Summary Chiral Aperiodic Monotiles and Spectres arxiv.org
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The article discusses chiral aperiodic monotiles and introduces a family of shapes called Spectres that can only be tiled in a non-periodic manner using a hierarchical substitution system, and explores the creation of these shapes using substitution rules on two base tiles.
Key Points
- Chiral aperiodic monotiles and spectres are tiles that can only be tiled in a non-periodic manner.
- Spectres are strictly chiral aperiodic monotiles that can be tiled within an infinite hierarchy of larger and larger supertiles, making it non-periodic.
- The study establishes a bijection between tilings by Tile(1, 1) and combinations of hats and turtles.
- Substitution rules can be used to create patches of spectres of any size, with practical drawing algorithms.
- The paper answers the Einstein problem in a world where sameness is restricted to orientation-preserving isometries.
- The lack of translation as a symmetry implies that the tiling by supertiles necessarily has the same symmetries as the original tiling by marked hexagons.
Summaries
160 word summary
The document discusses chiral aperiodic monotiles and spectres, which are tiles that can only be tiled in a non-periodic manner using translations and rotations. The authors introduce a family of shapes called Spectres that are strictly chiral aperiodic monotiles and admit only chiral non-periodic tilings based on a hierarchical substitution system. The Spectre is a chiral aperiodic monotile that can be tiled within an infinite hierarchy of larger and larger supertiles, making it non-periodic. The paper also provides a substitution system that can produce patches of Spectres of any size. The article explores the creation of chiral aperiodic monotiles using substitution rules on two base tiles: a single Spectre and a two-Spectre compound called a Mystic. The study establishes a bijection between tilings by Tile(1, 1) and combinations of hats and turtles. Finally, the document shows how changing the lengths of edges in a tiling by hats and turtles can result in a tiling by Tile(1,1) or a Laves tiling.
517 word summary
The document discusses chiral aperiodic monotiles and spectres, which are tiles that can only be tiled in a non-periodic manner. The hat is an example of a chiral aperiodic monotile, and the recently discovered "hat" aperiodic monotile mixes unreflected and reflected shapes, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. The authors introduce a family of shapes called Spectres that are strictly chiral aperiodic monotiles and admit only chiral non-periodic tilings based on a hierarchical substitution system. The Spectre is a chiral aperiodic monotile that can be tiled within an infinite hierarchy of larger and larger supertiles, making it non-periodic. The article explores the creation of chiral aperiodic monotiles using substitution rules on two base tiles: a single Spectre and a two-Spectre compound called a Mystic. The study establishes a bijection between tilings by Tile(1, 1) and combinations of hats and turtles. The document shows that any tiling by spectres is equivalent to a chiral tiling by hats and turtles, and that the Spectre tiling is periodic if and only if the hat-turtle tiling is. The article also provides a substitution system that can produce patches of Spectres of any size. Finally, the document shows how changing the lengths of edges in a tiling by hats and turtles can result in a tiling by Tile(1,1) or a Laves tiling. The document discusses chiral aperiodic monotiles and spectres. The authors construct a reduced list of 1-patches from a hat and T7H, and group seven hats to form a cluster. They discover that every 1-patch with T6H at its center must have hats and copies of T6H. Spectre is equivalent to one tiling where turtles are distributed sparsely among hats or vice versa. The authors also use the extended notion of reduced lists of k-patches for cent hexagons. The marked hexagons tile arbitrarily large finite regions of the plane, and the lack of translation as a symmetry implies that the tiling by supertiles necessarily has the same symmetries as the original tiling by marked hexagons. The supertiles are combinatorially equivalent to reflected versions of the hexagons, and the tiles in any tiling by the nine marked hexagons can be composed into the supertiles shown in Figure 5.1. Chiral Aperiodic Monotiles and Spectres is a document that discusses the construction of supertiles using 5-patches. The document discusses chiral aperiodic monotiles and spectres, which can be used to realize every combinatorial edge and vertex arrangement in a patch. The Spectre is a shape that tiles aperiodically if only translations and rotations are permitted. The paper answers the Einstein problem in a world where sameness is restricted to orientation-preserving isometries. The clusters T7H and T8H form the basis for substitution rules, and the methods used in the paper to prove aperiodicity rely heavily on constructing spectre tilings and the properties of the spectre in detail. The paper also discusses how substitution rules can be used to create patches of spectres of any size, with practical drawing algorithms. There are references to various papers and books on the topic of tiling and aperiodic monotiles.
