342 Sentences With "tilings" | Random Sentence Generator

And with that, the regular, monohedral, edge-to-edge tilings of the plane are completely understood.

When we remove the edge-to-edge restriction, we open up a whole new world of tilings.

Modular forms are functions that possess special symmetries like those in M.C. Escher's circular tilings of angels and devils.

The simple pentagon shows us that, even after thousands of years, questions about tilings still excite, inspire and astound us.

Twarock's tilings therefore applied to a wider range of viruses, including the polyomaviruses and papillomaviruses that had evaded Caspar and Klug's classification.

Remarkably, these three examples are the only regular, edge-to-edge, monohedral tilings of the plane: No other regular polygon will work.

We're starting to see that complicated relationships among the angles and sides make monohedral, edge-to-edge tilings with pentagons particularly complex.

The patterns generated by Penrose tilings do not repeat periodically, making it possible to piece together its two component shapes without leaving any gaps.

To understand the problem with pentagons, let's start with one of the simplest and most elegant of geometric structures: the regular tilings of the plane.

So Twarock turned to Penrose tilings, a mathematical technique developed in the 1970s to tile a plane with five-fold symmetry by fitting together four-sided figures called kites and darts.

For example, a simple 2-by-1 rectangle only admits one edge-to-edge tiling of the plane, but it admits infinitely many tilings of the plane that aren't edge-to-edge!

I personally can think of no better way to get students interested in monohedral disk tilings than relating it to pizza, so there could be more real world benefits than he realized.

Together with other existing knowledge, like the fact that no convex polygon with more than six sides can tile the plane, this finally settled an important question in the mathematical study of tilings.

In the middle of the 20th century, mathematicians discovered an astonishing link between reciprocity laws and what seemed like an entirely different subject: the "hyperbolic" geometry of patterns such as M.C. Escher's famous angel-devil tilings of a disk.

And with many open questions remaining in the field of mathematical tilings—like the search for a hypothetical concave "einstein" shape that can only tile the plane nonperiodically—we'll probably be putting the pieces together for a long time to come.

In their new paper, "Infinite families of monohedral disk tilings" (really gets your mouth watering for all that pizza slicing action) however, the pair prove that there is actually no limit to the number of equal slices a pie can be cut into.

Is it possible, they wondered, to divide a disc (a flat circle, like a pizza) into monohedral tilings (slices that are all the exact same shape and size) so that some of the slices don't touch the center AND not all sides of each slice follow the curve of the perimeter?

The 15 types of convex pentagons that admit tilings (not all edge-to-edge) of the plane were discovered by Karl Reinhardt in 1918, Richard Kershner in 1968, Richard James in 1975, Marjorie Rice in 1977, Rolf Stein in 1985, and Casey Mann, Jennifer McLoud-Mann and David Von Derau in 2015.

In the 1987 book, Tilings and Patterns, Branko Grünbaum calls the vertex-uniform tilings Archimedean in parallel to the Archimedean solids. Their dual tilings are called Laves tilings in honor of crystallographer Fritz Laves. They're also called Shubnikov–Laves tilings after Shubnikov, Alekseĭ Vasilʹevich.Encyclopaedia of Mathematics: Orbit - Rayleigh Equation , 1991 John Conway calls the uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra.

In general, the uniformity is greater than or equal to the number of types of vertices (m ≥ k), as different types of planigons necessarily have different orbits, but not vice versa. Setting m = n = k, there are 11 such dual tilings for n = 1; 20 such dual tilings for n = 2; 39 such dual tilings for n = 3; 33 such dual tilings for n = 4; 15 such dual tilings for n = 5; 10 such dual tilings for n = 6; and 7 such dual tilings for n = 7.

A tiling with rectangles is a tiling which uses rectangles as its parts. The domino tilings are tilings with rectangles of side ratio. The tilings with straight polyominoes of shapes such as , and tilings with polyominoes of shapes such as fall also into this category.

In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.

There are also other related unequivalent tilings, such as the hexagon-boat-star and Mikulla Roth tilings.

An example uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regularA New Kind of Science and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. John Conway calls these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra.

That is, each tile in the tiling must be congruent to one of these prototiles. A tiling that has no periods is non-periodic. A set of prototiles is said to be aperiodic if all of its tilings are non-periodic, and in this case its tilings are also called aperiodic tilings. Penrose tilings are among the simplest known examples of aperiodic tilings of the plane by finite sets of prototiles.

Every dual uniform tiling is in a 1:1 correspondence with the corresponding uniform tiling, by construction of the planigons above and superimposition. Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of planigons, a tiling is known as k-dual-uniform or k-isohedral; if there are t orbits of dual vertices, as t-isogonal; if there are e orbits of edges, as e-isotoxal. k-dual-uniform tilings with the same vertex figures can be further identified by their wallpaper group symmetry, which is identical to that of the corresponding k-uniform tiling. 1-dual-uniform tilings include 3 regular tilings, and 8 Laves tilings, with 2 or more types of regular degree vertices. There are 20 2-dual-uniform tilings, 61 3-dual-uniform tilings, 151 4-dual- uniform tilings, 332 5-dual-uniform tilings and 673 6--dualuniform tilings.

In geometry, the demiregular tilings are a set of Euclidean tessellations made from 2 or more regular polygon faces. Different authors have listed different sets of tilings. A more systematic approach looking at symmetry orbits are the 2-uniform tilings of which there are 20. Some of the demiregular ones are actually 3-uniform tilings.

These can be generalized to tessellations of other spaces, especially uniform tessellations, notably tilings of Euclidean space (honeycombs), which have exceptional objects, and tilings of hyperbolic space. There are various exceptional objects in dimension below 6, but in dimension 6 and above, the only regular polyhedra/tilings/hyperbolic tilings are the simplex, hypercube, cross- polytope, and hypercube lattice.

The simplest set of hyperbolic tilings are regular tilings {p,q}, which exist in a matrix with the regular polyhedra and Euclidean tilings. The regular tiling {p,q} has a dual tiling {q,p} across the diagonal axis of the table. Self-dual tilings {2,2}, {3,3}, {4,4}, {5,5}, etc. pass down the diagonal of the table.

The tilings by these marked tiles are necessarily aperiodic.C. Goodman-Strauss, Matching Rules and Substitution Tilings, Annals Math., 147 (1998), 181-223.Th. Fernique and N. Ollinger, Combinatorial substitutions and sofic tilings, Journees Automates Cellulaires 2010, J. Kari ed.

There are two tilings by all type 2 tiles, and one by all of each of the other four types. Fifteen of the other eighteen tilings are by special cases of type 1 tiles. Nine of the twenty-four tilings are edge-to-edge. There are also 2-isohedral tilings by special cases of type 1, type 2, and type 4 tiles, and 3-isohedral tilings, all edge-to-edge, by special cases of type 1 tiles.

In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations.

Algebraic theory of Penrose's nonperiodic tilings of the plane, I, II There is yet no complete (algebraic) characterization of cut and project tilings that can be enforced by matching rules, although numerous necessary or sufficient conditions are known.See, for example, the survey of T. T. Q. Le in Some tilings obtained by the cut and project method. The cutting planes are all parallel to the one which defines Penrose tilings (the fourth tiling on the third line). These tilings are all in different local isomorphism classes, that is, they are locally distinguishable.

There is only 1 alternation (snub) since all branch orders are odd. Many other hyperbolic families of uniform tilings can be seen at uniform tilings in hyperbolic plane.

A given set of tiles, in the Euclidean plane or some other geometric setting, admits a tiling if non- overlapping copies of the tiles in the set can be fitted together to cover the entire space. A given set of tiles might admit periodic tilings — that is, tilings which remain invariant after being shifted by a translation (for example, a lattice of square tiles is periodic). It is not difficult to design a set of tiles that admits non-periodic tilings as well as periodic tilings (for example, randomly arranged tilings using a 2×2 square and 2×1 rectangle will typically be non-periodic). However, an aperiodic set of tiles can only produce non-periodic tilings.

After the discovery of quasicrystals aperiodic tilings become studied intensively by physicists and mathematicians. The cut-and-project method of N.G. de Bruijn for Penrose tilings eventually turned out to be an instance of the theory of Meyer sets. Today there is a large amount of literature on aperiodic tilings.

The Wythoff symbol is able to handle the snub form, but not general alternations without all nodes ringed. The same constructions can be made on disjointed (orthogonal) Coxeter groups like the uniform prisms, and can be seen more clearly as tilings of dihedrons and hosohedrons on the sphere, like this [6]×[] or [6,2] family: In comparison, the [6,3], family produces a parallel set of 7 uniform tilings of the Euclidean plane, and their dual tilings. There are again 3 alternations and some half symmetry version. In the hyperbolic plane [7,3], family produces a parallel set of uniform tilings, and their dual tilings.

In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.

Selected tilings created by the Wythoff construction are given below.

The study of these tilings has been important in the understanding of physical materials that also form quasicrystals. Penrose tilings have also been applied in architecture and decoration, as in the floor tiling shown.

Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.

Penrose and John H. Conway investigated the properties of Penrose tilings, and discovered that a substitution property explained their hierarchical nature; their findings were publicized by Martin Gardner in his January 1977 "Mathematical Games" column in Scientific American. In 1981, N. G. De Bruijn provided two different methods to construct Penrose tilings. De Bruijn's "multigrid method" obtains the Penrose tilings as the dual graphs of arrangements of five families of parallel lines. In his "cut and project method", Penrose tilings are obtained as two-dimensional projections from a five-dimensional cubic structure.

In the Euclidean plane there are 3 regular forms equilateral triangle, squares, and regular hexagons; and 8 semiregular forms; and 4-demiregular forms which can tile the plane with other planigons. All angles of a planigon are whole divisors of 360°. Tilings are made by edge-to-edge connections by perpendicular bisectors of the edges of the original uniform lattice, or centroids along common edges (they coincide). Tilings made from planigons can be seen as dual tilings to the regular, semiregular, and demiregular tilings of the plane by regular polygons.

The {4,4} square tiling (black) with its dual (red). Tilings can also be self-dual. The square tiling, with Schläfli symbol {4,4}, is self-dual; shown here are two square tilings (red and black), dual to each other.

Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex. These 11 uniform tilings have 32 different uniform colorings.

He discovered several new aperiodic tilings, each among the simplest known examples of aperiodic sets of tiles. He also showed how to generate tilings using lines in the plane as guides for lines marked on the tiles, now called "Ammann bars". The discovery of quasicrystals in 1982 changed the status of aperiodic tilings and Ammann's work from mere recreational mathematics to respectable academic research.

The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n. The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.

Eleven prismatic tilings are obtained by stacking the eleven uniform plane tilings, shown below, in parallel layers. (One of these honeycombs is the cubic, shown above.) The vertex figure of each is an irregular bipyramid whose faces are isosceles triangles.

Subdivision rules give a sequence of tilings of a surface, and tilings give an idea of distance, length, and area (by letting each tile have length and area 1). In the limit, the distances that come from these tilings may converge in some sense to an analytic structure on the surface. The Combinatorial Riemann Mapping Theorem gives necessary and sufficient conditions for this to occur. Its statement needs some background.

The Penrose tiling is an example of an aperiodic tiling; every tiling it can produce lacks translational symmetry. An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings.

Spherical pentagonal hexecontahedron This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry.

There are 151 4-uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 33 4-uniform tilings with 4 distinct vertex types, as well as finding 85 of them with 3 vertex types, and 33 with 2 vertex types.

In terms of symbolic dynamics, this means that the pinwheel tilings form a sofic subshift.

It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals.

It is useful to classify the uniform polytopes by dimension. This is equivalent to the number of nodes on the Coxeter diagram, or the number of hyperplanes in the Wythoffian construction. Because (n+1)-dimensional polytopes are tilings of n-dimensional spherical space, tilings of n-dimensional Euclidean and hyperbolic space are also considered to be (n+1)-dimensional. Hence, the tilings of two-dimensional space are grouped with the three-dimensional solids.

The regular dodecahedron is topologically related to a series of tilings by vertex figure n3. The regular dodecahedron can be transformed by a truncation sequence into its dual, the icosahedron: The regular dodecahedron is a member of a sequence of otherwise non-uniform polyhedra and tilings, composed of pentagons with face configurations (V3.3.3.3.n). (For n > 6, the sequence consists of tilings of the hyperbolic plane.) These face-transitive figures have (n32) rotational symmetry.

An alternative approach is to apply Knuth's Algorithm X to enumerate valid tilings for the problem.

Dominos can tile the plane in a countably infinite number of ways. The number of tilings of a 2×n rectangle with dominoes is F_n, the nth Fibonacci number.Concrete Mathematics by Graham, Knuth and Patashnik, Addison- Wesley, 1994, p. 320, Domino tilings figure in several celebrated problems, including the Aztec diamond problem In which large diamond-shaped regions have a number of tilings equal to a power of two, with most tilings appearing random within a central circular region and having a more regular structure outside of this "arctic circle", and the mutilated chessboard problem, in which removing two opposite corners from a chessboard makes it impossible to tile with dominoes..

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n). This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram , progressing to infinity.

Every tile type subdivides into smaller tile types. Each edge also gets subdivided according to finitely many edge types. Finite subdivision rules can only subdivide tilings that are made up of polygons labelled by tile types. Such tilings are called subdivision complexes for the subdivision rule.

Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known.

A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic. The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are aperiodic. (The tiling itself is said to be "nonperiodic".) The first table explains the abbreviations used in the second table.

This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}. And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}. From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

Not all manifolds are finite. Where a polytope is understood as a tiling or decomposition of a manifold, this idea may be extended to infinite manifolds. plane tilings, space-filling (honeycombs) and hyperbolic tilings are in this sense polytopes, and are sometimes called apeirotopes because they have infinitely many cells. Among these, there are regular forms including the regular skew polyhedra and the infinite series of tilings represented by the regular apeirogon, square tiling, cubic honeycomb, and so on.

The sequence of tilings where b is centred at 1,2,4, \ldots,2^n,\ldots converges – in the local topology – to the periodic tiling consisting of as only. Thus T is not an aperiodic tiling, since its hull contains the periodic tiling For well-behaved tilings (e.g. substitution tilings with finitely many local patterns) holds: if a tiling is non-periodic and repetitive (i.e. each patch occurs in a uniformly dense way throughout the tiling), then it is aperiodic.

5, p.63 (illustration of this tiling), Theorem 2.9.1, p. 103 (classification of colored tilings), Figure 2.9.

There are 332 5-uniform tilings of the Euclidean plane. Brian Galebach's search identified 332 5-uniform tilings, with 2 to 5 types of vertices. There are 74 with 2 vertex types, 149 with 3 vertex types, 94 with 4 vertex types, and 15 with 5 vertex types.

Spherical pentagonal icositetrahedron This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry. The pentagonal icositetrahedron is second in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n. The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.

The snub tetrahexagonal tiling is fifth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

The snub tetrapentagonal tiling is fourth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

The snub tetraheptagonal tiling is sixth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

The snub tetraoctagonal tiling is seventh in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5n).

2, p. 105 (illustration of colored tilings), Figure 2.5.3(d), p. 83 (topologically equivalent star tiling), and Exercise 4.1.

The tetragonal trapezohedron is first in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (6n).

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5n).

Inflation and deflation yield a method for constructing kite and dart (P2) tilings, or rhombus (P3) tilings, known as up-down generation. The Penrose tilings, being non-periodic, have no translational symmetry - the pattern cannot be shifted to match itself over the entire plane. However, any bounded region, no matter how large, will be repeated an infinite number of times within the tiling. Therefore, no finite patch can uniquely determine a full Penrose tiling, nor even determine which position within the tiling is being shown.

Substitution tiling systems provide a rich source of aperiodic tilings. A set of tiles that forces a substitution structure to emerge is said to enforce the substitution structure. For example, the chair tiles shown below admit a substitution, and a portion of a substitution tiling is shown at right below. These substitution tilings are necessarily non- periodic, in precisely the same manner as described above, but the chair tile itself is not aperiodic—it is easy to find periodic tilings by unmarked chair tiles.

This tiling can be used to show that the hyperbolic plane has tilings by congruent tiles of arbitrarily small area.

The snub tetrapeirogonal tiling is last in an infinite series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedra), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

The other way we can start our tiling is by laying two horizontal tiles on top of each other, which leaves us with B_{n-1} that has F_{n-1} different tilings. By adding the two together, the number of tilings for B_{n+1} = F_{n} + F_{n-1} = F_{n+1}.

The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The tiles of the Laves tilings are called planigons. This includes the 3 regular tiles (triangle, square and hexagon) and 8 irregular ones. Each vertex has edges evenly spaced around it.

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

Even constrained in this manner, each variation yields infinitely many different Penrose tilings. alt= Penrose tilings are self-similar: they may be converted to equivalent Penrose tilings with different sizes of tiles, using processes called inflation and deflation. The pattern represented by every finite patch of tiles in a Penrose tiling occurs infinitely many times throughout the tiling. They are quasicrystals: implemented as a physical structure a Penrose tiling will produce diffraction patterns with Bragg peaks and five-fold symmetry, revealing the repeated patterns and fixed orientations of its tiles.

Given any subdivision complex for a subdivision rule, we can subdivide it over and over again to get a sequence of tilings. For instance, binary subdivision has one tile type and one edge type: The binary subdivision rule Since the only tile type is a quadrilateral, binary subdivision can only subdivide tilings made up of quadrilaterals. This means that the only subdivision complexes are tilings by quadrilaterals. The tiling can be regular, but doesn't have to be: We start with a complex with four quadrilaterals and subdivide twice.

In 2007, the physicists Peter J. Lu and Paul J. Steinhardt suggested that girih tilings possess properties consistent with self-similar fractal quasicrystalline tilings such as Penrose tilings, predating them by five centuries.Supplemental figures This finding was supported both by analysis of patterns on surviving structures, and by examination of 15th-century Persian scrolls. There is no indication of how much more the architects may have known about the mathematics involved. It is generally believed that such designs were constructed by drafting zigzag outlines with only a straightedge and a compass.

It is one of three regular tilings of the plane. The other two are the triangular tiling and the hexagonal tiling.

It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling.

In this way, setisets can produce non-periodic tilings. However, none of the non-periodic tilings thus far discovered qualify as aperiodic, because the prototiles can always be rearranged so as to yield a periodic tiling. Figure 5 shows the first two stages of inflation of an order 4 set leading to a non-periodic tiling.

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling. It can also be generated from the (4 3 3) hyperbolic tilings: This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and [n,3] Coxeter group symmetry.

Tilings combinatorially equivalent to the rhombille tiling can also be realized by parallelograms, and interpreted as axonometric projections of three dimensional cubic steps. There are only eight edge tessellations, tilings of the plane with the property that reflecting any tile across any one of its edges produces another tile; one of them is the rhombille tiling..

A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are not color-uniform) In addition to the 11 convex uniform tilings, there are also 14 nonconvex tilings, using star polygons, and reverse orientation vertex configurations.

The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings: From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Up-down generation yields one method to parameterize the tilings, but other methods use Ammann bars, pentagrids, or cut and project schemes.

The snub tetrapeirogonal tiling is last in an infinite series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.

The physics of this discovery has revived the interest in incommensurate structures and frequencies suggesting to link aperiodic tilings with interference phenomena.

The Schmitt–Conway–Danzer tile, a convex polyhedron that tiles space, is not a stereohedron because all of its tilings are aperiodic.

Islamic Girih tiles in Islamic architecture are self-similar tilings that can be modeled with finite subdivision rules. In 2007, Peter J. Lu of Harvard University and Professor Paul J. Steinhardt of Princeton University published a paper in the journal Science suggesting that girih tilings possessed properties consistent with self-similar fractal quasicrystalline tilings such as Penrose tilings (presentation 1974, predecessor works starting in about 1964) predating them by five centuries. Subdivision surfaces in computer graphics use subdivision rules to refine a surface to any given level of precision. These subdivision surfaces (such as the Catmull-Clark subdivision surface) take a polygon mesh (the kind used in 3D animated movies) and refines it to a mesh with more polygons by adding and shifting points according to different recursive formulas.

An alt= The subject of aperiodic tilings received new interest in the 1960s when logician Hao Wang noted connections between decision problems and tilings. In particular, he introduced tilings by square plates with colored edges, now known as Wang dominoes or tiles, and posed the "Domino Problem": to determine whether a given set of Wang dominoes could tile the plane with matching colors on adjacent domino edges. He observed that if this problem were undecidable, then there would have to exist an aperiodic set of Wang dominoes. At the time, this seemed implausible, so Wang conjectured no such set could exist.

A rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares. The same is true if the tiles are unequal isosceles right triangles. The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent polyaboloes.

This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram , progressing to infinity. This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.

This tiling is topologically related as a part of sequence of regular tilings with order-5 vertices with Schläfli symbol {n,5}, and Coxeter diagram , progressing to infinity. This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings, extending into the hyperbolic plane: {4,p}, p=3,4,5... This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

Non-commutative spaces arise naturally, even inevitably, from some constructions. For example, consider the non-periodic Penrose tilings of the plane by kites and darts. It is a theorem that, in such a tiling, every finite patch of kites and darts appears infinitely often. As a consequence, there is no way to distinguish two Penrose tilings by looking at a finite portion.

For a table of all uniform hyperbolic tilings with fundamental domains (p q r), where 2 ≤ p,q,r ≤ 8, and one or more as ∞.

Certain shapes of tiles, most obviously rectangles, can be replicated to cover a surface with no gaps. These shapes are said to tessellate (from the Latin tessella, 'tile') and such a tiling is called a tessellation. Geometric patterns of some Islamic polychrome decorative tilings are rather complicated (see Islamic geometric patterns and, in particular, Girih tiles), even up to supposedly quaziperiodic ones, similar to Penrose tilings.

An equivalent statement on homogeneous linear forms was originally conjectured by Hermann Minkowski. A consequence is Minkowski's conjecture on lattice tilings, which says that in any lattice tiling of space by cubes, there are two cubes that meet face to face. Keller's conjecture is the same conjecture for non-lattice tilings, which turns out to be false in high dimensions. Hajós's theorem was generalized by Tibor Szele.

To date, there is not a formal definition describing when a tiling has a hierarchical structure; nonetheless, it is clear that substitution tilings have them, as do the tilings of Berger, Knuth, Läuchli and Robinson. As with the term "aperiodic tiling" itself, the term "aperiodic hierarchical tiling" is a convenient shorthand, meaning something along the lines of "a set of tiles admitting only non-periodic tilings with a hierarchical structure". Each of these sets of tiles, in any tiling they admit, forces a particular hierarchical structure. (In many later examples, this structure can be described as a substitution tiling system; this is described below).

Applying twice aa, (rectifying a rectification) is Conway's expand operation, e, which is the same as Johnson's cantellation operation, t0,2 generated from regular polyhedral and tilings.

There are uncountably many different tilings of the hyperbolic plane by these tiles, even when they are modified by adding protrusions and indentations to force them to meet edge-to-edge. None of these different tilings are periodic (having a cocompact symmetry group), although some (such as the one in which there exists a line that is completely covered by tile edges) have a one-dimensional infinite symmetry group.

A variant tiling which is not a quasicrystal. It is not a Penrose tiling because it does not comply with the tile alignment rules. There are also other related unequivalent tilings, such as the hexagon-boat-star and Mikulla-Roth tilings. For instance, if the matching rules for the rhombus tiling are reduced to a specific restriction on the angles permitted at each vertex, a binary tiling is obtained.

The apeirogonal tiling is the arithmetic limit of the family of prisms t{2, p} or p.4.4, as p tends to infinity, thereby turning the prism into a Euclidean tiling. An alternation operation can create an apeirogonal antiprism composed of three triangles and one apeirogon at each vertex. : 320px Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling.

This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity. This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

Conway calls it a truncated hextille, constructed as a truncation operation applied to a hexagonal tiling (hextille). There are 3 regular and 8 semiregular tilings in the plane.

Each tiling in the sequence is indistinguishable from a true Ammann–Beenker tiling on a successively larger scale. Since some of these tilings are periodic, it follows that no decoration of the tiles which does force aperiodicity can be determined by looking at any finite patch of the tiling. The orientation of the vertex arrows which force aperiodicity, then, can only be deduced from the entire infinite tiling. The tiling has also an extremal property : among the tilings whose rhombuses alternate (that is, whenever two rhombuses are adjacent or separated by a row of square, they appear in different orientations), the proportion of squares is found to be minimal in the Ammann–Beenker tilings.

Daniel Wise obtained his Ph.D. from Princeton University in 1996 supervised by Martin Bridson His thesis was titled non-positively curved squared complexes, aperiodic tilings, and non- residually finite groups.

Charles Lewis Radin is an American mathematician, known for his work on aperiodic tilings and in particular for defining the pinwheel tiling and (with John Horton Conway) the quaquaversal tiling..

There are 61 3-uniform tilings of the Euclidean plane. 39 are 3-Archimedean with 3 distinct vertex types, while 22 have 2 identical vertex types in different symmetry orbits.

This tiling is uniform but not regular (it is by scalene triangles), and often regular tilings are used instead. A quotient of any tiling in the (2,3,7) family can be used (and will have the same automorphism group); of these, the two regular tilings are the tiling by 24 regular hyperbolic heptagons, each of degree 3 (meeting at 56 vertices), and the dual tiling by 56 equilateral triangles, each of degree 7 (meeting at 24 vertices). The order of the automorphism group is related, being the number of polygons times the number of edges in the polygon in both cases. :24 × 7 = 168 :56 × 3 = 168 The covering tilings on the hyperbolic plane are the order-3 heptagonal tiling and the order-7 triangular tiling.

Amman's A and B pair of A5 tiles, decorated with matching rules; any tiling by these tilings is necessarily non- periodic, and the tiles are therefore aperiodic. Ammann A5 substitution rules, used to prove that the A5 tiles can only form non-periodic hierarchical tilings and thus are aperiodic tiles. This tiling exists in a 2D orthogonal projection of a 4D 8-8 duoprism constructed from 16 octahedral prisms. Amman's A and B tiles in his pair A5 a 45-135-degree rhombus and a 45-45-90 degree triangle, decorated with matching rules that allowed only certain arrangements in each region, forcing the non-periodic, hierarchical, and quasiperiodic structures of each of the infinite number of individual Ammann–Beenker tilings.

