prototile submitted for publication; Preprint available at http://arxiv.org/abs/1009
- J E S Socolar
- J M Taylor
- J E S Socolar
- J M Taylor
PARI formula for A091072, Online Encyclopedia of Integer Sequences
- M Somos
M. Somos, PARI formula for A091072, Online Encyclopedia of Integer Sequences, 2005.
Forcing nonperiodicity with a single prototile submitted for publication
- J E S Socolar
- J M Taylor
J.E.S. Socolar, J.M. Taylor, Forcing nonperiodicity with a single prototile, Math. Intelligencer (September 2010), submitted for publication; Preprint available at http://arxiv.org/abs/1009.1419.
The diffraction pattern of self-similar tilings The Mathematics of Long-range Aperiodic Order
- F Gähler
- R Klitzing
F. Gähler, R. Klitzing, The diffraction pattern of self-similar tilings, in: R.V. Moody (Ed.), The Mathematics of Long-range Aperiodic Order, in: NATO ASI Ser. C, vol. 489, Kluwer Academic Publishers, Dordrecht, 1997, pp. 141–174.
Remarks on tiling: Details of a (1 + + 2 )-aperiodic set The Mathematics of Long-range Aperiodic Order
- R Penrose
R. Penrose, Remarks on tiling: Details of a (1 + + 2 )-aperiodic set, in: R.V. Moody (Ed.), The Mathematics of Long-range Aperiodic Order, in: NATO ASI Ser. C, vol. 489, Kluwer Academic Publishers, Dordrecht, 1997, pp. 467–497.
Wire bending, J. combin. Theory ser
- M France
- Mendès
- J O Shallit