1.
Egypt
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Egypt, officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia by a land bridge formed by the Sinai Peninsula. Egypt is a Mediterranean country bordered by the Gaza Strip and Israel to the northeast, the Gulf of Aqaba to the east, the Red Sea to the east and south, Sudan to the south, and Libya to the west. Across the Gulf of Aqaba lies Jordan, and across from the Sinai Peninsula lies Saudi Arabia, although Jordan and it is the worlds only contiguous Afrasian nation. Egypt has among the longest histories of any country, emerging as one of the worlds first nation states in the tenth millennium BC. Considered a cradle of civilisation, Ancient Egypt experienced some of the earliest developments of writing, agriculture, urbanisation, organised religion and central government. One of the earliest centres of Christianity, Egypt was Islamised in the century and remains a predominantly Muslim country. With over 92 million inhabitants, Egypt is the most populous country in North Africa and the Arab world, the third-most populous in Africa, and the fifteenth-most populous in the world. The great majority of its people live near the banks of the Nile River, an area of about 40,000 square kilometres, the large regions of the Sahara desert, which constitute most of Egypts territory, are sparsely inhabited. About half of Egypts residents live in areas, with most spread across the densely populated centres of greater Cairo, Alexandria. Modern Egypt is considered to be a regional and middle power, with significant cultural, political, and military influence in North Africa, the Middle East and the Muslim world. Egypts economy is one of the largest and most diversified in the Middle East, Egypt is a member of the United Nations, Non-Aligned Movement, Arab League, African Union, and Organisation of Islamic Cooperation. Miṣr is the Classical Quranic Arabic and modern name of Egypt. The name is of Semitic origin, directly cognate with other Semitic words for Egypt such as the Hebrew מִצְרַיִם, the oldest attestation of this name for Egypt is the Akkadian
2.
Symmetry
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Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
3.
Architecture
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Architecture is both the process and the product of planning, designing, and constructing buildings and other physical structures. Architectural works, in the form of buildings, are often perceived as cultural symbols. Historical civilizations are often identified with their surviving architectural achievements, Architecture can mean, A general term to describe buildings and other physical structures. The art and science of designing buildings and nonbuilding structures, the style of design and method of construction of buildings and other physical structures. A unifying or coherent form or structure Knowledge of art, science, technology, the design activity of the architect, from the macro-level to the micro-level. The practice of the architect, where architecture means offering or rendering services in connection with the design and construction of buildings. The earliest surviving work on the subject of architecture is De architectura. According to Vitruvius, a building should satisfy the three principles of firmitas, utilitas, venustas, commonly known by the original translation – firmness, commodity. An equivalent in modern English would be, Durability – a building should stand up robustly, utility – it should be suitable for the purposes for which it is used. Beauty – it should be aesthetically pleasing, according to Vitruvius, the architect should strive to fulfill each of these three attributes as well as possible. Leon Battista Alberti, who elaborates on the ideas of Vitruvius in his treatise, De Re Aedificatoria, saw beauty primarily as a matter of proportion, for Alberti, the rules of proportion were those that governed the idealised human figure, the Golden mean. The most important aspect of beauty was, therefore, an inherent part of an object, rather than something applied superficially, Gothic architecture, Pugin believed, was the only true Christian form of architecture. The 19th-century English art critic, John Ruskin, in his Seven Lamps of Architecture, Architecture was the art which so disposes and adorns the edifices raised by men. That the sight of them contributes to his health, power. For Ruskin, the aesthetic was of overriding significance and his work goes on to state that a building is not truly a work of architecture unless it is in some way adorned. For Ruskin, a well-constructed, well-proportioned, functional building needed string courses or rustication, but suddenly you touch my heart, you do me good. I am happy and I say, This is beautiful, le Corbusiers contemporary Ludwig Mies van der Rohe said Architecture starts when you carefully put two bricks together. The notable 19th-century architect of skyscrapers, Louis Sullivan, promoted an overriding precept to architectural design, function came to be seen as encompassing all criteria of the use, perception and enjoyment of a building, not only practical but also aesthetic, psychological and cultural
4.