If the tiling is properly scaled, it will close as an asymptopic limit at a single ideal point. These Euclidean tilings are inscribed in a horosphere just as polyhedra are inscribed in a sphere (which contains zero ideal points). The sequence extends when hyperbolic tilings are themselves used as facets of noncompact hyperbolic tessellations, as in the heptagonal tiling honeycomb {7,3,3}; they are inscribed in an equidistant surface (a 2-hypercycle), which has two ideal points.

This makes it impossible to assign the set of all tilings a topology in the traditional sense. Despite this, the Penrose tilings determine a non-commutative and consequently they can be studied by the techniques of non-commutative geometry. Another example, and one of great interest within differential geometry, comes from foliations of manifolds. These are ways of splitting the manifold up into smaller-dimensional submanifolds called leaves, each of which is locally parallel to others nearby.

This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.

There are infinitely many triangle group families including infinite orders. This article shows uniform tilings in 9 families: (∞ 3 2), (∞ 4 2), (∞ ∞ 2), (∞ 3 3), (∞ 4 3), (∞ 4 4), (∞ ∞ 3), (∞ ∞ 4), and (∞ ∞ ∞).

Nature provides examples of many kinds of pattern, including symmetries, trees and other structures with a fractal dimension, spirals, meanders, waves, foams, tilings, cracks and stripes.Stevens, Peter. Patterns in Nature, 1974. Page 3.

The term pentacle is used in Tilings and Patterns by Grumbaum and Shepard to indicate a five-pointed star composed of ten line segments, similar to a pentagram but containing no interior lines.

J. Amer. Math. Soc. 5 (1992), no. 1, 33–74. In the 2000s he wrote a series of papers on the divisible convex sets in projective space and periodic tilings by such sets.

In 1973 and 1974, Roger Penrose discovered a family of aperiodic tilings, now called Penrose tilings. The first description was given in terms of 'matching rules' treating the prototiles as jigsaw puzzle pieces. The proof that copies of these prototiles can be put together to form a tiling of the plane, but cannot do so periodically, uses a construction that can be cast as a substitution tiling of the prototiles. In 1977 Robert Ammann discovered a number of sets of aperiodic prototiles, i.e.

In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{3,6}.

There are several different variations of Penrose tilings with different tile shapes. The original form of Penrose tiling used tiles of four different shapes, but this was later reduced to only two shapes: either two different rhombi, or two different quadrilaterals called kites and darts. The Penrose tilings are obtained by constraining the ways in which these shapes are allowed to fit together. This may be done in several different ways, including matching rules, substitution tiling or finite subdivision rules, cut and project schemes, and coverings.

The term aperiodic has been used in a wide variety of ways in the mathematical literature on tilings (and in other mathematical fields as well, such as dynamical systems or graph theory, with altogether different meanings). With respect to tilings the term aperiodic was sometimes used synonymously with the term non-periodic. A non-periodic tiling is simply one that is not fixed by any non-trivial translation. Sometimes the term described – implicitly or explicitly – a tiling generated by an aperiodic set of prototiles.

Conway, 2008, p288 table It can be considered a cantellated by Norman Johnson's terminology or an expanded hexagonal tiling by Alicia Boole Stott's operational language. There are 3 regular and 8 semiregular tilings in the plane.

An 8th represents an alternation operation, deleting alternate vertices from the highest form with all mirrors active. Families with r = 2 contain regular hyperbolic tilings, defined by a Coxeter group such as [7,3], [8,3], [9,3], ... [5,4], [6,4], .... Hyperbolic families with r = 3 or higher are given by (p q r) and include (4 3 3), (5 3 3), (6 3 3) ... (4 4 3), (5 4 3), ... (4 4 4).... Hyperbolic triangles (p q r) define compact uniform hyperbolic tilings. In the limit any of p, q or r can be replaced by ∞ which defines a paracompact hyperbolic triangle and creates uniform tilings with either infinite faces (called apeirogons) that converge to a single ideal point, or infinite vertex figure with infinitely many edges diverging from the same ideal point. More symmetry families can be constructed from fundamental domains that are not triangles.

For two perpendicular families of parallel lines this construction just gives the familiar square tiling of the plane, and for three families of lines at 120-degree angles from each other (themselves forming a trihexagonal tiling) this produces the rhombille tiling. However, for more families of lines this construction produces aperiodic tilings. In particular, for five families of lines at equal angles to each other (or, as de Bruijn calls this arrangement, a pentagrid) it produces a family of tilings that include the rhombic version of the Penrose tilings. The tetrakis square tiling is an infinite arrangement of lines forming a periodic tiling that resembles a multigrid with four parallel families, but in which two of the families are more widely spaced than the other two, and in which the arrangement is simplicial rather than simple.

The apeirogonal tiling is the arithmetic limit of the family of dihedra {p, 2}, as p tends to infinity, thereby turning the dihedron into a Euclidean tiling. Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and cantellated forms are duplicated, and as two times infinity is also infinity, the truncated and omnitruncated forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal tiling, the apeirogonal hosohedron, the apeirogonal prism, and the apeirogonal antiprism.

Patterns may be elaborated by the use of two levels of design, as at the 1453 Darb-e Imam shrine. Square repeating units of known patterns can be copied as templates, and historic pattern books may have been intended for use in this way. The 15th century Topkapı Scroll explicitly shows girih patterns together with the tilings used to create them. A set of tiles consisting of a dart and a kite shape can be used to create aperiodic Penrose tilings, though there is no evidence that such a set was used in medieval times.

This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity. This tiling is topologically related to regular polyhedra with vertex figure n3, as a part of sequence that continues into the hyperbolic plane. It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6. This tiling is also a part of a sequence of truncated rhombic polyhedra and tilings with [n,3] Coxeter group symmetry.

Geometric patterns occur in a variety of forms in Islamic art and architecture including kilim carpets, Persian girih and Moroccan zellige tilework, muqarnas decorative vaulting, jali pierced stone screens, ceramics, leather, stained glass, woodwork, and metalwork. Interest in Islamic geometric patterns is increasing in the West, both among craftsmen and artists including M. C. Escher in the twentieth century, and among mathematicians and physicists including Peter J. Lu and Paul Steinhardt who controversially claimed in 2007 that tilings at the Darb-e Imam shrine in Isfahan could generate quasi-periodic patterns like Penrose tilings.

It is related to two star- tilings by the same vertex arrangement: the order-7 heptagrammic tiling, {7/2,7}, and heptagrammic-order heptagonal tiling, {7,7/2}. This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {3,p}. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

There are no regular plane tilings of star polygons. There are many enumerations that fit in the plane (1/p + 1/q = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc., but none repeat periodically.

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,6}, and Coxeter diagram , with n progressing to infinity.

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,5}, and Coxeter diagram , with n progressing to infinity.

Other names used for this pattern include Mediterranean tiling and octagonal tiling, which is often represented by smaller squares, and nonregular octagons which alternate long and short edges. There are 3 regular and 8 semiregular tilings in the plane.

Conway, 2008, p.288 table There are 3 regular and 8 semiregular tilings in the plane. This tiling is similar to the snub square tiling which also has 3 triangles and two squares on a vertex, but in a different order.

They are all locally undistinguishable (i.e., they have the same finite patches). They all share with the Conway tiling the property that tiles appear in infinitely many orientations (and vertices have rational coordinates). The main result proven by Radin is that there is a finite (though very large) set of so-called prototiles, with each being obtained by coloring the sides of T, so that the pinwheel tilings are exactly the tilings of the plane by isometric copies of these prototiles, with the condition that whenever two copies intersect in a point, they have the same color in this point.

A tile substitution is described by a set of prototiles (tile shapes) T_1,T_2,\dots, T_m, an expanding map Q and a dissection rule showing how to dissect the expanded prototiles Q T_i to form copies of some prototiles T_j. Intuitively, higher and higher iterations of tile substitution produce a tiling of the plane called a substitution tiling. Some substitution tilings are periodic, defined as having translational symmetry. Every substitution tiling (up to mild conditions) can be "enforced by matching rules"—that is, there exist a set of marked tiles that can only form exactly the substitution tilings generated by the system.

Rhombitriheptagonal tiling of the hyperbolic plane, seen in the Poincaré disk model Like the Euclidean plane it is also possible to tessellate the hyperbolic plane with regular polygons as faces. There are an infinite number of uniform tilings based on the Schwarz triangles (p q r) where 1/p + 1/q + 1/r < 1, where p, q, r are each orders of reflection symmetry at three points of the fundamental domain triangle, the symmetry group is a hyperbolic triangle group. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains.

This effect has come to be called the Terrell rotation or Penrose–Terrell rotation. A alt= In 1967, Penrose invented the twistor theory which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature (2,2). Penrose is well known for his 1974 discovery of Penrose tilings, which are formed from two tiles that can only tile the plane nonperiodically, and are the first tilings to exhibit fivefold rotational symmetry. Penrose developed these ideas based on the article Deux types fondamentaux de distribution statistiqueJaromír Korčák (1938): Deux types fondamentaux de distribution statistique.

In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures. Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors (the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice). The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone sets.

Example Wythoff construction with right triangles (r = 2) and the 7 generator points. Lines to the active mirrors are colored red, yellow, and blue with the 3 nodes opposite them as associated by the Wythoff symbol. There are an infinite number of uniform tilings based on the Schwarz triangles (p q r) where + + < 1, where p, q, r are each orders of reflection symmetry at three points of the fundamental domain triangle – the symmetry group is a hyperbolic triangle group. Each symmetry family contains 7 uniform tilings, defined by a Wythoff symbol or Coxeter-Dynkin diagram, 7 representing combinations of 3 active mirrors.

The Heritage Floor, which sits underneath the table, features the names of 998 women (and one man, Kresilas, mistakenly included as he was thought to have been a woman called Cresilla) inscribed on white handmade porcelain floor tilings. The tilings cover the full extent of the triangular table area, from the footings at each place setting, continues under the tables themselves and fills the full enclosed area within the three tables. There are 2304 tiles with names spread across more than one tile. The names are written in the Palmer cursive script, a twentieth-century American form.

But in 1982 Beenker published a similar mathematical explanation for the octagonal case which became known as the Ammann–Beenker tiling.Beenker FPM, "Algebraic theory of non periodic tilings of the plane by two simple building blocks: a square and a rhombus", TH Report 82-WSK-04 (1982), Technische Hogeschool, Eindhoven In 1987 Wang, Chen and Kuo announced the discovery of a quasicrystal with octagonal symmetry. The decagonal covering of the Penrose tiling was proposed in 1996 and two years later F. Gahler proposed an octagonal variant for the Ammann–Beenker tilingS. Ben Abraham and F. Gahler, Phys. Rev.

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

The prismatic pentagonal tiling is a dual uniform tiling in the Euclidean plane. It is one of 15 known isohedral pentagon tilings. It can be seen as a stretched hexagonal tiling with a set of parallel bisecting lines through the hexagons. Conway calls it an .

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

It is a chiral pattern, meaning that it is impossible to superpose it on top of its mirror image using only translations and rotations. A uniform tiling is a tiling in which each tile is a regular polygon and in which every vertex can be mapped to every other vertex by a symmetry of the tiling. Usually, uniform tilings additionally are required to have tiles that meet edge-to-edge, but if this requirement is relaxed then there are eight additional uniform tilings. Four are formed from infinite strips of squares or equilateral triangles, and three are formed from equilateral triangles and regular hexagons.

Infinitely many distinct tilings may be obtained from a single aperiodic set of tiles.A set of aperiodic prototiles can always form uncountably many different tilings, even up to isometry, as proven by Nikolaï Dolbilin in his 1995 paper The Countability of a Tiling Family and the Periodicity of a Tiling The best-known examples of an aperiodic set of tiles are the various Penrose tiles. The known aperiodic sets of prototiles are seen on the list of aperiodic sets of tiles. The underlying undecidability of the domino problem implies that there exists no systematic procedure for deciding whether a given set of tiles can tile the plane.

There are infinitely many general triangle group families (p q r). This article shows uniform tilings in 9 families: (4 3 3), (4 4 3), (4 4 4), (5 3 3), (5 4 3), (5 4 4), (6 3 3), (6 4 3), and (6 4 4).

A square tiling (which would resemble an infinitely large fenestrane) would suffer form the same problem as octahedrane, and the triangular tiling icosahedrane. No generalisations to hyperbolic tilings seem to be known. The regular convex 4-polytopes may also have hydrocarbon analogues; hypercubane has been proposed.

Lu's most widely publicized work involves his discovery of the girih tiles, a set of fundamental geometric tiles used to create a wide range of patterns in medieval Islamic architecture. In collaboration with Paul Steinhardt, he demonstrated their use to create quasicrystal tilings on the walls of Darb-i Imam shrine (1453 A.D.) in Isfahan, Iran. The finding was considered a significant breakthrough by demonstrating a simple and straightforward method that could have been used by common workers to create extremely complicated patterns using girih tiles, and by identifying a medieval example of quasicrystalline patterns, which were not widely known to or understood by the West until the discovery of Penrose tilings by Roger Penrose in the 1970s. For its timely scientific and political implications, Lu and Steinhardt's work on medieval Islamic architectural tilings received substantial worldwide coverage on the front pages of a number of major newspapers, on the radio, and in magazines; the finding was identified as among the top 100 scientific discoveries of 2007 by Discover magazine.