2121 word summary
This excerpt is a list of references to various papers and books on the topic of tiling and aperiodic monotiles. One paper mentioned is about forcing nonperiodicity with a single tile, while another presents an aperiodic hexagonal tile. There is also a reference to an article on polyomino, polyhex, and polyiamond tiling. The text mentions an eigenvalue computation that shows areas increase by a factor of 4 + 15 with each stage of the substitution, and the ratio of the number of "even" spectres to the number of "odd" spectres is also 4 + 15. The text notes that aperiodic monotiles are similar to metatiles in that they quickly converge to limit shapes. The paper discusses chiral aperiodic monotiles and spectres, and how they can be used to construct patches of spectre clusters. The substitution rules used to replace spectres and mystics with clusters of reflected tiles can be used to create patches of spectres of any size. The process involves identifying key points on the boundary of the spectres, using them to guide the placement of neighboring tiles, and iterating the process to build superclusters. The rules are combinatorial and can be used to develop a practical drawing algorithm. The paper emphasizes that these substitution rules can produce patches of spectre clusters and that they are not related by rigid motions and uniform scalings that pack tiles into supertiles. The document discusses chiral aperiodic monotiles and spectres. The clusters T7H and T8H form the basis for substitution rules. The odd spectres play a role similar to the even spectres. The sparse turtle tiling can be composed into hats and turtles. The methods used in the paper to prove aperiodicity rely heavily on constructing spectre tilings and the properties of the spectre in detail. The chiral marked hexagons disprove the existence of an aperiodic monotile with bilateral reflection symmetry. The construction works generally for C1 curves, which makes chirality moot. The document presents a family of strictly chiral aperiodic monotiles that tile aperiodically using tiles of a single handedness, even when reflections are allowed. The Spectre is a shape that tiles aperiodically if only translations and rotations are permitted. The paper answers the Einstein problem in a world where sameness is restricted to orientation-preserving isometries. The Spectre enforces tiling substitution rules on marked hexagons, and erasing the markings restores the same tile and edge markings as the associated marked hexagons. The clusters have combinatorially equivalent match-preprint, and the Spectre is a strictly chiral aperiodic monotile. The document discusses chiral aperiodic monotiles and spectres. The geometry of marked hexagons can be used to realize every combinatorial edge and vertex arrangement in a patch. There exists an unbounded combinatorial patch, which can be used to derive combinatorial patches of Spectres. Labelled clusters define substitution rules on marked Spectres that are combinatorially equivalent to the substitution rules on marked hexagons. The families of shapes turtles and tilings by Spectres are non-periodic. Theorem 5.1 allows for the composition of any tiling by marked hexagons into a combinatorially equivalent tiling by supertiles. A large patch of marked hexagons is shown in Figure 5.2. Chiral Aperiodic Monotiles and Spectres is a document that discusses the construction of supertiles using 5-patches. The unique choice of the locations of the marked hexagons within the 5-patches is used to assign neighbouring hexagons to their own supertiles, which determines the locations of the vertices of the supertile. There are exactly nine illustrated supertiles resulting from these assignment rules. Every unassigned hexagon is assigned to the supertile where it is in the right position relative to that supertile's marked hexagon, and this assignment places each marked hexagon in any tiling to at most one supertile. The document discusses chiral aperiodic monotiles and spectres. The marked hexagons tile arbitrarily large finite regions of the plane, and the lack of translation as a symmetry implies that the tiling by supertiles necessarily has the same symmetries as the original tiling by marked hexagons. The supertiles are combinatorially equivalent to reflected versions of the hexagons, and the tiles in any tiling by the nine marked hexagons can