Decorative arts
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The decorative arts are arts or crafts concerned with the design and manufacture of beautiful objects that are also functional. It includes interior design, but not usually architecture, the decorative arts are often categorized in opposition to the fine arts, namely, painting, drawing, photography, and large-scale sculpture, which generally have no function other than to be seen. The distinction between the decorative and the arts has essentially arisen from the post-Renaissance art of the West. This distinction is less meaningful when considering the art of other cultures and periods. For example, Islamic art in many periods and places consists entirely of the arts, often using geometric and plant forms. The distinction between decorative and fine arts is not very useful for appreciating Chinese art, and neither is it for understanding Early Medieval art in Europe, large-scale wall-paintings were much less regarded, crudely executed, and rarely mentioned in contemporary sources. They were probably seen as a substitute for mosaic, which for this period must be viewed as a fine art. The term ars sacra is sometimes used for medieval Christian art done in metal, ivory, textiles, illuminated manuscripts have a much higher survival rate, especially in the hands of the church, as there was little value in the materials and they were easy to store. Most European art during the Middle Ages had been produced under a different set of values. The lower status given to works of art in contrast to fine art narrowed with the rise of the Arts and Crafts Movement. This aesthetic movement of the half of the 19th century was born in England and inspired by William Morris. The movement represented the beginning of an appreciation of the decorative arts throughout Europe. Many converts, both professional artists ranks and from among the intellectual class as a whole, helped spread the ideas of the movement. The influence of the Arts and Crafts Movement led to the arts being given a greater appreciation and status in society. Until the enactment of the Copyright Act 1911 only works of art had been protected from unauthorised copying. The 1911 Act extended the definition of a work to include works of artistic craftsmanship
5.
Textile
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A textile or cloth is a flexible material consisting of a network of natural or artificial fibres. Yarn is produced by spinning raw fibres of wool, flax, cotton, hemp, Textiles are formed by weaving, knitting, crocheting, knotting, or felting. The words fabric and cloth are used in textile assembly trades as synonyms for textile, however, there are subtle differences in these terms in specialized usage. Textile refers to any material made of interlacing fibres, a fabric is a material made through weaving, knitting, spreading, crocheting, or bonding that may be used in production of further goods. Cloth may be used synonymously with fabric but is often a piece of fabric used for a specific purpose. The word textile is from Latin, from the adjective textilis, meaning woven, from textus, the word cloth derives from the Old English clað, meaning a cloth, woven or felted material to wrap around one, from Proto-Germanic kalithaz. The discovery of dyed flax fibres in a cave in the Republic of Georgia dated to 34,000 BCE suggests textile-like materials were made even in prehistoric times. The production of textiles is a craft whose speed and scale of production has been altered almost beyond recognition by industrialization, however, for the main types of textiles, plain weave, twill, or satin weave, there is little difference between the ancient and modern methods. Textiles have an assortment of uses, the most common of which are for clothing and for such as bags. In the household they are used in carpeting, upholstered furnishings, window shades, towels, coverings for tables, beds, and other flat surfaces, in the workplace they are used in industrial and scientific processes such as filtering. Textiles are used in traditional crafts such as sewing, quilting. Textiles for industrial purposes, and chosen for other than their appearance, are commonly referred to as technical textiles. Technical textiles include textile structures for applications, medical textiles, geotextiles, agrotextiles. In all these applications stringent performance requirements must be met, woven of threads coated with zinc oxide nanowires, laboratory fabric has been shown capable of self-powering nanosystems using vibrations created by everyday actions like wind or body movements. Fashion designers commonly rely on textile designs to set their fashion collections apart from others, armani, the late Gianni Versace, and Emilio Pucci can be easily recognized by their signature print driven designs. Textiles can be made from many materials and these materials come from four main sources, animal, plant, mineral, and synthetic. In the past, all textiles were made from natural fibres, including plant, animal, in the 20th century, these were supplemented by artificial fibres made from petroleum. Textiles are made in various strengths and degrees of durability, from the finest gossamer to the sturdiest canvas, microfibre refers to fibres made of strands thinner than one denier
6.