In 1998, Madden and Aimee Johnson won the George Pólya Award for their joint paper on aperiodic tiling, "Putting the Pieces Together: Understanding Robinson's Nonperiodic Tilings". In 2017, Madden, Johnson, and their co-author Ayşe Şahin published the textbook Discovering Discrete Dynamical Systems through the Mathematical Association of America.

3, p. 171 (topological equivalence of trihexagonal and two-triangle tilings). It consists of equilateral triangles and regular hexagons, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular hexagonal tiling and a regular triangular tiling.

The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille. It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling.

Three dimensional analogues of the planigons are called stereohedrons. These tilings are listed by their face configuration, the number of faces at each vertex of a face. For example V4.8.8 (or V4.82) means isosceles triangle tiles with one corner with four triangles, and two corners containing eight triangles.

Girih tile subdivision found in the decagonal girih pattern on a spandrel from the Darb-e Imam shrine Peter Lu and Paul Steinhardt have studied Islamic tiling patterns, called girih tiles. They strongly resemble Penrose tilings, to which the designs on the Darb-e Imam shrine are almost identical.

The apeirogonal hosohedron is the arithmetic limit of the family of hosohedra {2,p}, as p tends to infinity, thereby turning the hosohedron into a Euclidean tiling. All the vertices have then receded to infinity and the digonal faces are no longer defined by closed circuits of finite edges. Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and cantellated forms are duplicated, and as two times infinity is also infinity, the truncated and omnitruncated forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal tiling, the apeirogonal hosohedron, the apeirogonal prism, and the apeirogonal antiprism.

Visualization of the map (2,3,∞) → (2,3,7) by morphing the associated tilings.Platonic tilings of Riemann surfaces: The Modular Group, Gerard Westendorp Extending the scalars from Q(η) to R (via the standard imbedding), one obtains an isomorphism between the quaternion algebra and the algebra M(2,R) of real 2 by 2 matrices. Choosing a concrete isomorphism allows one to exhibit the (2,3,7) triangle group as a specific Fuchsian group in SL(2,R), specifically as a quotient of the modular group. This can be visualized by the associated tilings, as depicted at right: the (2,3,7) tiling on the Poincaré disc is a quotient of the modular tiling on the upper half-plane.

Moorish tessellations including this one at the Alhambra inspired Escher's work with tilings of the plane. He made sketches of this and other Alhambra patterns in 1936.Locher, 1974. p. 17 In 1922, an important year of his life, Escher traveled through Italy, visiting Florence, San Gimignano, Volterra, Siena, and Ravello.

There are many tools used throughout tiling projects, but two useful ones are GeoGebra and program created by Jim Propp, Greg Kuperberg, and David Wilson in SageMath to count the tilings of a shape. The link to this specific program is located in external links, under Tiling Program in Sage.

As an antiprism, the square antiprism belongs to a family of polyhedra that includes the octahedron (which can be seen as a triangle-capped antiprism), the pentagonal antiprism, the hexagonal antiprism, and the octagonal antiprism. The square antiprism is first in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

Conway calls it a snub hextille, constructed as a snub operation applied to a hexagonal tiling (hextille). There are 3 regular and 8 semiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry. There is only one uniform coloring of a snub trihexagonal tiling.

Each can be grouped by the number m of distinct vertex figures, which are also called m-Archimedean tilings.k-uniform tilings by regular polygons Nils Lenngren, 2009 Finally, if the number of types of planigons is the same as the uniformity (m = k below), then the tiling is said to be Krotenheerdt.

The Danzer cube is example 8.9 in the book "Lectures on Polytopes" by G.M. Ziegler. Danzer also found many new tilings. Ludwig Danzer worked at the Technical University of Dortmund and died on December 3, 2011 after a long illness. Danzer had at least ten students, the most prominent one being Egon Schulte.

The Socolar-Taylor tile forms two-dimensional aperiodic tilings, but is defined by combinatorial matching conditions rather than purely by its shape. In higher dimensions, the problem is solved: the Schmitt-Conway-Danzer tile is the prototile of a monohedral aperiodic tiling of three-dimensional Euclidean space, and cannot tile space periodically.

In general, a honeycomb in n dimensions is an infinite example of a polytope in n + 1 dimensions. Tilings of the plane and close-packed space-fillings of polyhedra are examples of honeycombs in two and three dimensions respectively. A line divided into infinitely many finite segments is an example of an apeirogon.

In a 1976 visit, Gardner kept him for a week, pumping him for information on the Penrose tilings which had just been announced. Conway had discovered many (if not most) of the major properties of the tilings.Interview with Martin Gardner Notices of the AMS, Vol. 52, No. 6, June/July 2005, pp.

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 octagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,4] symmetry, and 7 with subsymmetry.

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [6,4] symmetry, and 7 with subsymmetry.

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-5 hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [6,5] symmetry, and 3 with subsymmetry.

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.

Motivated by physical systems involving dimers, in 1961, Kasteleyn and Temperley and Fisher independently found the number of domino tilings for the m-by-n rectangle. This is equivalent to counting the number of perfect matchings for the m-by-n lattice graph. By 1967, Kasteleyn had generalized this result to all planar graphs.

He classified the possible patterns on the surface of an Adidas Telstar soccer ball, i.e. specialThe sides of the pentagons may only encounter hexagons; the hexagons must alternately bifurcate with pentagons and hexagons. tilings with pentagons and hexagons on the sphere.Kolumne Mathematische Unterhaltungen, Spektrum der Wissenschaft, Juli 2006Braungardt, Kotschick Die Klassifikation von Fußballmustern, Math.

These tilings displayed instances of fivefold symmetry. One year later Alan Mackay showed experimentally that the diffraction pattern from the Penrose tiling had a two-dimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern. Around the same time, Robert Ammann created a set of aperiodic tiles that produced eightfold symmetry.

The classical convex polytopes may be considered tessellations, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.

Cellular automaton processors are physical implementations of CA concepts, which can process information computationally. Processing elements are arranged in a regular grid of identical cells. The grid is usually a square tiling, or tessellation, of two or three dimensions; other tilings are possible, but not yet used. Cell states are determined only by interactions with adjacent neighbor cells.

In geometry, the Cairo pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is given its name because several streets in Cairo are paved in this design... It is one of 15 known monohedral pentagon tilings. It is also called MacMahon's net. after Percy Alexander MacMahon and his 1921 publication New Mathematical Pastimes.. PDF p.

An example of such combinatorial objects are the domino tilings of a given region in the plane. In this case, a flip can be performed when two adjacent dominos cover a square: it consists in rotating these dominos by 90 degrees around the center of the square, resulting in a different domino tiling of the same region.

The disproof of Keller's conjecture, for sufficiently high dimensions, has progressed through a sequence of reductions that transform it from a problem in the geometry of tilings into a problem in group theory, and from there into a problem in graph theory. first reformulated Keller's conjecture in terms of factorizations of abelian groups. He shows that, if there is a counterexample to the conjecture, then it can be assumed to be a periodic tiling of cubes with an integer side length and integer vertex positions; thus, in studying the conjecture, it is sufficient to consider tilings of this special form. In this case, the group of integer translations, modulo the translations that preserve the tiling, forms an abelian group, and certain elements of this group correspond to the positions of the tiles.

Conversely, abstract patterns in science, mathematics, or language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature and in art. Visual patterns in nature are often chaotic, never exactly repeating, and often involve fractals. Natural patterns include spirals, meanders, waves, foams, tilings, cracks, and those created by symmetries of rotation and reflection.

The isoperimetric problem, a recurring concept in convex geometry, was studied by the Greeks as well, including Zenodorus. Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied convex polytopes and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, Gaussian curvature, algorithms, tilings and lattices.

Taylor and Socolar remark that the 3D monotile aperiodically tiles three-dimensional space. However the tile does allow tilings with a period, shifting one (non-periodic) two dimensional layer to the next, and so the tile is only "weakly aperiodic". Physical copies of the three-dimensional tile could not be fitted together without allowing reflections, which would require access to four-dimensional space.

Like the uniform polyhedra there are eight uniform tilings that can be based from the regular square tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three topologically distinct forms: square tiling, truncated square tiling, snub square tiling.

In geometry, the floret pentagonal tiling or rosette pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is one of 15 known isohedral pentagon tilings. It is given its name because its six pentagonal tiles radiate out from a central point, like petals on a flower.Five space-filling polyhedra by Guy Inchbald Conway calls it a 6-fold pentille.

In addition to that Regel wrote: "It is mainly Dr. Tiling to whom we owe the introduction of many excellent Siberian plants" "" – . The review of Lange and Gumprecht go far above a normal book review. They add information about Tilings voyage that is not contained in the book they review. The size of the review with 21 full pages is remarkable as well.

See Base 36. The truncated cube and the truncated octahedron are Archimedean solids with 36 edges. The number of domino tilings of a 4×4 checkerboard is 36. Since it is possible to find sequences of 36 consecutive integers such that each inner member shares a factor with either the first or the last member, 36 is an Erdős–Woods number.

The unexpected existence of aperiodic tilings, although not Berger's explicit construction of them, follows from another result proved by Berger: that the so-called domino problem is undecidable, disproving a conjecture of Hao Wang, Berger's advisor. The result is analogous to a 1962 construction used by Kahr, Moore, and Wang, to show that a more constrained version of the domino problem was undecidable.

Both platforms have cream-colored tiles and a pink trim line with "79TH ST" written on it in black sans serif font at regular intervals. These tilings were installed during a 1970s renovation that covered most of the original mosaics and cartouches. Some of these as well as the decorated ceiling beams can still be seen by the fare control areas.

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Fritz Henning Emil Paul Berndt Laves (27 February 1906 – 12 August 1978) was a German crystallographer who served as the president of the German Mineralogical Society from 1956 to 1958. He is the namesake of Laves phases and the Laves tilings; the Laves graph, a highly-symmetrical three-dimensional crystal structure that he studied, was named after him by H. S. M. Coxeter...

As Boris Delaunay proved in 1929, every parallelohedron can be made into a plesiohedron by an affine transformation, but this remains open in higher dimensions, and in three dimensions there also exist other plesiohedra that are not parallelohedra. The tilings of space by plesiohedra have symmetries taking any cell to any other cell, but unlike for the parallelohedra, these symmetries may involve rotations, not just translations.

This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.

If a sequence T_1,T_2,\ldots of tilings of a surface is conformal (K) in the above sense, then there is a conformal structure on the surface and a constant K' depending only on K in which the classical moduli and approximate moduli (from T_i for i sufficiently large) of any given annulus are K'-comparable, meaning that they lie in a single interval [r,K'r].

Area-based visualizations have existed for decades. For example, mosaic plots (also known as Marimekko diagrams) use rectangular tilings to show joint distributions (i.e., most commonly they are essentially stacked column plots where the columns are of different widths). The main distinguishing feature of a treemap, however, is the recursive construction that allows it to be extended to hierarchical data with any number of levels.

The hexagonal tiling honeycomb, {6,3,3}, has hexagonal tiling, {6,3}, facets with vertices on a horosphere. One such facet is shown in as seen in this Poincaré disk model. In H3 hyperbolic space, paracompact regular honeycombs have Euclidean tiling facets and vertex figures that act like finite polyhedra. Such tilings have an angle defect that can be closed by bending one way or the other.

This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.

Originally, the new form of matter was dubbed "Shechtmanite". The term "quasicrystal" was first used in print by Steinhardt and Levine shortly after Shechtman's paper was published. The adjective quasicrystalline had already been in use, but now it came to be applied to any pattern with unusual symmetry. 'Quasiperiodical' structures were claimed to be observed in some decorative tilings devised by medieval Islamic architects.

There are infinitely many (p q 2) triangle group families. This article shows the regular tiling up to p, q = 8, and uniform tilings in 12 families: (7 3 2), (8 3 2), (5 4 2), (6 4 2), (7 4 2), (8 4 2), (5 5 2), (6 5 2) (6 6 2), (7 7 2), (8 6 2), and (8 8 2).

Several properties and common features of the Penrose tilings involve the golden ratio = (1+)/2 (approximately 1.618). This is the ratio of chord lengths to side lengths in a regular pentagon, and satisfies = 1 + 1/. alt= Consequently, the ratio of the lengths of long sides to short sides in the (isosceles) Robinson triangles is :1. It follows that the ratio of long side lengths to short in both kite and dart tiles is also :1, as are the length ratios of sides to the short diagonal in the thin rhomb t, and of long diagonal to sides in the thick rhomb T. In both the P2 and P3 tilings, the ratio of the area of the larger Robinson triangle to the smaller one is :1, hence so are the ratios of the areas of the kite to the dart, and of the thick rhomb to the thin rhomb.

During much of the 20th century, the converse was also taken for granted – until the discovery of quasicrystals in 1982 showed that there are perfectly deterministic tilings that do not possess lattice periodicity. Besides structural order, one may consider charge ordering, spin ordering, magnetic ordering, and compositional ordering. Magnetic ordering is observable in neutron diffraction. It is a thermodynamic entropy concept often displayed by a second-order phase transition.