be composed into the supertiles shown in Figure 5.1. Each supertile contains a ? hexagon and either six or seven other hexagons. The tiles in any tiling admitted by the marked hexagons can be uniquely composed into the supertiles of Figure 5.1, and this composition yields a unique hierarchy of level-n supertiles for all n. The tiles can be composed into copies of T7H and T8H in every chiral tiling by hats and turtles, and copies of T7H and T8H cannot overlap in a tiling by hats and turtles. The document discusses chiral aperiodic monotiles and spectres. The authors construct a reduced list of 1-patches from a hat and T7H, and group seven hats to form a cluster. They discover that every 1-patch with T6H at its center must have hats and copies of T6H. The unique 1-patch consists of a turtle surrounded by hats. Spectre is equivalent to one tiling where turtles are distributed sparsely among hats or vice versa. Every tiling by the surrounded hat has no other turtle neighbor. In reduced lists of 1-patches, every reduced 1-patch with a hat at the center has at most one turtle in its corona, except for a single 1-patch containing a hat surrounded by turtles. By computing the reduced 1-patches of a set consisting of an unreflected hat and an unre- flected turtle, the resulting list may be generated from a list of (k ? 1)-patches by considering ways to surround the tiles in the k > 1. The computation is analogous to that of reduced lists of 1-patches. The authors also use the extended notion of reduced lists of k-patches for cent hexagons. The document discusses Chiral Aperiodic Monotiles and Spectres. The authors use computer-assisted case analysis to establish the aperiodicity of the hat. To construct a reduced list of 1-patches for a given set of tiles, first generate a list of legal 1-patches, and then "reduce" the list by eliminating those patches that cannot occur in tilings. The final list contains all 1-patches that occur together in tilings. The clusters define supertiles for substitution rules on the marked Spectres, which may be colored the same way as the combinatorially equivalent rules of Figure 5.1. To prove Theorem 4.1, the authors assume without loss of generality that the tiling by Spectres and the corresponding tiling by hats and turtles consist entirely of unreflected tiles. The clusters taken together can be additional hat. The document discusses chiral aperiodic monotiles and spectres, which are made up of clusters of tiles arranged in specific patterns. The clusters have matching conditions and symmetries that allow them to be combined into larger tilings. The document shows that any tiling by spectres is equivalent to a chiral tiling by hats and turtles, and that the Spectre tiling is periodic if and only if the hat-turtle tiling is. The document also provides a substitution system that can produce patches of Spectres of any size. The next section discusses the relationship between a tiling by Tile(1,1) and a combinatorially equivalent tiling by hats and turtles, which can be used to enumerate patches of tiles that can occur in chiral tilings. Finally, the document shows how changing the lengths of edges in a tiling by hats and turtles can result in a tiling by Tile(1,1) or a Laves tiling. The text describes a mathematical study of chiral aperiodic monotiles and spectres. The tiles are described as having six families of parallel edges, with even and odd edges labeled according to their parity. The study involves modifying the lengths of the edges and producing a family of combinatorially equivalent monotiles. The tiling can be manipulated discretely by associating information with the cells of the underlying kite grid. The study establishes a bijection between tilings by Tile(1, 1) and combinations of hats and turtles. The study also explores the translational symmetry of the tiles and how they can be separated by vectors. The article discusses the combinatorial equivalence of tilings by sets of tiles A and B. If T A is periodic, then T B is also periodic. The article introduces the concept of an edge patch and the local consistency condition for combinatorial equivalence. The article provides a lemma for testing combinatorial equivalence. The article also discusses an algorithm for laying out Spectres and includes an appendix with larger rules for substituting tiles. The article notes that the rules of Figure 2.1 reverse