Tile
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A tile is a manufactured piece of hard-wearing material such as ceramic, stone, metal, or even glass, generally used for covering roofs, floors, walls, showers, or other objects such as tabletops. Alternatively, tile can sometimes refer to units made from lightweight materials such as perlite, wood. In another sense, a tile is a tile or similar object. The word is derived from the French word tuile, which is, in turn, from the Latin word tegula, Tiles are often used to form wall and floor coverings, and can range from simple square tiles to complex mosaics. Tiling stone is marble, onyx, granite or slate. Thinner tiles can be used on walls than on floors, which require more durable surfaces that will resist impacts, the earliest evidence of glazed brick is the discovery of glazed bricks in the Elamite Temple at Chogha Zanbil, dated to the 13th century BC. Glazed and colored bricks were used to make low reliefs in Ancient Mesopotamia, most famously the Ishtar Gate of Babylon, now reconstructed in Berlin. Mesopotamian craftsmen were imported for the palaces of the Persian Empire such as Persepolis, tiling was used in the second century by the Sinhalese kings of ancient Sri Lanka, using smoothed and polished stone laid on floors and in swimming pools. Historians consider the techniques and tools for tiling as well advanced, evidenced by the fine workmanship, tiling from this period can be seen Ruwanwelisaya and Kuttam Pokuna in the city of Anuradhapura. The Achaemenid Empire decorated buildings with glazed tiles, including Darius the Greats palace at Susa. The succeeding Sassanid Empire used tiles patterned with geometric designs, flowers, plants, birds and human beings, early Islamic mosaics in Iran consist mainly of geometric decorations in mosques and mausoleums, made of glazed brick. Typical turquoise tiling becomes popular in 10th-11th century and is used mostly for Kufic inscriptions on mosque walls, seyyed Mosque in Isfahan, Dome of Maraqeh and the Jame Mosque of Gonabad are among the finest examples. The dome of Jame Atiq Mosque of Qazvin is also dated to this period, the golden age of Persian tilework began during the reign the Timurid Empire. In the moraq technique, single-color tiles were cut into small geometric pieces, after hardening, these panels were assembled on the walls of buildings. But the mosaic was not limited to flat areas, Tiles were used to cover both the interior and exterior surfaces of domes. Prominent Timurid examples of this include the Jame Mosque of Yazd, Goharshad Mosque, the Madrassa of Khan in Shiraz. Other important tile techniques of time include girih tiles, with their characteristic white girih. Mihrabs, being the points of mosques, were usually the places where most sophisticated tilework was placed
7.
Wallpaper
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It is usually sold in rolls and is put onto a wall using wallpaper paste. Wallpapers can come plain as lining paper, textured, with a repeating pattern design, or, much less commonly today. Wallpaper printing techniques include surface printing, gravure printing, silk screen-printing, rotary printing, Wallpaper is made in long rolls, which are hung vertically on a wall. The number of times the pattern repeats horizontally across a roll does not matter for this purpose, a single pattern can be issued in several different colorways. The worlds most expensive wallpaper, Les Guerres DIndependence, was priced at £24,896.50 for a set of 32 panels, the wallpaper was designed by Zuber in France and is very popular in the United States. The main historical techniques are, hand-painting, woodblock printing, stencilling, the first three all date back to before 1700. Wallpaper, using the technique of woodcut, gained popularity in Renaissance Europe amongst the emerging gentry. The social elite continued to hang large tapestries on the walls of their homes and these tapestries added color to the room as well as providing an insulating layer between the stone walls and the room, thus retaining heat in the room. However, tapestries were extremely expensive and so only the rich could afford them. Less well-off members of the elite, unable to buy tapestries due either to prices or wars preventing international trade, turned to wallpaper to brighten up their rooms. Prints were very often pasted to walls, instead of being framed and hung, and the largest sizes of prints, some important artists made such pieces - notably Albrecht Dürer, who worked on both large picture prints and also ornament prints - intended for wall-hanging. The largest picture print was The Triumphal Arch commissioned by the Holy Roman Emperor Maximilian I, very few samples of the earliest repeating pattern wallpapers survive, but there are a large number of old master prints, often in engraving of repeating or repeatable decorative patterns. These are called ornament prints and were intended as models for wallpaper makers, England and France were leaders in European wallpaper manufacturing. Among the earliest known samples is one found on a wall from England and is printed on the back of a London proclamation of 1509, without any tapestry manufacturers in England, English gentry and aristocracy alike turned to wallpaper. During the Protectorate under Oliver Cromwell, the manufacture of wallpaper, in 1712, during the reign of Queen Anne, a wallpaper tax was introduced which was not abolished until 1836. By the mid-eighteenth century, Britain was the wallpaper manufacturer in Europe. However this trade was disrupted in 1755 by the Seven Years War and later the Napoleonic Wars. In 1748 the British Ambassador to Paris decorated his salon with blue flock wallpaper, in the 1760s the French manufacturer Jean-Baptiste Réveillon hired designers working in silk and tapestry to produce some of the most subtle and luxurious wallpaper ever made
8.