Walter Borho (born 17 December 1945, in Hamburg) is a German mathematician, who works on algebra and number theory. Borho received his PhD in 1973 from the University of Hamburg under the direction of Ernst Witt with thesis Wesentliche ganze Erweiterungen kommutativer Ringe. He is a professor at the University of Wuppertal. Borho does research on representation theory, Lie algebras, ring theory and also on number theory (amicable numbers) and tilings.

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry. From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

There are 11 circle packings based on the 11 uniform tilings of the plane.[7] In these packings, every circle can be mapped to every other circle by reflections and rotations. The hexagonal gaps can be filled by one circle and the dodecagonal gaps can be filled with 7 circles, creating 3-uniform packings. The truncated trihexagonal tiling with both types of gaps can be filled as a 4-uniform packing.

Although ' was published anonymously the reviewers did know the author and supported his views. At least among the German speaking people "in the know" it was clear who the author of ' was. There must have been intensive contact between author and reviewers previously. Specifically intensive was Tilings exchange with the director of the Botanical Gardens in St Petersburg, Eduard August von Regel with whom he published Florula Ajanensis.

Following Allen H. Brady's initial work of turmites on a triangular grid, hexagonal tilings have also been explored. Much of this work is due to Tim Hutton, and his results are on the Rule Table Repository. He has also considered Turmites in three dimensions, and collected some preliminary results. Allen H. Brady and Tim Hutton have also investigated one-dimensional relative turmites on the integer lattice, which Brady termed flippers.

Acta Crystallogr. A71, 569-582E. Zappa, E.C. Dykeman & R. Twarock (2014) On the subgroup structure of the hyperoctahedral group in six dimensions, Acta Cryst A 70, 417-428 More insights were gained using the "cut and project" method of generating penrose tilings. Her models can be thought of as squashed-down three dimensional pictures of the 6-demicubic honeycomb tiling, a "six-dimensional version" of the three- dimensional Tetrahedral-octahedral honeycomb.

In geometry, isotoxal polyhedra and tilings are defined by the property that they have symmetries taking any edge to any other edge.Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, , p. 371 Polyhedra with this property can also be called "edge-transitive", but they should be distinguished from edge-transitive graphs, where the symmetries are combinatorial rather than geometric. Regular polyhedra are isohedral (face- transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are eight forms.

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and [n,3] Coxeter group symmetry. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Increasing the number of atoms that comprise the carbon skeleton leads to a geometry that increasingly approximates a sphere, and the space enclosed in the carbon "cage" increases. This trend continues with buckyballs or spherical fullerene (C60). Although not a Platonic hydrocarbon, buckminsterfullerene has the shape of a truncated icosahedron, an Archimedean solid. The concept can also be extended to regular Euclidean tilings, with the hexagonal tiling producing graphane.

Karl August Reinhardt (27 January 1895 Frankfurt am Main – 27 April 1941 Berlin) was a German mathematician who discovered the 5 tile-transitive pentagon tilings, solved the odd case of the biggest little polygon problem, and constructed the smoothed octagon conjectured to be the worst-packing point-symmetric planar convex shape. He also gave a partial solution to Hilbert's eighteenth problem by discovering an anisohedral tiling in three dimensions.

Figure 5: A setiset of order 4 using octominoes. Two stages of inflation are shown. The properties of setisets mean that their pieces form substitution tilings, or tessellations in which the prototiles can be dissected or combined so as to yield smaller or larger duplicates of themselves. Clearly, the twin actions of forming still larger and larger copies (known as inflation), or still smaller and smaller dissections (deflation), can be repeated indefinitely.

For 2-dimensional tilings, they can be given by a vertex configuration listing the sequence of faces around every vertex. For example 4.4.4.4 represents a regular tessellation, a square tiling, with 4 squares around each vertex. In general an n-dimensional uniform tessellation vertex figures are define by an (n-1)-polytope with edges labeled with integers, representing the number of sides of the polygonal face at each edge radiating from the vertex.

Knuth's Algorithm X is an algorithm that finds all solutions to an exact cover problem. DLX is the name given to Algorithm X when it is implemented efficiently using Donald Knuth's Dancing Links technique on a computer. The standard exact cover problem can be generalized slightly to involve not only "exactly one" constraints but also "at-most-one" constraints. Finding Pentomino tilings and solving Sudoku are noteworthy examples of exact cover problems.

The chair substitution tiling system. However, the tiles shown below force the chair substitution structure to emerge, and so are themselves aperiodic. Trilobite and Cross tiles enforce the chair substitution structure—they can only admit tilings in which the chair substitution can be discerned and so are aperiodic. The Penrose tiles, and shortly thereafter Amman's several different sets of tiles, were the first example based on explicitly forcing a substitution tiling structure to emerge.

Non-periodic tilings can also be obtained by projection of higher-dimensional structures into spaces with lower dimensionality and under some circumstances there can be tiles that enforce this non-periodic structure and so are aperiodic. The Penrose tiles are the first and most famous example of this, as first noted in the pioneering work of de Bruijn.N. G. de Bruijn, Nederl. Akad. Wetensch. Indag. Math. 43, 39–52, 53–66 (1981).

Locher, 1974. p. 18 His first study of mathematics began with papers by George Pólya and by the crystallographer Friedrich Haag on plane symmetry groups, sent to him by his brother Berend, a geologist. He carefully studied the 17 canonical wallpaper groups and created periodic tilings with 43 drawings of different types of symmetry. From this point on, he developed a mathematical approach to expressions of symmetry in his artworks using his own notation.

Grünbaum's rotationally symmetrical 5-set Venn diagram, 1975 Grünbaum also devised a multi-set generalisation of Venn diagrams. He was an editor and a frequent contributor to Geombinatorics. Grünbaum's classic monograph Convex Polytopes, first published in 1967, became the main textbook on the subject. His monograph Tilings and Patterns, coauthored with G. C. Shephard, helped to rejuvenate interest in this classic field, and has proved popular with nonmathematical audiences, as well as with mathematicians.

An alternate set of tiles, also discovered by Ammann, and labelled "Ammann 4" in Grünbaum and Shephard, consists of two nonconvex right-angle- edged pieces. One consists of two squares overlapping on a smaller square, while the other consists of a large square attached to a smaller square. The diagrams below show the pieces and a portion of the tilings. Image:ammannbeenkerreplace2.svg This is the substitution rule for the alternate tileset. Image:ammannbeenker.

The chair substitution (left) and a portion of a chair tiling (right). In geometry, a chair tiling (or L tiling) is a nonperiodic substitution tiling created from L-tromino prototiles. These prototiles are examples of rep-tiles and so an iterative process of decomposing the L tiles into smaller copies and then rescaling them to their original size can be used to cover patches of the plane. Chair tilings do not possess translational symmetry, i.e.

The mathematical counterpart of physical diffraction is the Fourier transform and the qualitative description of a diffraction picture as 'clear cut' or 'sharp' means that singularities are present in the Fourier spectrum. There are different methods to construct model quasicrystals. These are the same methods that produce aperiodic tilings with the additional constraint for the diffractive property. Thus, for a substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers.

Wythoff is known in combinatorial game theory and number theory for his study of Wythoff's game, whose solution involves the Fibonacci numbers. The Wythoff array, a two-dimensional array of numbers related to this game and to the Fibonacci sequence, is also named after him... In geometry, Wythoff is known for the Wythoff construction of uniform tilings and uniform polyhedra and for the Wythoff symbol used as a notation for these geometric objects.

In this case, the prototile is an elongated irregular nonagon, or nine-sided figure. Because it has no translational symmetries, the Voderberg tiling is technically non-periodic, even though it exhibits an obvious repeating pattern. This tiling was the first spiral tiling to be devised, preceding later work by Branko Grünbaum and Geoffrey C. Shephard in the 1970s. A Voderberg tiling is depicted on the cover of Grünbaum and Shephard's 1987 book Tilings and Patterns..

After more than ten years of coaxing, he agreed to meet various professionals in person, and eventually even went to two conferences and delivered a lecture at each. Afterwards, Ammann dropped out of sight, and died of a heart attack a few years later. News of his death did not reach the research community for a few more years. Five sets of tiles discovered by Ammann were described in Tilings and PatternsB.

By overlaying a square grid of side length c onto the Pythagorean tiling, it may be used to generate a five-piece dissection of two unequal squares of sides a and b into a single square of side c, showing that the two smaller squares have the same area as the larger one. Similarly, overlaying two Pythagorean tilings may be used to generate a six-piece dissection of two unequal squares into a different two unequal squares..

The station was overhauled in the late 1970s. The Metropolitan Transportation Authority (MTA) fixed the station's structure and overhauled its appearance. It refurbished the wall tilings and replaced the old signs and incandescent lighting with modern fixtures. It also fixed the staircases and platform edges. On January 16, 1978, the ex-IRT and ex-BMT stations were connected into a single station complex, eliminating a "double fare" that formerly was required to transfer between the Atlantic Ave.

There is no upper bound on k for k-isohedral tilings by certain tiles that are both type 1 and type 2, and hence neither on the number of tiles in a primitive unit. The wallpaper group symmetry for each tiling is given, with orbifold notation in parentheses. A second lower symmetry group is given if tile chirality exists, where mirror images are considered distinct. These are shown as yellow and green tiles in those cases.

Although there is evidence that some ancient girih tilings used a subdivision rule to draw a two-level pattern, there are no known historic examples that can be repeated an infinite level of times. For example, the pattern used in the spandrel of the Darb-i Imam shrine (see figure) consists only of decagons and bowties, while the subdivision rule uses an elongated hexagon tile alongside these two shapes. Therefore, this design lacks self-similarity between the two levels.

These quasicrystal tilings contain shapes with five-fold symmetry that repeat periodically inbetween other shapes that do not repeat. One way to create quasi-periodic patterns is to create a Penrose tiling. Girih tiles can be subdivided into Penrose tiles called "dart" and "kite", but there is no evidence that this approach was used by medieval artisans. Another way to create quasiperiodic patterns is by subdividing girih tiles repeatedly into smaller tiles using a subdivision rule.

In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}. Conway calls it a deltille, named from the triangular shape of the Greek letter delta (Δ).

Grünbaum and Shephard, section 11.1. An even smaller set of six aperiodic tiles (based on Wang tiles) was discovered by Raphael M. Robinson in 1971. Roger Penrose discovered three more sets in 1973 and 1974, reducing the number of tiles needed to two, and Robert Ammann discovered several new sets in 1977. The aperiodic Penrose tilings can be generated not only by an aperiodic set of prototiles, but also by a substitution and by a cut-and-project method.

Shahar Mozes (שחר מוזס) is an Israeli mathematician. Mozes received in 1991 his doctorate from the Hebrew University of Jerusalem with thesis Actions of Cartan subgroups under the supervision of Hillel Fürstenberg. (doctoral dissertation) At the Hebrew University of Jerusalem, Mozes became in 1993 a senior lecturer, in 1996 associate professor, and in 2002 a full professor. Moses does research on Lie groups and discrete subgroups of Lie groups, geometric group theory, ergodic theory, and aperiodic tilings.

Grünbaum authored over 200 papers, mostly in discrete geometry, an area in which he is known for various classification theorems. He wrote on the theory of abstract polyhedra. His paper on line arrangements may have inspired a paper by N. G. de Bruijn on quasiperiodic tilings (the most famous example of which is the Penrose tiling of the plane). This paper is also cited by the authors of a monograph on hyperplane arrangements as having inspired their research.

Wall tableau of one of Escher's bird tessellations at the Princessehof Ceramics Museum in Leeuwarden Doris Schattschneider identifies eleven strands of mathematical and scientific research anticipated or directly inspired by Escher. These are the classification of regular tilings using the edge relationships of tiles: two- color and two-motif tilings (counterchange symmetry or antisymmetry); color symmetry (in crystallography); metamorphosis or topological change; covering surfaces with symmetric patterns; Escher's algorithm (for generating patterns using decorated squares); creating tile shapes; local versus global definitions of regularity; symmetry of a tiling induced by the symmetry of a tile; orderliness not induced by symmetry groups; the filling of the central void in Escher's lithograph Print Gallery by H. Lenstra and B. de Smit. The Pulitzer Prize-winning 1979 book Gödel, Escher, Bach by Douglas Hofstadter discusses the ideas of self-reference and strange loops, drawing on a wide range of artistic and scientific sources including Escher's art and the music of J. S. Bach. The asteroid 4444 Escher was named in Escher's honor in 1985.

Adrian Fisher is a pioneer, inventor, designer and creator of mazes, puzzles, public art, tessellations, tilings, patterns and networks of many kinds. He is responsible for more than 700 mazes in 42 countries since 1979. Fisher has created 63 mirror mazes, and pioneered the extensive use of thematic chambers within mirror mazes, to achieve Mirror Maze Adventures. He has created 44 hedge mazes, and pioneered the use of Folly Towers, Tunnels, Walk-through Parting Waterfalls and Foaming Fountain Gates in mazes.

In 2007, the physicists Peter Lu and Paul Steinhardt argued that girih from the 15th century resembled quasicrystalline Penrose tilings. "Although they were probably unaware of the mathematical properties and consequences of the construction rule they devised, they did end up with something that would lead to what we understand today to be a quasi-crystal." Elaborate geometric zellige tilework is a distinctive element in Moroccan architecture. Muqarnas vaults are three- dimensional but were designed in two dimensions with drawings of geometrical cells.

