In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker; because all tilings obtained with the tiles are non-periodic, Ammann–Beenker tilings are considered aperiodic tilings. They are one of the five sets of tilings described in Tilings and Patterns; the Ammann–Beenker tilings have many properties similar to the more famous Penrose tilings, most notably: They are nonperiodic, which means that they lack any translational symmetry. Their non-periodicity is implied by their hierarchical structure: the tilings are substitution tilings arising from substitution rules for growing larger and larger patches; this substitution structure implies that: Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. Thus, the infinite tilings all look similar to one another, they are quasicrystalline: implemented as a physical structure an Ammann–Beenker tiling will produce Bragg diffraction.
This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called "deflation" or "inflation." All of this infinite global structure is forced through local matching rules on a pair of tiles, among the simplest aperiodic sets of tiles found, Ammann's A5 set. Various methods to describe the tilings have been proposed: matching rules, substitutions and project schemes and coverings. In 1987 Wang and Kuo announced the discovery of a quasicrystal with octagonal symmetry. 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 and quasiperiodic structures of each of the infinite number of individual Ammann-Beenker tilings. An alternate set of tiles discovered by Ammann, 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 a portion of the tilings. This is the substitution rule for the alternate tileset; 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, 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; 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 can only be deduced from the entire infinite tiling.
The tiling has an extremal property: among the tilings whose rhombuses alternate, the proportion of squares is found to be minimal in the Ammann–Beenker tilings. The Ammann–Beenker tilings are related to the silver ratio and the Pell numbers; the substitution scheme R → R r R. The eigenvalues of the substitution matrix are 1 + 2 and 1 − 2. In the alternate tileset, the long edges have 1 + 2 times longer sides than the short edges. One set of Conway worms, formed by the short and long diagonals of the rhombs, forms the above strings, with r as the short diagonal and R as the long diagonal. Therefore, the Ammann bars form Pell ordered grids; the Ammann bars for the usual tileset. If the bold outer lines are taken to have length 2 2, the bars split the edges into segments of length 1 + 2 and 2 − 1; the Ammann bars for the alternate tileset. Note that the bars for the asymmetric tile extend outside it; the tesseractic honeycomb has an eightfold rotational symmetry, corresponding to an eightfold rotational symmetry of the tesseract.
A rotation matrix representing this symmetry is: A = [ 0 0 0 − 1 1 0 0 0 0 − 1
A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur in architecture and decorative art in textiles and tiles as well as wallpaper. A proof that there were only 17 distinct groups of possible patterns was first carried out by Evgraf Fedorov in 1891 and derived independently by George Pólya in 1924; the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done. The seventeen possible wallpaper groups are listed below in § The seventeen groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are different in style, scale or orientation may belong to the same group. Consider the following examples: Examples A and B have the same wallpaper group.
Example C has a different wallpaper group, called p4g or 4*2. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it looks the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe; the pattern is unchanged. Speaking, a true symmetry only exists in patterns that repeat and continue indefinitely. A set of only, five stripes does not have translational symmetry—when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, classification is applied to finite patterns, small imperfections may be ignored. Sometimes two categorizations are meaningful, one based on shapes alone and one including colors.
When colors are ignored there may be more symmetry. In black and white there are 17 wallpaper groups; the types of transformations that are relevant here are called Euclidean plane isometries. For example: If we shift example B one unit to the right, so that each square covers the square, adjacent to it the resulting pattern is the same as the pattern we started with; this type of symmetry is called a translation. Examples A and C are similar. If we turn example B clockwise by 90°, around the centre of one of the squares, again we obtain the same pattern; this is called a rotation. Examples A and C have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can flip example B across a horizontal axis that runs across the middle of the image; this is called a reflection. Example B has reflections across a vertical axis, across two diagonal axes; the same can be said for A. However, example C is different, it only has reflections in vertical directions, not across diagonal axes.
If we flip across a diagonal line, we do not get the same pattern back. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C. Another transformation is "Glide", a combination of reflection and translation parallel to the line of reflection. Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations. Two such isometry groups are of the same type if they are the same up to an affine transformation of the plane, thus e.g. a translation of the plane does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry. Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those that preserve orientation, it follows from the Bieberbach theorem that all wallpaper groups are different as abstract groups.