Mathematical proof
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In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
9.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
10.
Evgraf Fedorov
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Evgraf Stepanovich Fedorov was a Russian mathematician, crystallographer and mineralogist. Fedorov was born in the Russian city of Orenburg into a family of engineers, the family later moved to Saint Petersburg. From the age of fifteen he was interested in the theory of polytopes. He was a graduate of the Gorny Institute, which he joined at the age of 26. He contributed to the identification of conditions under which a group of Euclidean motions must have a subgroup whose vectors span the Euclidean space. His best-known result is his 1891 proof that there are only 17 possible wallpaper groups which can tile a Euclidean plane and this was then proved independently by George Pólya in 1924. The proof that the list of groups was complete only came after the much harder case of space groups had been settled. In 1895, he became a professor of geology at the Moscow Agricultural Institute, Fedorov died from pneumonia in 1919 during the Russian Civil War in Petrograd, RSFSR. His first book, Basics of Polytopes, was finished in 1879 and it offers a classification of polytopes and derives Fedorov polytopes, congruent polytopes that can completely fill space. He wrote the classic The Symmetry of Regular Systems of Figures in 1891, the same year the equivalent results were presented by German mathematician Schönflies. He published his classic work The Theodolite Method in Mineralogy and Petrography in 1893, group theory List of Russian material scientists Fedorov session 2010. Small universal stage after Federow for mineralogical microscopy, Berlin approx,1900 Universal stage after Federow for mineralogical microscopy, Berlin approx. 1925 Large microscope after Brandão-Leiß with integrated universal stage after Fedorow, Berlin approx
11.
Symmetry group
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In abstract algebra, the symmetry group of an object is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, the objects may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be more precise by specifying what is meant by image or pattern. For symmetry of objects, one may also want to take their physical composition into account. The group of isometries of space induces an action on objects in it. The symmetry group is also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the figure invariant is called its symmetry group. The proper symmetry group of an object is equal to its symmetry group if. The proper symmetry group is then a subgroup of the orthogonal group SO. A discrete symmetry group is a group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances, the group of all symmetries of a sphere O is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups, for example, two 3D figures have mirror symmetry, but with respect to different mirror planes. Two 3D figures have 3-fold rotational symmetry, but with respect to different axes, two 2D patterns have translational symmetry, each in one direction, the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also involved in continuous symmetries. A figure with this group is non-drawable and up to arbitrarily fine detail homogeneous. The group generated by all translations, this group cannot be the group of a pattern, it would be homogeneous
12.
Frieze group
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A frieze group is a mathematical concept used to classify designs on two-dimensional surfaces that are repetitive in one direction, according to the symmetries of the pattern. Such patterns occur frequently in architecture and decorative art, the mathematical study of such patterns reveals that exactly seven types of symmetry can occur. Frieze groups are two-dimensional line groups, having repetition in only one direction, formally, a frieze group is a class of infinite discrete symmetry groups of patterns on a strip, hence a class of groups of isometries of the plane, or of a strip. There are seven frieze groups, listed in the summary table, many authors present the frieze groups in a different order. Thus there are two degrees of freedom for group 1, three for groups 2,3, and 4, and four for groups 5,6, and 7. For two of the seven groups the symmetry groups are singly generated, for four they have a pair of generators. A symmetry group in frieze group 1,2,3, a symmetry group in frieze group 4 or 6 is a subgroup of a symmetry group in the last frieze group with half the translational distance. This last frieze group contains the groups of the simplest periodic patterns in the strip. Therefore, in a way, this frieze group contains the largest symmetry groups, the inclusion of the discrete condition is to exclude the group containing all translations, and groups containing arbitrarily small translations. Even apart from scaling and shifting, there are many cases. The inclusion of the condition is to exclude groups that have no translations. The group consisting of the identity and reflection in the horizontal axis, each of these subgroups is the symmetry group of a frieze pattern, and sample patterns are shown in Fig.1. The seven different groups correspond to the 7 infinite series of point groups in three dimensions, with n = ∞. As we have seen, up to isomorphism, there are four groups, the groups can be classified by their type of two-dimensional grid or lattice. The lattice being oblique means that the second direction need not be orthogonal to the direction of repeat, symmetry groups in one dimension Line group Rod group Wallpaper group Space group There exist software graphic tools that create 2D patterns using frieze groups. Usually, the pattern is updated automatically in response to edits of the original strip. Kali, a free and open source application for wallpaper, frieze. Kali, free downloadable Kali for Windows and Mac Classic, tess, a nagware tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups, as well as Heesch tilings