A tiling is called periodic when it has periods that shift the tiling in two different directions.General references for this article include , , and . The tiles in the square tiling have only one shape, and it is common for other tilings to have only a finite number of shapes. These shapes are called prototiles, and a set of prototiles is said to admit a tiling or tile the plane if there is a tiling of the plane using only these shapes.

These concepts have been used and adapted by artists from Michelangelo to modern comic book artists. The golden ratio is a particular proportion that has had a controversial role in art. Often claimed to be the most aesthetically pleasing ratio of lengths, it is frequently stated to be incorporated into famous works of art, though the most reliable and unambiguous examples were made deliberately by artists aware of this legend. Tilings, or tessellations, have been used in art throughout history.

In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Projective polyhedra are also referred to as elliptic tessellations or elliptic tilings, referring to the projective plane as (projective) elliptic geometry, by analogy with spherical tiling, a synonym for "spherical polyhedron". However, the term elliptic geometry applies to both spherical and projective geometries, so the term carries some ambiguity for polyhedra.

The dihedral symmetry of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polyhedra, the hosohedra and dihedra which exist as tilings on the sphere. The dihedral symmetry is represented by a fundamental triangle (p 2 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group I2(p) or [n,2], as well as a prismatic Coxeter diagram: . Below are the first five dihedral symmetries: D2 ... D6.

This idea was invented by professor Ben Shneiderman at the University of Maryland Human – Computer Interaction Lab in the early 1990s. Shneiderman and his collaborators then deepened the idea by introducing a variety of interactive techniques for filtering and adjusting treemaps. These early treemaps all used the simple "slice-and-dice" tiling algorithm. Despite many desirable properties (it is stable, preserves ordering, and is easy to implement), the slice-and-dice method often produces tilings with many long, skinny rectangles.

Two hexagons and two triangles alternate around each vertex, and its edges form an infinite arrangement of lines. Its dual is the rhombille tiling. This pattern, and its place in the classification of uniform tilings, was already known to Johannes Kepler in his 1619 book Harmonices Mundi.. The pattern has long been used in Japanese basketry, where it is called kagome. The Japanese term for this pattern has been taken up in physics, where it is called a Kagome lattice.

There are a few constructions of aperiodic tilings known. Some constructions are based on infinite families of aperiodic sets of tiles. Those constructions which have been found are mostly constructed in a few ways, primarily by forcing some sort of non-periodic hierarchical structure. Despite this, the undecidability of the Domino Problem ensures that there must be infinitely many distinct principles of construction, and that in fact, there exist aperiodic sets of tiles for which there can be no proof of their aperiodicity.

Michael Tye (born 1960) is a mosaic artist specialising in the design, fabrication and installation of unique mosaic and decorative tilings. He also works as a community artist and has worked on community mosaic projects throughout South Australia. Starting out as a graphic designer, Michael spent a few years in the printing and advertising industry before moving onto ceramic design. From 1990 to 1996 he ran a pottery studio, making a range of wheelthrown and sculptural ceramic objects, including functional tableware.

A single 30-tetrahedron ring Boerdijk–Coxeter helix within the 600-cell, seen in stereographic projection Regular tetrahedra can be stacked face-to-face in a chiral aperiodic chain called the Boerdijk–Coxeter helix. In four dimensions, all the convex regular 4-polytopes with tetrahedral cells (the 5-cell, 16-cell and 600-cell) can be constructed as tilings of the 3-sphere by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface.

"Du-sum-oh" (a.k.a. "geometry number place") puzzles replace the 3×3 (or R×C) regions of Sudoku with irregular shapes of a fixed size. Bob Harris has proved that it is always possible to create (N − 1)-clue du-sum-ohs on an N×N grid, and has constructed several examples. Johan de Ruiter has proved that for any N>3 there exist polyomino tilings that can not be turned into a Sudoku puzzle with N irregular shapes of size N.

Islamic art makes use of geometric patterns and symmetries in many of its art forms, notably in girih tilings. These are formed using a set of five tile shapes, namely a regular decagon, an elongated hexagon, a bow tie, a rhombus, and a regular pentagon. All the sides of these tiles have the same length; and all their angles are multiples of 36° (π/5 radians), offering fivefold and tenfold symmetries. The tiles are decorated with strapwork lines (girih), generally more visible than the tile boundaries.

"... any finite patch that we choose in a tiling will lie inside a single inflated tile if we continue moving far enough up in the inflation hierarchy. This means that anywhere that tile occurs at that level in the hierarchy, our original patch must also occur in the original tiling. Therefore, the patch will occur infinitely often in the original tiling and, in fact, in every other tiling as well." This shows in particular that the number of distinct Penrose tilings (of any type) is uncountably infinite.

Gethner has two doctorates. She completed her first, a PhD in mathematics from Ohio State University, in 1992; her dissertation, Rational Period Functions For The Modular Group And Related Discrete Groups, was supervised by L. Alayne Parson. She completed a second PhD in computer science from the University of British Columbia in 2002, with a dissertation Computational Aspects of Escher Tilings supervised by Nick Pippenger and David G. Kirkpatrick. Gethner is an associate professor in the Department of Computer Science and Engineering at University of Colorado Denver.

In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle, (p q r), defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles (p q 2). Uniform solutions are constructed by a single generator point with 7 positions within the funamental triangle, the 3 corners, along the 3 edges, and the triangle interior. All vertices exist at the generator, or a reflected copy of it.

101 Conway calls it a 4-fold pentille.John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table) As a 2-dimensional crystal net, it shares a special feature with the honeycomb net. Both nets are examples of standard realization, the notion introduced by M. Kotani and T. Sunada for general crystal nets.T. Sunada, Topological Crystallography ---With a View Towards Discrete Geometric Analysis---, Surveys and Tutorials in the Applied Mathematical Sciences, Vol.

This shows that the Penrose tiling has a scaling self- similarity, and so can be thought of as a fractal. Penrose originally discovered the P1 tiling in this way, by decomposing a pentagon into six smaller pentagons (one half of a net of a dodecahedron) and five half- diamonds; he then observed that when he repeated this process the gaps between pentagons could all be filled by stars, diamonds, boats and other pentagons. By iterating this process indefinitely he obtained one of the two P1 tilings with pentagonal symmetry.

According to Keller's conjecture, any tiling of the plane by congruent squares must include two squares that meet edge-to-edge.The truth of his conjecture for two-dimensional tilings was known already to Keller, but it was since proven false for dimensions eight and above. For a recent survey on results related to this conjecture, see . None of the squares in the Pythagorean tiling meet edge-to-edge, but this fact does not violate Keller's conjecture because the tiles have different sizes, so they are not all congruent to each other.

In the first part of the 20th century, Coxeter and Petrie discovered three infinite structures {4, 6}, {6, 4} and {6, 6}. They called them regular skew polyhedra, because they seemed to satisfy the definition of a regular polyhedron -- all the vertices, edges and faces are alike, all the angles are the same, and the figure has no free edges. Nowadays, they are called infinite polyhedra or apeirohedra. The regular tilings of the plane {4, 4}, {3, 6} and {6, 3} can also be regarded as infinite polyhedra.

The (6,4,2) triangular hyperbolic tiling that inspired M. C. Escher M. C. Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter around 1956 inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's wood engravings Circle Limit I–IV demonstrate this concept between 1958 and 1960, the final one being Circle Limit IV: Heaven and Hell in 1960.Escher's Circle Limit Exploration According to Bruno Ernst, the best of them is Circle Limit III.

"Plan Early Opening of New Arcade," Los Angeles Times, January 9, 1924, page A-10 > The general decorative scheme . . . lends itself to beautification of the > shops[,] which will be framed, as in a picture, by the soft silvery blues > and modulated reds of walls and tilings. Inside the frame, each shop may > devise its own decorative effects. . . . The true spaciousness of the > building cannot be fully realized until one stands on the bridge at the > third floor level, when the immensity of the structure is borne in upon the > eye. . . .

Tessellation Tango, The Mathematical Tourist, Drexel University, retrieved 2012-05-23. It appears in ancient Greek floor mosaics from Delos. and from Italian floor tilings from the 11th century,. although the tiles with this pattern in Siena Cathedral are of a more recent vintage.. In quilting, it has been known since the 1850s as the "tumbling blocks" pattern, referring to the visual dissonance caused by its doubled three-dimensional interpretation... This is a mystery novel, but it also includes a brief description of the tumbling blocks quilt pattern in its front matter.

Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k−1)-polytopes in common. For example, a two- dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes.

Truncations of the cube beyond rectification When "truncation" applies to platonic solids or regular tilings, usually "uniform truncation" is implied, which means truncating until the original faces become regular polygons with twice as many sides as the original form. 320px This sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron. The middle image is the uniform truncated cube; it is represented by a Schläfli symbol t{p,q,...}.

It is also possible to restrict the classes of point sets that may be Danzer sets in other ways than by their densities. In particular, they cannot be the union of finitely many lattices, they cannot be generated by choosing a point in each tile of a substitution tiling (in the same position for each tile of the same type), and they cannot be generated by the cut-and-project method for constructing aperiodic tilings. Therefore, the vertices of the pinwheel tiling and Penrose tiling are not Danzer sets.

Frequently the term aperiodic was just used vaguely to describe the structures under consideration, referring to physical aperiodic solids, namely quasicrystals, or to something non-periodic with some kind of global order. The use of the word "tiling" is problematic as well, despite its straightforward definition. There is no single Penrose tiling, for example: the Penrose rhombs admit infinitely many tilings (which cannot be distinguished locally). A common solution is to try to use the terms carefully in technical writing, but recognize the widespread use of the informal terms.

PNG The relationship between the two tilesets. In addition to the edge arrows in the usual tileset, the matching rules for both tilesets can be expressed by drawing pieces of large arrows at the vertices, and requiring them to piece together into full arrows. Katz has studied the additional tilings allowed by dropping the vertex constraints and imposing only the requirement that the edge arrows match. Since this requirement is itself preserved by the substitution rules, any new tiling has an infinite sequence of "enlarged" copies obtained by successive applications of the substitution rule.

A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two-, three-, four-, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders—for instance, five- fold. Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of natural quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography.

Selected families of uniform tilings are shown below (using the Poincaré disk model for the hyperbolic plane). Three of them – (7 3 2), (5 4 2), and (4 3 3) – and no others, are minimal in the sense that if any of their defining numbers is replaced by a smaller integer the resulting pattern is either Euclidean or spherical rather than hyperbolic; conversely, any of the numbers can be increased (even to infinity) to generate other hyperbolic patterns. Each uniform tiling generates a dual uniform tiling, with many of them also given below.

The Conway operation of dual interchanges faces and vertices. In Archimedean solids and k-uniform tilings alike, the new vertex coincides with the center of each regular face, or the centroid. In the Euclidean (plane) case; in order to make new faces around each original vertex, the centroids must be connected by new edges, each of which must intersect exactly one of the original edges. Since regular polygons have dihedral symmetry, we see that these new centroid- centroid edges must be perpendicular bisectors of the common original edges (e.g.

Robert Ammann (October 1, 1946 – May, 1994) was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings. Ammann–Beenker tiling Ammann attended Brandeis University, but generally did not go to classes, and left after three years. He worked as a programmer for Honeywell. After ten years, his position was eliminated as part of a routine cutback, and Ammann ended up working as a mail sorter for a post office. In 1975, Ammann read an announcement by Martin Gardner of new work by Roger Penrose.

Penrose had discovered two simple sets of aperiodic tiles, each consisting of just two quadrilaterals. Since Penrose was taking out a patent, he wasn't ready to publish them, and Gardner's description was rather vague. Ammann wrote a letter to Gardner, describing his own work, which duplicated one of Penrose's sets, plus a foursome of "golden rhombohedra" that formed aperiodic tilings in space."The Mysterious Mr. Ammann" The Mathematical Intelligencer, September 2004, Volume 26, Issue 4, pp 10–21 More letters followed, and Ammann became a correspondent with many of the professional researchers.

Grünbaum and G.C. Shephard, Tilings and Patterns, Freemann, NY 1986 and later, in collaboration with the authors of the book, he published a paperR.Ammann, B. Grünbaum and G.C. Shephard, Aperiodic Tiles, Discrete Comput Geom 8 (1992),1–25 proving the aperiodicity for four of them. Ammann's discoveries came to notice only after Penrose had published his own discovery and gained priority. In 1981 de Bruijn exposed the cut and project method and in 1984 came the sensational news about Shechtman quasicrystals which promoted the Penrose tiling to fame.

There are few constructions of aperiodic tilings known, even forty years after Berger's groundbreaking construction. Some constructions are of infinite families of aperiodic sets of tiles. Those constructions which have been found are mostly constructed in a few ways, primarily by forcing some sort of non-periodic hierarchical structure. Despite this, the undecidability of the Domino Problem ensures that there must be infinitely many distinct principles of construction, and that in fact, there exist aperiodic sets of tiles for which there can be no proof of their aperiodicity.