2D patterns with double translational symmetry can be categorized according to their symmetry group type. Isometries of the Euclidean plane fall into four categories. Translations, denoted by Tv, where v is a vector in R2; this has the effect of shifting the plane applying displacement vector v. Rotations, denoted by Rc,θ, where c is a point in the plane, θ is the angle of rotation. Reflections, or mirror isometries, denoted by FL, where L is a line in R2.. This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror. Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance; this is a combination of a reflection in the line L and a translation along L by a distance d. The condition
Valérie Berthé is a French mathematician who works as a director of research for the Centre national de la recherche scientifique at the Institut de Recherche en Informatique Fondamentale, a joint project between CNRS and Paris Diderot University. Her research involves symbolic dynamics, combinatorics on words, discrete geometry, numeral systems and fractals. Berthé completed her baccalauréat at age 16, studied at the École Normale Supérieure from 1988 to 1993, she earned a licentiate and master's degree in pure mathematics from Pierre and Marie Curie University in 1989, a Diplôme d'études approfondies from University of Paris-Sud in 1991, completed her agrégation in 1992, was recruited by CNRS in 1993. Continuing her graduate studies, she defended a doctoral thesis in 1994 at the University of Bordeaux 1, her dissertation, Fonctions de Carlitz et automates: Entropies conditionnelles was supervised by Jean-Paul Allouche. She completed a habilitation in 1999, again under the supervision of Allouche, at the University of the Mediterranean Aix-Marseille II.
Berthé is a vice-president of the Société mathématique de France, director of publications for the SMF. She has played an active role in mathématiques. Berthé has been associated with the M. Lothaire pseudonymous mathematical collaboration on combinatorics on words and the Pythias Fogg pseudonymous collaboration on substitution systems. In 2013, Berthé was elevated to the Legion of Honour. Valérie Berthé publications indexed by Google Scholar
Robert Ammann was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings. Ammann attended Brandeis University, but did not go to classes, left after three years, he worked as a programmer for Honeywell. After ten years, his position was eliminated as part of a routine cutback, 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, 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. More letters followed, Ammann became a correspondent with many of the professional researchers, he discovered several new aperiodic tilings, each among the simplest known examples of aperiodic sets of tiles.
He 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. After more than ten years of coaxing, he agreed to meet various professionals in person, even went to two conferences and delivered a lecture at each. Afterwards, Ammann dropped out of sight, 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 Patterns and in collaboration with the authors of the book, he published a paper 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.
But in 1982 Beenker published a similar mathematical explanation for the octagonal case which became known as the Ammann–Beenker tiling. In 1987 Wang 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 F. Gahler proposed an octagonal variant for the Ammann–Beenker tiling Ammann's name became that of the perennial second, it is acknowledged however that Robert Ammann first proposed the construction of rhombic prisms, the three-dimensional model of Shechtman's quasicrystals. Senechal, Marjorie, "The Mysterious Mr. Ammann", The Mathematical Intelligencer, 26: 10–21, doi:10.1007/BF02985414, MR 2104463. Amman tilings and references at the Tilings encyclopedia
Combinatorics is an area of mathematics concerned with counting, both as a means and an end in obtaining results, certain properties of finite structures. It is related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. To understand the scope of combinatorics requires a great deal of further amplification, the details of which are not universally agreed upon. According to H. J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions. Insofar as an area can be described by the types of problems it addresses, combinatorics is involved with the enumeration of specified structures, sometimes referred to as arrangements or configurations in a general sense, associated with finite systems, the existence of such structures that satisfy certain given criteria, the construction of these structures in many ways, optimization, finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other optimality criterion.
Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge in their objectives, their methods, the degree of coherence they have attained." One way to define combinatorics is to describe its subdivisions with their problems and techniques. This is the approach, used below. However, there are purely historical reasons for including or not including some topics under the combinatorics umbrella. Although concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite but discrete setting. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory and geometry, as well as in its many application areas. Many combinatorial questions have been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the twentieth century, however and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right.
One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist. Basic combinatorial concepts and enumerative results appeared throughout the ancient world. In the 6th century BCE, ancient Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc. thus computing all 26 − 1 possibilities. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle. In the Middle Ages, combinatorics continued to be studied outside of the European civilization; the Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, these formulas may have been familiar to Indian mathematicians as early as the 6th century CE.