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Space group
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In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct, Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space. In crystallography, space groups are called the crystallographic or Fedorov groups. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography, in 1879 Leonhard Sohncke listed the 65 space groups whose elements preserve the orientation. More accurately, he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were really the same, the space groups in 3 dimensions were first enumerated by Fedorov, and shortly afterwards were independently enumerated by Schönflies. The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies, burckhardt describes the history of the discovery of the space groups in detail. The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, the combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries. The elements of the space group fixing a point of space are rotations, reflections, the identity element, the translations form a normal abelian subgroup of rank 3, called the Bravais lattice. There are 14 possible types of Bravais lattice, the quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups. Translation is defined as the moves from one point to another point. A glide plane is a reflection in a plane, followed by a parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, the latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of the cell, the glides occur in two directions simultaneously, i. e. the same glide plane can be called b or c, a or b. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb, in 1992, it was suggested to use symbol e for such planes. The symbols for five groups have been modified, A screw axis is a rotation about an axis. These are noted by a number, n, to describe the degree of rotation, the degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So,21 is a rotation followed by a translation of 1/2 of the lattice vector
14.
Tahiti
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Tahiti is the largest island in the Windward group of French Polynesia, this overseas collectivity of the French Republic is sometimes referred to as a French overseas country. The island was formed from volcanic activity and is high and mountainous with surrounding coral reefs, the population is 183,645 inhabitants, making it the most populous island of French Polynesia and accounting for 68.5 percent of its total population. Tahiti is the economic, cultural and political centre of French Polynesia, the capital, Papeete, is located on the northwest coast with the only international airport in the region, Faaā International Airport, situated 5 km from the town centre. Tahiti was originally settled by Polynesians between 300 and 800 CE and they represent about 70 percent of the islands population with the rest made up of Europeans, Chinese and those of mixed heritage. The island was part of the Kingdom of Tahiti until its annexation by France in 1880, when it was proclaimed a colony of France, French is the only official language although the Tahitian language is widely spoken. Tahiti is the highest and largest island in French Polynesia lying close to Moorea island and it is located 4,400 kilometres south of Hawaii,7,900 km from Chile and 5,700 km from Australia. The island is 45 km across at its widest point and covers an area of 1,045 km2, the highest peak is Mont Orohena. Mount Roonui, or Mount Ronui in the southeast rises to 1,332 m, the island consists of two roughly round portions centred on volcanic mountains and connected by a short isthmus named after the small town of Taravao, situated there. The northwestern portion is known as Tahiti Nui, while the much smaller portion is known as Tahiti Iti or Taiarapū. Tahiti Nui is heavily populated along the coast, especially around the capital, the interior of Tahiti Nui is almost entirely uninhabited. Tahiti Iti has remained isolated, as its half is accessible only to those travelling by boat or on foot. The rest of the island is encircled by a road which cuts between the mountains and the sea. A scenic and winding road climbs past dairy farms and citrus groves with panoramic views. Tahitis landscape features lush rainforests and many rivers and waterfalls, including the Papenoo River on the side. November to April is the wet season, the wettest month of which is January with 13.2 in of rain in Papeetē, August is the driest with 1.9 inches. The average temperature ranges between 21 and 31 °C with little seasonal variation, the lowest and highest temperatures recorded in Papeete are 16 and 34 °C, respectively. The first Tahitians arrived from Southeast Asia in about 200 BCE and this hypothesis of an emigration from Southeast Asia is supported by a number of linguistic, biological and archaeological proofs. For example, the languages of Fiji and Polynesia all belong to the same Oceanic sub-group, Fijian-Polynesian, in 1769, for instance, James Cook mentions a great traditional ship in Tahiti that was 33 m long, and could be propelled by sail or paddles
15.
