Tiling with rectangles
A tiling with rectangles is a tiling which uses rectangles as its parts. The domino tilings are tilings with rectangles of 1 × 2 side ratio; the tilings with straight polyominoes of shapes such as 1 × 3, 1 × 4 and tilings with polyominoes of shapes such as 2 × 3 fall into this category. Some tiling of rectangles include: The smallest square that can be cut into rectangles, such that all m and n are different integers, is the 11 × 11 square, the tiling uses five rectangles; the smallest rectangle that can be cut into rectangles, such that all m and n are different integers, is the 9 × 13 rectangle, the tiling uses five rectangles. Squaring the square Tessellation Tiling puzzle
In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular n-gonal hosohedron has Schläfli symbol, with each spherical lune having internal angle 2π/n radians. For a regular polyhedron whose Schläfli symbol is, the number of polygonal faces may be found by: N 2 = 4 n 2 m + 2 n − m n The Platonic solids known to antiquity are the only integer solutions for m ≥ 3 and n ≥ 3; the restriction m ≥ 3 enforces. When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons can be represented as spherical lunes, having non-zero area. Allowing m = 2 admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron is represented as n abutting lunes, with interior angles of 2π/n. All these lunes share two common vertices; the digonal faces of a 2n-hosohedron, represents the fundamental domains of dihedral symmetry in three dimensions: Cnv, order 2n.
The reflection domains can be shown as alternately colored lunes as mirror images. Bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry Dnh, order 4n; the tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles. The dual of the n-gonal hosohedron is the n-gonal dihedron; the polyhedron is self-dual, is both a hosohedron and a dihedron. A hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation; the truncated n-gonal hosohedron is the n-gonal prism. In the limit the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation: Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol has each with a vertex figure; the two-dimensional hosotope, is a digon. The term “hosohedron” was coined by H. S. M. Coxeter, derives from the Greek ὅσος “as many”, the idea being that a hosohedron can have “as many faces as desired”.
Polyhedron Polytope McMullen, Peter. S. M. Dover Publications Inc. ISBN 0-486-61480-8 Weisstein, Eric W. "Hosohedron". MathWorld
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 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
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
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows. Uniform polyhedra may be quasi-regular or semi-regular; the faces and vertices need not be convex, so many of the uniform polyhedra are star polyhedra. There are two infinite classes of uniform polyhedra together with 75 others. Infinite classes prisms antiprisms Convex exceptional 5 Platonic solids – regular convex polyhedra 13 Archimedean solids – 2 quasiregular and 11 semiregular convex polyhedra Star exceptional 4 Kepler–Poinsot polyhedra – regular nonconvex polyhedra 53 uniform star polyhedra – 5 quasiregular and 48 semiregularhence 5 + 13 + 4 + 53 = 75. There are many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the great disnub dirhombidodecahedron. Dual polyhedra to uniform polyhedra are face-transitive and have regular vertex figures, are classified in parallel with their dual polyhedron; the dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid.
The concept of uniform polyhedron is a special case of the concept of uniform polytope, which applies to shapes in higher-dimensional space. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces, they define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space. There are some generalizations of the concept of a uniform polyhedron. If the connectedness assumption is dropped we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate we get the so-called degenerate uniform polyhedra; these require a more general definition of polyhedra. Grunbaum gave a rather complicated definition of a polyhedron, while McMullen & Schulte gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization.
Here an abstract polytope is a poset of its "faces" satisfying various condition, a realization is a function from its vertices to some space, the realization is called non-degenerate if any two distinct faces of the abstract polytope have distinct realizations. Some of the ways they can be degenerate are as follows: Hidden faces; some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are not counted as uniform polyhedra. Degenerate compounds; some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. Double covers. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces and vertices, they are not counted as uniform polyhedra. Double faces. There are several polyhedra with doubled faces produced by Wythoff's construction.
Most authors do not remove them as part of the construction. Double edges. Skilling's figure has the property that it has double edges but its faces cannot be written as a union of two uniform polyhedra; the Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Theaetetus, Timaeus of Locri and Euclid. The Etruscans discovered the regular dodecahedron before 500 BC; the cuboctahedron was known by Plato. Archimedes discovered all of the 13 Archimedean solids, his original book on the subject was lost, but Pappus of Alexandria mentioned Archimedes listed 13 polyhedra. Piero della Francesca rediscovered the five truncation of the Platonic solids: truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, truncated icosahedron. Luca Pacioli republished Francesca's work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, drawn by Leonardo da Vinci. Johannes Kepler was the first to publish the complete list of Archimedean solids, in 1619, as well as identified the infinite families of uniform prisms and antiprisms.
Kepler discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot discovered the other two. The set of four were named by Arthur Cayley. Of the remaining 53, Edmund Hess discovered two, Albert Badoureau discovered 36 more, Pitsch independently discovered 18, of which 3 had not been discovered. Together these gave 41 polyhedra; the geometer H. S. M. Coxeter did not publish. M. S. Longuet-Higgins and H. C. Longuet-Higgins independently discovered eleven of these. Lesavre and Mercier rediscovered five of them in 1947. Coxeter, Longuet-Higgins & Miller published the list of uniform polyhedra. Sopov (19
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