The philosopher and astronomer Rabbi Abraham ibn Ezra established the symmetry of binomial coefficients, while a closed formula was obtained by the talmudist and mathematician Levi ben Gerson, in 1321. The arithmetical triangle— a graphical diagram showing relationships among the binomial coefficients— was presented by mathematicians in treatises dating as far back as the 10th century, would become known as Pascal's triangle. In Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for algebraic combinatorics. Graph theory enjoyed an explosion of interest at the same time in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject.
In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics; the twelvefold way provides a unified framework for counting permutations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theo
A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, it is used as a pattern for floor tiles. When used for this, it is known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern; this tiling has four-way rotational symmetry around each of its squares. When the ratio of the side lengths of the two squares is an irrational number such as the golden ratio, its cross-sections form aperiodic sequences with a similar recursive structure to the Fibonacci word. Generalizations of this tiling to three dimensions have been studied; the Pythagorean tiling is the unique tiling by squares of two different sizes, both unilateral and equitransitive. 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, 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.
Uniform tilings additionally are required to have tiles that meet edge-to-edge, but if this requirement is relaxed there are eight additional uniform tilings. Four are formed from infinite strips of squares or equilateral triangles, three are formed from equilateral triangles and regular hexagons; the remaining one is the Pythagorean tiling. This tiling is called the Pythagorean tiling because it has been used as the basis of proofs of the Pythagorean theorem by the ninth-century Islamic mathematicians Al-Nayrizi and Thābit ibn Qurra, by the 19th-century British amateur mathematician Henry Perigal. If the sides of the two squares forming the tiling are the numbers a and b the closest distance between corresponding points on congruent squares is c, where c is the length of the hypotenuse of a right triangle having sides a and b. For instance, in the illustration to the left, the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple.
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. Overlaying two Pythagorean tilings may be used to generate a six-piece dissection of two unequal squares into a different two unequal squares. Although the Pythagorean tiling is itself periodic its cross sections can be used to generate one-dimensional aperiodic sequences. In the "Klotz construction" for aperiodic sequences, 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. One chooses a line parallel to the sides of the squares, 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. If x is chosen as the golden ratio, the sequence of 0s and 1s generated in this way has the same recursive structure as the Fibonacci word: it can be split into substrings of the form "01" and "0" and if these two substrings are replaced by the shorter strings "0" and "1" another string with the same structure results. According to Keller's conjecture, any tiling of the plane by congruent squares must include two squares that meet edge-to-edge. 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; the Pythagorean tiling may be generalized to a three-dimensional tiling of Euclidean space by cubes of two different sizes, unilateral and equitransitive. Attila Bölcskei calls this three-dimensional tiling the Rogers filling, he conjectures that, in any dimension greater than three, there is again a unique unilateral and equitransitive way of tiling space by hypercubes of two different sizes.
Burns and Rigby found several prototiles, including the Koch snowflake, that may be used to tile the plane only by using copies of the prototile in two or more different sizes. An earlier paper by Danz
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to a variety of geometries. A periodic tiling has a repeating pattern; some special kinds include regular tilings with regular polygonal tiles all of the same shape, semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons; such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings.
Tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles. Decorative mosaic tilings made of small squared blocks called tesserae were employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made an early documented study of tessellations, he wrote about semiregular tessellations in his Harmonices Mundi. Some two hundred years in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries.
Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Aleksei Shubnikov and Nikolai Belov, Heinrich Heesch and Otto Kienzle. In Latin, tessella is a small cubical piece of stone or glass used to make mosaics; the word "tessella" means "small square". It corresponds to the everyday term tiling, which refers to applications of tessellations made of glazed clay. Tessellation in two dimensions called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules; these rules can be varied. Common ones are that there must be no gaps between tiles, that no corner of one tile can lie along the edge of another; the tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile.
There are only three shapes that can form such regular tessellations: the equilateral triangle and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps. Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. Irregular tessellations can be made from other shapes such as pentagons, polyominoes and in fact any kind of geometric shape; the artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, these can be used to decorate physical surfaces such as church floors. More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries.
These tiles may be any other shapes. Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane; the Conway criterion is a sufficient but not necessary set of rules for deciding if a given shape tiles the plane periodically without reflections: some tiles fail the criterion but still tile the plane. No general rule has been found for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations. Mathematically, tessellations can be extended to spaces other than the Euclidean plane; the Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more dimensions, he further defined the Schläfli symbol notation to make it easy to describe polytopes.
For example, the Schläfli symbol for an equilateral triangle is. The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is. Other methods exist for describing polygonal tilings; when the tessellation