Nineveh
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Nineveh was an ancient Assyrian city of Upper Mesopotamia, located on the outskirts of Mosul in modern-day northern Iraq. It is on the bank of the Tigris River, and was the capital of the Neo-Assyrian Empire. It is also a name for the half of Mosul which lies on the eastern bank of the Tigris in the modern day. Its ruins are across the river from the major city of Mosul. The two main tells, or mound-ruins, within the walls are Kouyunjik, the Northern Palace, large amounts of Assyrian sculpture and other artifacts have been excavated and are now located in museums around the world. Site remains suffered in the 2010s from the occupation of the area by ISIS, Iraqi forces recaptured the area in January 2017. The English placename Nineveh comes from Latin Ninive and Septuagint Greek Nineuḗ under influence of the Biblical Hebrew Nīnewēh, the original meaning of the name is unclear but may have referred to a patron goddess. The cuneiform for Ninâ is a fish within a house and this may have simply intended Place of Fish or may have indicated a goddess associated with fish or the Tigris, possibly originally of Hurrian origin. The city was said to be devoted to the Ishtar of Nineveh. The city was known as Ninii or Ni in Ancient Egyptian, Ninuwa in Mari, Ninawa in Aramaic, ܢܸܢܘܵܐ in Syriac. Nabī Yūnus is the Arabic for Prophet Jonah, Kouyunjik was, according to Layard, a Turkish name, and it was known as Armousheeah by the Arabs, and is thought to have some connection with the Kara Koyunlu dynasty. This whole extensive space is now one immense area of ruins overlaid in parts by new suburbs of the city of Mosul, Nineveh was one of the oldest and greatest cities in antiquity. The area was settled as early as 6000 BC and, by 3000 BC, had become an important religious center for the Mesopotamian goddess Ishtar, the early city was constructed on a fault line and, consequently, suffered damage from a number of earthquakes. One such event destroyed the first temple of Ishtar, which was rebuilt in 2260 BC by the Akkadian king Manishtushu. Texts from the Hellenistic period later offered an eponymous Ninus as the founder of Nineveh, the regional influence of Nineveh became particularly pronounced during the archaeological period known as Ninevite 5, or Ninevite V. This period is defined primarily by the pottery that is found widely throughout northern Mesopotamia. Also, for the northern Mesopotamian region, the Early Jezirah chronology has been developed by archaeologists, according to this regional chronology, Ninevite 5 is equivalent to the Early Jezirah I–II period. Ninevite 5 was preceded by the Late Uruk period, Ninevite 5 pottery is roughly contemporary to the Early Transcaucasian culture ware, and the Jemdet Nasr ware
16.
Assyria
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Assyria was a major Mesopotamian East Semitic-speaking kingdom and empire of the ancient Near East and the Levant. Centered on the Tigris in Upper Mesopotamia, the Assyrians came to rule powerful empires at several times. Assyria is named after its capital, the ancient city of Aššur. In the 25th and 24th centuries BC, Assyrian kings were pastoral leaders, Assyria can also refer to the geographic region or heartland where Assyria, its empires and the Assyrian people were centered. The indigenous modern Eastern Aramaic-speaking Assyrian Christian ethnic minority in northern Iraq, north east Syria, southeast Turkey, in prehistoric times, the region that was to become known as Assyria was home to a Neanderthal culture such as has been found at the Shanidar Cave. The earliest Neolithic sites in Assyria were the Jarmo culture c.7100 BC and Tell Hassuna, during the 3rd millennium BC, a very intimate cultural symbiosis developed between the Sumerians and the Akkadians throughout Mesopotamia, which included widespread bilingualism. The influence of Sumerian on Akkadian, and vice versa, is evident in all areas, from lexical borrowing on a scale, to syntactic, morphological. This has prompted scholars to refer to Sumerian and Akkadian in the third millennium BC as a sprachbund and it is highly likely that the city was named in honour of its patron Assyrian god with the same name. The city of Aššur, together with a number of other Assyrian cities, however it is likely that they were initially Sumerian-dominated administrative centres. In the late 26th century BC, Eannatum of Lagash, then the dominant Sumerian ruler in Mesopotamia, similarly, in c. the early 25th century BC, Lugal-Anne-Mundu the king of the Sumerian state of Adab lists Subartu as paying tribute to him. Of the early history of the kingdom of Assyria, little is known, in the Assyrian King List, the earliest king recorded was Tudiya. According to Georges Roux he would have lived in the mid 25th century BC, Tudiya was succeeded on the list by Adamu, the first known reference to the Semitic name Adam and then a further thirteen rulers. The earliest kings, such as Tudiya, who are recorded as kings who lived in tents, were independent semi-nomadic pastoralist rulers and these kings at some point became fully urbanised and founded the city state of Ashur in the mid 21st century BC. During the Akkadian Empire, the Assyrians, like all the Mesopotamian Semites, became subject to the dynasty of the city state of Akkad, the Akkadian Empire founded by Sargon the Great claimed to encompass the surrounding four quarters. Assyrian rulers were subject to Sargon and his successors, and the city of Ashur became an administrative center of the Empire. On those tablets, Assyrian traders in Burushanda implored the help of their ruler, Sargon the Great, the name Hatti itself even appears in later accounts of his grandson, Naram-Sin, campaigning in Anatolia. Assyrian and Akkadian traders spread the use of writing in the form of the Mesopotamian cuneiform script to Asia Minor, the Akkadian Empire was destroyed by economic decline and internal civil war, followed by attacks from barbarian Gutian people in 2154 BC. The rulers of Assyria during the period between c.2154 BC and 2112 BC once again fully independent, as the Gutians are only known to have administered southern Mesopotamia
17.