all tile orientations and produces chiral patches of alternating handedness. The article also discusses the unique infinite hierarchy of supertiles in a tiling by Spectres. The article discusses the creation of chiral aperiodic monotiles using substitution rules on two base tiles: a single Spectre and a two-Spectre compound called a Mystic. Two substitution rules are illustrated in Figure 2.1, and these can be iterated to tile the plane with Spectres. The marked hexagons of Figure 4.2 and the substitution rules defined for them in Section 5 can be used to partition the tiles into congruent copies of clusters of eight and nine Spectres. In any tiling by the nine marked hexagons, the generation of patches cannot be reduced to discrete computations on an underlying lattice. The proof relies on clusters (Figure 4.1), and that each category may be viewed combinatorially as a regular hexagon. At the end of Section 5, the article transcribes a preprint with a "Spectre cluster" containing a Mystic and seven Spectres; the second replaces a Mystic by a "Mystic cluster" containing a Mystic and six Spectres. The Spectre is a chiral aperiodic monotile that can be tiled within an infinite hierarchy of larger and larger supertiles, making it non-periodic. The main result concerns the context of strictly chiral aperiodicity, excluding Tile(1, 1). The Spectre can be constructed by replacing every edge of Tile(1, 1) by copies of a smooth, non-straight s-curve symmetric under 180 rotation about its centre. This results in a topological disk that can only correspond to chiral tilings by the same set of tiles. The s-curves are the minimally invasive way to "chiralize" a polygon and preserve legal adjacencies while forcing chiral tilings. The tiles are nested within an infinite hierarchical superstructure in every tiling they admit. This document discusses chiral aperiodic monotiles and spectres, which are shapes that can be tiled in a non-periodic manner. The tiles in a tiling have a length of 1 or 2, and any maximal line segment in the union of the interior of an edge of an adjacent tile introduces a vertex with an interior angle of 180 degrees. Tile(1, 1) is regarded as an equilateral polygon with 14 unit-length edges and vertices, and copies of the tile may fit together in more ways than generic equilateral polygon monotiles, with three exceptions. The document presents two explicit constructions that yield families of Spectres, where the first is a weakly chiral aperiodic monotile. The set of Spectres admits only non-periodic tilings and can only have tilings of a single handedness, even when reflections are permitted. The paper provides examples of non-periodic tilings by Tile(1, 1) and a Spectre, and modifies the edges of Tile(1, 1) to obtain strictly chiral aperiodic monotiles called "Spectres" that admit only non-periodic tilings. A chiral aperiodic monotile is a tile that only admits chiral non-periodic tilings. The hat is an example of a chiral aperiodic monotile, but it is unclear whether there exist others. A tile is a closed topological disk in the plane that admits monohedral tilings, which are tilings where every tile is congruent to every other tile and the union of all tiles covers the entire plane. A weakly chiral aperiodic monotile is a tile that admits only weakly chiral non-periodic tilings. The strict case remains both chiral and aperiodic in the absence of reflections. The discovery of the hat highlights how little is known about the possibilities and subtleties of monohedral tilings. The document discusses chiral aperiodic monotiles and their isometries in the plane. Orientation-preserving isometries map left-handed shapes to left-handed and right-handed to right-handed, while orientation-reversing isometries exchange left- and right-handedness. Tilings formed by copies of a hat cut from paper or plastic can easily be turned over in three dimensions to obtain their reflection, but a glazed ceramic tile cannot. The hat is asymmetric and cannot be brought into perfect correspondence with its own mirror reflections. The recently discovered "hat" aperiodic monotile mixes unreflected and reflected shapes, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. The authors introduce a family of shapes called Spectres that are strictly chiral aperiodic monotiles and admit only chiral non-periodic tilings based on a hierarchical substitution system.