Together with the fact that they don't overlap, this implies that the cubes placed in this way tile space. However, the condition that any two clique vertices differ in at least two coordinates implies that no two cubes have a face in common. disproved Keller's conjecture by finding a clique of size 210 in the Keller graph of dimension 10. This clique leads to a non-face-to-face tiling in dimension 10, and copies of it can be stacked (offset by half a unit in each coordinate direction) to produce non-face-to-face tilings in any higher dimension.

Penrose's first tiling uses pentagons and three other shapes: a five- pointed "star" (a pentagram), a "boat" (roughly 3/5 of a star) and a "diamond" (a thin rhombus). To ensure that all tilings are non-periodic, there are matching rules that specify how tiles may meet each other, and there are three different types of matching rule for the pentagonal tiles. Treating these three types as different prototiles gives a set of six prototiles overall. It is common to indicate the three different types of pentagonal tiles using three different colors, as in the figure above right.

Tie and Navette tiling (in red on a Penrose background) The three variants of the Penrose tiling are mutually locally derivable. Selecting some subsets from the vertices of a P1 tiling allows to produce other non-periodic tilings. If the corners of one pentagon in P1 are labeled in succession by 1,3,5,2,4 an unambiguous tagging in all the pentagons is established, the order being either clockwise or counterclockwise. Points with the same label define a tiling by Robinson triangles while points with the numbers 3 and 4 on them define the vertices of a Tie-and-Navette tiling.

It is a polyhedra in a sequence defined by the face configuration V4.6.2n. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any n ≥ 7\. With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors. Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex.

Following this, three chapters concern lattice tilings by polycubes. The question here is to determine, from the shape of the polycube, whether all cubes in the tiling meet face-to-face or, equivalently, whether the lattice of symmetries must be a subgroup of the integer lattice. After a chapter on the general version of this problem, two chapters consider special classes of cross and "semicross"-shaped polycubes, both with regard to tiling and then, when these shapes do not tile, with regard to how densely they can be packed. In three dimensions, this is the notorious tripod packing problem.

In 1994 Mountaz Hascoet and Michel Beaudouin-Lafon invented a "squarifying" algorithm, later popularized by Jarke van Wijk, that created tilings whose rectangles were closer to square. In 1999 Martin Wattenberg used a variation of the "squarifying" algorithm that he called "pivot and slice" to create the first Web-based treemap, the SmartMoney Map of the Market, which displayed data on hundreds of companies in the U.S. stock market. Following its launch, treemaps enjoyed a surge of interest, especially in financial contexts. A third wave of treemap innovation came around 2004, after Marcos Weskamp created the Newsmap, a treemap that displayed news headlines.

The rhombille tiling is the dual of the trihexagonal tiling. It is one of many different ways of tiling the plane by congruent rhombi. Others include a diagonally flattened variation of the square tiling (with translational symmetry on all four sides of the rhombi), the tiling used by the Miura-ori folding pattern (alternating between translational and reflectional symmetry), and the Penrose tiling which uses two kinds of rhombi with 36° and 72° acute angles aperiodically. When more than one type of rhombus is allowed, additional tilings are possible, including some that are topologically equivalent to the rhombille tiling but with lower symmetry.

For points in Euclidean space, a set X is a Meyer set if it is relatively dense and its difference set X − X is uniformly discrete. Equivalently, X is a Meyer set if both X and X − X are Delone. Meyer sets are named after Yves Meyer, who introduced them (with a different but equivalent definition based on harmonic analysis) as a mathematical model for quasicrystals. They include the point sets of lattices, Penrose tilings, and the Minkowski sums of these sets with finite sets.. The Voronoi cells of symmetric Delone sets form space-filling polyhedra called plesiohedra..

Aperiodic tilings were considered as mathematical artefacts until 1984, when physicist Dan Shechtman announced the discovery of a phase of an aluminium- manganese alloy which produced a sharp diffractogram with an unambiguous fivefold symmetry – so it had to be a crystalline substance with icosahedral symmetry. In 1975 Robert Ammann had already extended the Penrose construction to a three-dimensional icosahedral equivalent. In such cases the term 'tiling' is taken to mean 'filling the space'. Photonic devices are currently built as aperiodical sequences of different layers, being thus aperiodic in one direction and periodic in the other two.

The modular curve X(7) is the Klein quartic of genus 3 with 24 cusps. It can be interpreted as a surface with three handles tiled by 24 heptagons, with a cusp at the center of each face. These tilings can be understood via dessins d'enfants and Belyi functions – the cusps are the points lying over ∞ (red dots), while the vertices and centers of the edges (black and white dots) are the points lying over 0 and 1. The Galois group of the covering X(7) → X(1) is a simple group of order 168 isomorphic to PSL(2, 7).

In Circle Limit III, for example, one can see that the number of fishes within a distance of n from the center rises exponentially. The fishes have an equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n. The art of crochet has been used to demonstrate hyperbolic planes (pictured above) with the first being made by Daina Taimiņa, whose book Crocheting Adventures with Hyperbolic Planes won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year. HyperRogue is a roguelike game set on various tilings of the hyperbolic plane.

Gathering for Gardner Celebration of Mind Presenter During 1995 he did research at The Geometry Center, a mathematics research and education center at the University of Minnesota, where he investigated aperiodic tilings of the plane.Chaim Goodman- Strauss: Activities at the Geometry Center University Of Minnesota Goodman- Strauss has been fascinated by patterns and mathematical paradoxes for as long as he can remember. He attended a lecture about the mathematician Georg Cantor when he was 17 and says, "I was already doomed to be a mathematician, but that lecture sealed my fate."The Shape of Everyday Things by Melissa Lutz Blouin.

Almeida–Portela–Pinto's TAMS paper exhibits new tilings determined by circle diffeomorphisms that are low smoothness fixed points of renormalization. Alves–Pinheiro–Pinto's paper in JLMS proved that if a topological conjugacy between multimodal maps is smooth at a point in the expanding set then the conjugacy is smooth in a renormalization interval. Carvalho–Peixoto–Pinheiro–Pinto's TAMS paper makes a clear connection between the otherwise distant concepts of focal decomposition, renormalization and semiclassical physics. Pinto's paper in JDG created new models to study the appearance of sudden social and political disruptions using the replicator equation in the theory of planned behavior.

Inner section of Kepler's Platonic solid model of planetary spacing in the Solar System from Mysterium Cosmographicum (1596) Sacred geometry ascribes symbolic and sacred meanings to certain geometric shapes and certain geometric proportions.dartmouth.edu: Paul Calter, Polygons, Tilings, & Sacred Geometry It is associated with the belief that a god is the geometer of the world. The geometry used in the design and construction of religious structures such as churches, temples, mosques, religious monuments, altars, and tabernacles has sometimes been considered sacred. The concept applies also to sacred spaces such as temenoi, sacred groves, village greens, pagodas and holy wells, and the creation of religious art.

Alice Guionnet is known for her work on large random matrices. In this context, she established principles of large deviations for the empirical measurements of the eigenvalues of large random matrices with Gérard Ben Arous and Ofer Zeitouni, applied the theory of concentration of measure, initiated the rigorous study of matrices with a heavy tail, and obtained the convergence of spectral measurement of non-normal matrices. She developed the analysis of Dyson-Schwinger equations to obtain topological asymptotic expansions, and studied changes in beta-models and random tilings. In collaboration with Alessio Figalli, she introduced the concept of approximate transport to demonstrate the universality of local fluctuations.

A periodic tiling of the plane is the regular repetition of a "unit cell", in the manner of a wallpaper, without any gaps. Such tilings can be seen as a two-dimensional crystal, and because of the crystallographic restriction theorem, the unit cell is restricted to a rotational symmetry of 2-fold, 3-fold, 4-fold, and 6-fold. It is therefore impossible to tile the plane periodically with a figure that has five-fold rotational symmetry, such as a five-pointed star or a decagon. Patterns with infinite perfect quasi- periodic translational order can have crystallographically forbidden rotational symmetries such as pentagonal or decagonal shapes.

Unable to get her own faculty position at Arizona because of the anti-nepotism rules then in place, she and her husband visited Brazil, supported by a Fulbright Scholarship. They then moved to Massachusetts, where she took the faculty position at Smith that she would keep for the rest of her career. She eventually divorced Senechal, and married photographer Stan Sherer in 1989. She retired in 2007; a festival in 2006 honoring her impending retirement included the performance of a musical play that she wrote with The Talking Band member Ellen Maddow, loosely centered around the theme of aperiodic tilings and the life of amateur mathematician Robert Ammann...

Polyominoes tiling the plane have been classified by the symmetries of their tilings and by the number of aspects (orientations) in which the tiles appear in them.Grünbaum and Shephard, section 9.4 The two tiling nonominoes not satisfying the Conway criterion. The study of which polyominoes can tile the plane has been facilitated using the Conway criterion: except for two nonominoes, all tiling polyominoes up to order 9 form a patch of at least one tile satisfying it, with higher-order exceptions more frequent. Several polyominoes can tile larger copies of themselves, and repeating this process recursively gives a rep-tile tiling of the plane.

Conway calls it a kisdeltille,John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table) constructed as a kis operation applied to a triangular tiling (deltille). In Japan the pattern is called asanoha for hemp leaf, although the name also applies to other triakis shapes like the triakis icosahedron and triakis octahedron. It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex. :320px It is one of eight edge tessellations, tessellations generated by reflections across each edge of a prototile..

Conway calls it a kisrhombille for his kis vertex bisector operation applied to the rhombille tiling. More specifically it can be called a 3-6 kisrhombille, to distinguish it from other similar hyperbolic tilings, like 3-7 kisrhombille. The related rhombille tiling becomes the kisrhombille by cutting each rhombic face along its diagonals into four triangular faces It can be seen as an equilateral hexagonal tiling with each hexagon divided into 12 triangles from the center point. (Alternately it can be seen as a bisected triangular tiling divided into 6 triangles, or as an infinite arrangement of lines in six parallel families.) It is labeled V4.6.

Specific ideas from Penrose's work include: the idea that the human mind operates in certain fundamental ways as a quantum computer, espoused in Penrose's The Emperor's New Mind; Platonic realism as a philosophical basis for works of fiction, as in stories from Penrose's The Road to Reality; and the theory of aperiodic tilings, which appear in the Teglon puzzle in the novel. Stephenson also cites as an influence the works of Kurt Gödel and Edmund Husserl, both of whom the character Durand mentions by name in the novel. Much of the Geometers' technology seen in the novel reflects existing scientific concepts. The alien ship moves by means of nuclear pulse propulsion.

In 2007, R. E. Schwartz showed that outer billiards has some unbounded orbits when defined relative to the Penrose Kite, thus answering the original Moser-Neumann question in the affirmative. The Penrose kite is the convex quadrilateral from the kites-and-darts Penrose tilings. Subsequently, Schwartz showed that outer billiards has unbounded orbits when defined relative to any irrational kite. An irrational kite is a quadrilateral with the following property: One of the diagonals of the quadrilateral divides the region into two triangles of equal area and the other diagonal divides the region into two triangles whose areas are not rational multiples of each other.

Two-dimensional examples are helpful in order to get some understanding about the origin of the competition between local rules and geometry in the large. Consider first an arrangement of identical discs (a model for a hypothetical two-dimensional metal) on a plane; we suppose that the interaction between discs is isotropic and locally tends to arrange the disks in the densest way as possible. The best arrangement for three disks is trivially an equilateral triangle with the disk centers located at the triangle vertices. The study of the long range structure can therefore be reduced to that of plane tilings with equilateral triangles.

A family of closed sets called tiles forms a tessellation or tiling of a Euclidean space if their union is the whole space and every two distinct sets in the family have disjoint interiors. A tiling is said to be monohedral if all of the tiles are congruent to each other. Keller's conjecture concerns monohedral tilings in which all of the tiles are hypercubes of the same dimension as the space. As formulates the problem, a cube tiling is a tiling by congruent hypercubes in which the tiles are additionally required to all be translations of each other, without any rotation, or equivalently to have all of their sides parallel to the coordinate axes of the space.

Similarly, reduced the dimension in which a counterexample to the conjecture is known by finding a clique of size 28 in the Keller graph of dimension eight. Subsequently, showed that the Keller graph of dimension seven has a maximum clique of size 124 < 27. Because this is less than 27, the graph-theoretic version of Keller's conjecture is true in seven dimensions. However, the translation from cube tilings to graph theory can change the dimension of the problem, so this result doesn't settle the geometric version of the conjecture in seven dimensions. Finally, a 200-gigabyte computer- assisted proof in 2019 used Keller graphs to establish that the conjecture holds true in seven dimensions.

The puzzle was solved on May 15, 2000, before the first deadline, by two Cambridge mathematicians, Alex Selby and Oliver Riordan. Key to their success was the mathematical rigour with which they approached the problem of determining the tileability of individual pieces and of empty regions within the board. These provided measures of the probability that a given piece could help to fill or 'tile' a given region, and the probability that a given region could be tiled by some combination of pieces. In the search for a solution, these probabilities were used to identify which partial tilings, out of a vast number explored by the computer program, were most likely to lead to a solution.

Wang tiles have recently become a popular tool for procedural synthesis of textures, heightfields, and other large and nonrepeating bidimensional data sets; a small set of precomputed or hand-made source tiles can be assembled very cheaply without too obvious repetitions and without periodicity. In this case, traditional aperiodic tilings would show their very regular structure; much less constrained sets that guarantee at least two tile choices for any two given side colors are common because tileability is easily ensured and each tile can be selected pseudorandomly.. Introduces the idea of using Wang tiles for texture variation, with a deterministic substitution system.. Introduces stochastic tiling.. . Applies Wang Tiles for real-time texturing on a GPU.. . Shows advanced applications.