Porcelain
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Porcelain /ˈpɔːrsəlᵻn, ˈpɔːrslᵻn/ is a ceramic material made by heating materials, generally including kaolin, in a kiln to temperatures between 1,200 and 1,400 °C. Porcelain was first developed in China around 2,000 years ago, then spread to other East Asian countries, and finally Europe. It combines well with both glazes and paint, and can be modelled very well, allowing a range of decorative treatments in tablewares, vessels. It also has uses in technology and industry. The European name, porcelain in English, come from the old Italian porcellana because of its resemblance to the translucent surface of the shell, Porcelain is also referred to as china or fine china in some English-speaking countries, as it was first seen in imports from China. Porcelain has been described as being completely vitrified, hard, impermeable, white or artificially coloured, translucent, however, the term porcelain lacks a universal definition and has been applied in a very unsystematic fashion to substances of diverse kinds which have only certain surface-qualities in common. Terms such as porcellaneous or near-porcelain may be used in such cases, a high proportion of modern porcelain is made of the variant bone china. Kaolin is the material from which porcelain is made, even though clay minerals might account for only a small proportion of the whole. The word paste is an old term for both the unfired and fired material, a more common terminology these days for the unfired material is body, for example, when buying materials a potter might order an amount of porcelain body from a vendor. The composition of porcelain is highly variable, but the mineral kaolinite is often a raw material. Other raw materials can include feldspar, ball clay, glass, bone ash, steatite, quartz, petuntse, the clays used are often described as being long or short, depending on their plasticity. Long clays are cohesive and have high plasticity, short clays are cohesive and have lower plasticity. Clays used for porcelain are generally of lower plasticity and are shorter than many other pottery clays and they wet very quickly, meaning that small changes in the content of water can produce large changes in workability. Thus, the range of content within which these clays can be worked is very narrow. The following section provides information on the methods used to form, decorate, finish, glaze. Many types of glaze, such as the iron-containing glaze used on the wares of Longquan, were designed specifically for their striking effects on porcelain. Porcelain wares may be decorated under the glaze using pigments that include cobalt and copper or over the glaze using coloured enamels. Like many earlier wares, modern porcelains are often biscuit-fired at around 1,000 °C, coated with glaze, another early method is once-fired where the glaze is applied to the unfired body and the two fired together in a single operation
18.
