Wythoff symbol

In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. A Wythoff symbol consists of a vertical bar, it represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with Oh symmetry, 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D 2 h symmetry. With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space. In three dimensions, Wythoff's construction begins by choosing a generator point on the triangle. If the distance of this point from each of the sides is non-zero, the point must be chosen to be an equal distance from each edge.

A perpendicular line is dropped between the generator point and every face that it does not lie on. The three numbers in Wythoff's symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, which are π / p, π / q and π / r radians respectively; the triangle is represented with the same numbers, written. The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following: p | q r indicates that the generator lies on the corner p, p q | r indicates that the generator lies on the edge between p and q, p q r | indicates that the generator lies in the interior of the triangle. In this notation the mirrors are labeled by the reflection-order of the opposite vertex; the p, q, r values are listed before the bar. The one impossible symbol | p q r implies the generator point is on all mirrors, only possible if the triangle is degenerate, reduced to a point; this unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored.

The resulting figure has rotational symmetry only. The generator point can either be off each mirror, activated or not; this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. The Wythoff symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. A node is circled. There are seven generator points with each set of p, q, r: There are three special cases: p q | – This is a mixture of p q r | and p q s |, containing only the faces shared by both. | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isn't Wythoff-constructible. There are 4 symmetry classes of reflection on the sphere, three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are listed. Point groups: dihedral symmetry, p = 2, 3, 4 … tetrahedral symmetry octahedral symmetry icosahedral symmetry Euclidean groups: *442 symmetry: 45°-45°-90° triangle *632 symmetry: 30°-60°-90° triangle *333 symmetry: 60°-60°-60° triangleHyperbolic groups: *732 symmetry *832 symmetry *433 symmetry *443 symmetry *444 symmetry *542 symmetry *642 symmetry...

The above symmetry groups only include the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, determine the full set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a fundamental domain, colored by and odd reflections. Selected tilings created by the Wythoff con

Geometry

Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.

While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.

The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.

Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.

He studied the sp

Dual polyhedron

In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.

There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.

R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.

The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.

When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form

Wallpaper group

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

Rhombille tiling

In geometry, the rhombille tiling known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 120 ° angles. Sets of three rhombi meet at their 120° angles and sets of six rhombi meet at their 60° angles; the rhombille tiling can be seen as a subdivision of a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon. This subdivision represents a regular compound tiling, it can be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi. The diagonals of each rhomb are in the ratio 1:√3; this is the dual tiling of the trihexagonal kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, in the face configuration for monohedral tilings it is denoted, it is one of 56 possible isohedral tilings by quadrilaterals, one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.

It is possible to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points with |x + y + z| ≤ 1, in such a way that two vertices are adjacent if and only if the corresponding lattice points are at unit distance from each other, more such that the number of edges in the shortest path between any two vertices of the tiling is the same as the Manhattan distance between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite unit distance graph and partial cube; the rhombille tiling can be interpreted as an isometric projection view of a set of cubes in two different ways, forming a reversible figure related to the Necker Cube. In this context it is known as the "reversible cubes" illusion. In the M. C. Escher artworks Metamorphosis I, Metamorphosis II, Metamorphosis III Escher uses this interpretation of the tiling as a way of morphing between two- and three-dimensional forms. In another of his works, Escher played with the tension between the two-dimensionality and three-dimensionality of this tiling: in it he draws a building that has both large cubical blocks as architectural elements and an upstairs patio tiled with the rhombille tiling.

A human figure descends from the patio past the cubes, becoming more stylized and two-dimensional as he does so. These works involve only a single three-dimensional interpretation of the tiling, but in Convex and Concave Escher experiments with reversible figures more and includes a depiction of the reversible cubes illusion on a flag within the scene; the rhombille tiling is used as a design for parquetry and for floor or wall tiling, sometimes with variations in the shapes of its rhombi. 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; as a quilting pattern it has many other names including cubework, heavenly stairs, Pandora's box. It has been suggested that the tumbling blocks quilt pattern was used as a signal in the Underground Railroad: when slaves saw it hung on a fence, they were to box up their belongings and escape.

See Quilts of the Underground Railroad. In these decorative applications, the rhombi may appear in multiple colors, but are given three levels of shading, brightest for the rhombs with horizontal long diagonals and darker for the rhombs with the other two orientations, to enhance their appearance of three-dimensionality. There is a single known instance of implicit rhombille and trihexagonal tiling in English heraldry – in the Geal/e arms; the rhombille tiling may be viewed as the result of overlaying two different hexagonal tilings, translated so that some of the vertices of one tiling land at the centers of the hexagons of the other tiling. Thus, it can be used to define block cellular automata in which the cells of the automaton are the rhombi of a rhombille tiling and the blocks in alternating steps of the automaton are the hexagons of the two overlaid hexagonal tilings. In this context, it is called the "Q*bert neighborhood", after the video game Q*bert which featured an isometric view of a pyramid of cubes as its playing field.

The Q*bert neighborhood may be used to support universal computation via a simulation of billiard ball computers. In condensed matter physics, the rhombille tiling is known as the dice lattice, diced lattice, or dual kagome lattice, it is one of several repeating structures used to investigate Ising models and related systems of spin interactions in diatomic crystals, it has been studied in percolation theory. The rhombille tiling has *632 symmetry, but vertices can be colored with alternating colors on the inner points leading to a *333 symmetry; the rhombille tiling is the dual of the trihexagonal tiling, as such is part of a set of uniform dual tilings. It is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry, starting from the cube, which can be seen as a rhombic hexahedron where the rhombi are squares; the nth element in this sequence has a face configuration of V3.n.3.n. The rhombille tiling is one of many different ways of tiling the plane by congruent rhombi.

Others include a diagonally flattened variation of the square tiling, the tiling used by the Miura-ori folding pattern, the Penrose tiling which

Tessellation

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

Pythagorean tiling

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