In dynamical systems, Johnson is known for her work on a conjecture of Hillel Furstenberg on the classification of invariant measures for the action of two independent modular multiplication operations on an interval. In 1998, Johnson and Kathleen Madden won the George Pólya Award for their joint paper on aperiodic tiling, "Putting the Pieces Together: Understanding Robinson's Nonperiodic Tilings". In 2017, Madden, Johnson, and their co-author Ayşe Şahin published the textbook Discovering Discrete Dynamical Systems through the Mathematical Association of America. With Joseph Auslander and Cesar E. Silva she is also the co-editor of Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby (Contemporary Mathematics 678, American Mathematical Society, 2016).

By the early years of the twentieth century, mathematicians had moved on and geometry was little studied. Coxeter's analysis in The Fifty-Nine Icosahedra introduced modern ideas from graph theory and combinatorics into the study of polyhedra, signalling a rebirth of interest in geometry. Coxeter himself went on to enumerate the star uniform polyhedra for the first time, to treat tilings of the plane as polyhedra, to discover the regular skew polyhedra and to develop the theory of complex polyhedra first discovered by Shephard in 1952, as well as making fundamental contributions to many other areas of geometry. In the second part of the twentieth century, Grünbaum published important works in two areas.

His work features mathematical objects and operations including impossible objects, explorations of infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations. Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya, Roger Penrose, Harold Coxeter and crystallographer Friedrich Haag, and conducted his own research into tessellation. Early in his career, he drew inspiration from nature, making studies of insects, landscapes, and plants such as lichens, all of which he used as details in his artworks. He traveled in Italy and Spain, sketching buildings, townscapes, architecture and the tilings of the Alhambra and the Mezquita of Cordoba, and became steadily more interested in their mathematical structure.

Twarock's study of these viruses lead her to believe that there was much more insight into virology that could be gotten from mathematics. Mathematical virology had previously only studied the surfaces of virus, using models that were tilings of the 2-sphere; Twarock hoped to go further than this, to illuminate three-dimensional protein structure and genome packaging. It was known that, using rotations, simple capsid patterns could be "generated" from a single shape by making copies of it and moving them around in ways that preserve the symmetry. Twarock decided to consider adding an outward translation to this generating process, which created a quite complex patterns of points in 3D space.

In the theory of random surfaces, it is also called the harmonic crystal. It is also the starting point for many constructions in quantum field theory, where it is called the Euclidean bosonic massless free field. A key property of the 2-dimensional GFF is conformal invariance, which relates it in several ways to the Schramm-Loewner Evolution, see and . Similarly to Brownian motion, which is the scaling limit of a wide range of discrete random walk models (see Donsker's theorem), the continuum GFF is the scaling limit of not only the discrete GFF on lattices, but of many random height function models, such as the height function of uniform random planar domino tilings, see .

The seven chapters of the book are largely self-contained, and consider different problems combining tessellations and algebra. Throughout the book, the history of the subject as well as the state of the art is discussed, and there are many illustrations. The first chapter concerns a conjecture of Hermann Minkowski that, in any lattice tiling of a Euclidean space by unit hypercubes (a tiling in which a lattice of translational symmetries takes any hypercube to any other hypercube) some two cubes must meet face-to-face. This result was resolved positively by Hajós's theorem in group theory, but a generalization of this question to non-lattice tilings (Keller's conjecture) was disproved shortly before the publication of the book, in part by using similar group-theoretic methods.

There are an infinite number of uniform tilings of the hyperbolic plane by kites, the simplest of which is the deltoidal triheptagonal tiling. Kites and darts in which the two isosceles triangles forming the kite have apex angles of 2π/5 and 4π/5 represent one of two sets of essential tiles in the Penrose tiling, an aperiodic tiling of the plane discovered by mathematical physicist Roger Penrose. Face-transitive self- tesselation of the sphere, Euclidean plane, and hyperbolic plane with kites occurs as uniform duals: for Coxeter group [p,q], with any set of p,q between 3 and infinity, as this table partially shows up to q=6. When p=q, the kites become rhombi; when p=q=4, they become squares.

In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of √2. Conway calls it a kisquadrille,John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table) represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille).

In Britain, The Prince's School of Traditional Arts runs a range of courses in Islamic art including geometry, calligraphy, and arabesque (vegetal forms), tile-making, and plaster carving. Seljuk princes at Kharaghan, Qazvin province, Iran, covered with many different brick patterns like those that inspired Ahmad Rafsanjani to create auxetic materials Computer graphics and computer-aided manufacturing make it possible to design and produce Islamic geometric patterns effectively and economically. Craig S. Kaplan explains and illustrates in his Ph.D. thesis how Islamic star patterns can be generated algorithmically. Two physicists, Peter J. Lu and Paul Steinhardt, attracted controversy in 2007 by claiming that girih designs such as that used on the Darb-e Imam shrine in Isfahan were able to create quasi-periodic tilings resembling those discovered by Roger Penrose in 1973.

It contains as an induced subgraph the Grötzsch graph, the smallest triangle-free four-chromatic graph, and every four-chromatic induced subgraph of the Clebsch graph is a supergraph of the Grötzsch graph. More strongly, every triangle-free four-chromatic graph with no induced path of length six or more is an induced subgraph of the Clebsch graph and an induced supergraph of the Grötzsch graph.. The 5-regular Clebsch graph is the Keller graph of dimension two, part of a family of graphs used to find tilings of high- dimensional Euclidean spaces by hypercubes no two of which meet face-to-face. The 5-regular Clebsch graph can be embedded as a regular map in the orientable manifold of genus 5, forming pentagonal faces; and in the non-orientable surface of genus 6, forming tetragonal faces.

A patch of 25 monotiles, showing the triangular hierarchical structure The Socolar–Taylor tile is a single non-connected tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane (due to the Sierpinski's triangle-like tiling that occurs), with rotations and reflections of the tile allowed.. It is the first known example of a single aperiodic tile, or "einstein". The basic version of the tile is a simple hexagon, with printed designs to enforce a local matching rule, regarding how the tiles may be placed. It is currently unknown whether this rule may be geometrically implemented in two dimensions while keeping the tile a connected set. This is, however, confirmed to be possible in three dimensions, and, in their original paper, Socolar and Taylor suggest a three-dimensional analogue to the monotile.

The tetrakis square tiling : The tetrakis square tiling is the tiling of the Euclidean plane dual to the truncated square tiling. It can be constructed square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of . Conway calls it a kisquadrille,John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table) represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille).

However, despite the growing use of stick-and-ball molecular models, the use of graphical edges or line segments to represent chemical bonds in specific crystals have become popular more recently, and the publication of see encouraged efforts to determine graphical structures of known crystals, to generate crystal nets of as yet unknown crystals, and to synthesize crystals of these novel crystal nets. The coincident expansion of interest in tilings and tessellations, especially those modeling quasicrystals, and the development of modern Nanotechnology, all facilitated by the dramatic increase in computational power, enabled the development of algorithms from computational geometry for the construction and analysis of crystal nets. Meanwhile, the ancient association between models of crystals and tessellations has expanded with Algebraic topology. There is also a thread of interest in the very-large-scale integration (VLSI) community for using these crystal nets as circuit designs.

In Escher's woodcut, the sides of the squares and triangles are formed by arcs of hypercycles, which are not straight in hyperbolic geometry, but which connect smoothly to each other without corners. The points at the centers of the squares, where four fish meet at their fins, form the vertices of an order-8 triangular tiling, while the points where three fish fins meet and the points where three white lines cross together form the vertices of its dual, the octagonal tiling. Similar tessellations by lines of fish may be constructed for other hyperbolic tilings formed by polygons other than triangles and squares, or with more than three white curves at each crossing.. Euclidean coordinates of circles containing the three most prominent white curves in the woodcut may be obtained by calculations in the field of rational numbers extended by the square roots of two and three..

The first specific occurrence of aperiodic tilings arose in 1961, when logician Hao Wang tried to determine whether the Domino Problem is decidable — that is, whether there exists an algorithm for deciding if a given finite set of prototiles admits a tiling of the plane. Wang found algorithms to enumerate the tilesets that cannot tile the plane, and the tilesets that tile it periodically; by this he showed that such a decision algorithm exists if every finite set of prototiles that admits a tiling of the plane also admits a periodic tiling. In 1964 Robert Berger found an aperiodic set of prototiles from which he demonstrated that the tiling problem is in fact not decidable.. This first such set, used by Berger in his proof of undecidability, required 20,426 Wang tiles. Berger later reduced his set to 104, and Hans Läuchli subsequently found an aperiodic set requiring only 40 Wang tiles.

Kim's Living Tones CD features her signature bi- cultural compositions Nong Rock for string quartet and komungo, Tchong for flute and daegum, Piri Quartet for oboe/English horn with three piri(s) and Yoeum for kagok singer and baritone. In 1986 she began to be recognized as a composer when she was commissioned by the Kronos Quartet for her work Linking. Jin Hi Kim is both composer and soloist for the following compositions: Nong Rock for the Kronos Quartet premiered at Alice Tully Hall, Lincoln Center in 1992; Voices of Sigimse for Chamber Music Society of Lincoln Center premiered at the Lincoln Center Summer Festival 1996 with Tan Dun conducting; Eternal Rock (2001) for American Composers Orchestra premiered at Carnegie Hall; and Tilings (2013) for Either/Or Ensemble conducted by Richard Carrick premiered at The Kitchen (NYC). Kim also has introduced Korean tall and colorful barrel drums in the orchestra.

A Penrose tiling In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it periodically (hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student Robert Berger constructed a set of some 20,000 square tiles (now called "Wang tiles") that can tile the plane but not in a periodic fashion.A New Kind of Science As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1976 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane.

The tiling of the quartic by reflection domains is a quotient of the 3-7 kisrhombille. The Klein quartic admits tilings connected with the symmetry group (a "regular map"), and these are used in understanding the symmetry group, dating back to Klein's original paper. Given a fundamental domain for the group action (for the full, orientation-reversing symmetry group, a (2,3,7) triangle), the reflection domains (images of this domain under the group) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group (reflections in the lines of a given fundamental triangle give a set of 3 generating reflections). This tiling is a quotient of the order-3 bisected heptagonal tiling of the hyperbolic plane (the universal cover of the quartic), and all Hurwitz surfaces are tiled in the same way, as quotients.

The Pythagorean tiling is the unique tiling by squares of two different sizes that is both unilateral (no two squares have a common side) and equitransitive (each two squares of the same size can be mapped into each other by a symmetry of the tiling).. Topologically, the Pythagorean tiling has the same structure as the truncated square tiling by squares and regular octagons.. The smaller squares in the Pythagorean tiling are adjacent to four larger tiles, as are the squares in the truncated square tiling, while the larger squares in the Pythagorean tiling are adjacent to eight neighbors that alternate between large and small, just as the octagons in the truncated square tiling. However, the two tilings have different sets of symmetries, because the truncated square tiling is symmetric under mirror reflections whereas the Pythagorean tiling isn't. Mathematically, this can be explained by saying that the truncated square tiling has dihedral symmetry around the center of each tile, while the Pythagorean tiling has a smaller cyclic set of symmetries around the corresponding points, giving it p4 symmetry., p. 42.

An aperiodic sequence generated from tilings by two squares whose side lengths form the golden ratio Although the Pythagorean tiling is itself periodic (it has a square lattice of translational symmetries) its cross sections can be used to generate one-dimensional aperiodic sequences.. In the "Klotz construction" for aperiodic sequences (Klotz is a German word for a block), one forms a Pythagorean tiling with two squares whose sizes are chosen to make the ratio between the two side lengths be an irrational number x. Then, one chooses a line parallel to the sides of the squares, and forms a sequence of binary values from the sizes of the squares crossed by the line: a 0 corresponds to a crossing of a large square and a 1 corresponds to a crossing of a small square. In this sequence, the relative proportion of 0s and 1s will be in the ratio x:1. This proportion cannot be achieved by a periodic sequence of 0s and 1s, because it is irrational, so the sequence is aperiodic.

Hajós's theorem is named after Hajós, and concerns factorizations of Abelian groups into Cartesian products of subsets of their elements.. This result in group theory has consequences also in geometry: Hajós used it to prove a conjecture of Hermann Minkowski that, if a Euclidean space of any dimension is tiled by hypercubes whose positions form a lattice, then some pair of hypercubes must meet face-to-face. Hajós used similar group- theoretic methods to attack Keller's conjecture on whether cube tilings (without the lattice constraint) must have pairs of cubes that meet face to face; his work formed an important step in the eventual disproof of this conjecture.. Hajós's conjecture is a conjecture made by Hajós that every graph with chromatic number contains a subdivision of a complete graph . However, it is now known to be false: in 1979, Paul A. Catlin found a counterexample for ,. and Paul Erdős and Siemion Fajtlowicz later observed that it fails badly for random graphs.. The Hajós construction is a general method for constructing graphs with a given chromatic number, also due to Hajós.. As cited by .