China
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China, officially the Peoples Republic of China, is a unitary sovereign state in East Asia and the worlds most populous country, with a population of over 1.381 billion. The state is governed by the Communist Party of China and its capital is Beijing, the countrys major urban areas include Shanghai, Guangzhou, Beijing, Chongqing, Shenzhen, Tianjin and Hong Kong. China is a power and a major regional power within Asia. Chinas landscape is vast and diverse, ranging from forest steppes, the Himalaya, Karakoram, Pamir and Tian Shan mountain ranges separate China from much of South and Central Asia. The Yangtze and Yellow Rivers, the third and sixth longest in the world, respectively, Chinas coastline along the Pacific Ocean is 14,500 kilometers long and is bounded by the Bohai, Yellow, East China and South China seas. China emerged as one of the worlds earliest civilizations in the basin of the Yellow River in the North China Plain. For millennia, Chinas political system was based on hereditary monarchies known as dynasties, in 1912, the Republic of China replaced the last dynasty and ruled the Chinese mainland until 1949, when it was defeated by the communist Peoples Liberation Army in the Chinese Civil War. The Communist Party established the Peoples Republic of China in Beijing on 1 October 1949, both the ROC and PRC continue to claim to be the legitimate government of all China, though the latter has more recognition in the world and controls more territory. China had the largest economy in the world for much of the last two years, during which it has seen cycles of prosperity and decline. Since the introduction of reforms in 1978, China has become one of the worlds fastest-growing major economies. As of 2016, it is the worlds second-largest economy by nominal GDP, China is also the worlds largest exporter and second-largest importer of goods. China is a nuclear weapons state and has the worlds largest standing army. The PRC is a member of the United Nations, as it replaced the ROC as a permanent member of the U. N. Security Council in 1971. China is also a member of numerous formal and informal multilateral organizations, including the WTO, APEC, BRICS, the Shanghai Cooperation Organization, the BCIM, the English name China is first attested in Richard Edens 1555 translation of the 1516 journal of the Portuguese explorer Duarte Barbosa. The demonym, that is, the name for the people, Portuguese China is thought to derive from Persian Chīn, and perhaps ultimately from Sanskrit Cīna. Cīna was first used in early Hindu scripture, including the Mahābhārata, there are, however, other suggestions for the derivation of China. The official name of the state is the Peoples Republic of China. The shorter form is China Zhōngguó, from zhōng and guó and it was then applied to the area around Luoyi during the Eastern Zhou and then to Chinas Central Plain before being used as an occasional synonym for the state under the Qing
19.
Orbifold notation
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Groups representable in this notation include the point groups on the sphere, the frieze groups and wallpaper groups of the Euclidean plane, and their analogues on the hyperbolic plane. e. All translations which occur are assumed to form a subgroup of the group symmetries being described. The symbol ×, which is called a miracle and represents a topological crosscap where a pattern repeats as an image without crossing a mirror line. A string written in boldface represents a group of symmetries of Euclidean 3-space, a string not written in boldface represents a group of symmetries of the Euclidean plane, which is assumed to contain two independent translations. By abuse of language, we say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way, the exceptional symbol o indicates that there are precisely two linearly independent translations. An orbifold symbol is called if it is not one of the following, p, pq, *p, *pq, for p, q>=2. An object is chiral if its symmetry group contains no reflections, the corresponding orbifold is orientable in the chiral case and non-orientable otherwise. The Euler characteristic of an orbifold can be read from its Conway symbol, as follows. Each feature has a value, n without or before an asterisk counts as n −1 n n after an asterisk counts as n −12 n asterisk, subtracting the sum of these values from 2 gives the Euler characteristic. If the sum of the values is 2, the order is infinite. Indeed, Conways Magic Theorem indicates that the 17 wallpaper groups are exactly those with the sum of the feature values equal to 2, otherwise, the order is 2 divided by the Euler characteristic. The following groups are isomorphic, 1* and *1122 and 221 *22 and *221 2* and this is because 1-fold rotation is the empty rotation. The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a dimension to the object which does not add or spoil symmetry. The bullet is added on one- and two-dimensional groups to imply the existence of a fixed point, thus the discrete symmetry groups in one dimension are *•, *1•, ∞• and *∞•. Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the Cartesian product of the object, on Three-dimensional Orbifolds and Space Groups. Contributions to Algebra and Geometry,42, 475-507,2001, J. H. Conway, D. H. Huson. The Orbifold Notation for Two-Dimensional Groups, structural Chemistry,13, 247-257, August 2002
20.
Translational symmetry
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In geometry, a translation slides a thing by a, Ta = p + a. In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation, discrete translational symmetry is invariant under discrete translation. More precisely it must hold that ∀ δ A f = A, laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noethers theorem, space translational symmetry of a system is equivalent to the momentum conservation law. Translational symmetry of a means that a particular translation does not change the object. Fundamental domains are e. g. H + a for any hyperplane H for which a has an independent direction. This is in 1D a line segment, in 2D an infinite strip, Note that the strip and slab need not be perpendicular to the vector, hence can be narrower or thinner than the length of the vector. In spaces with higher than 1, there may be multiple translational symmetry. For each set of k independent translation vectors the symmetry group is isomorphic with Zk, in particular the multiplicity may be equal to the dimension. This implies that the object is infinite in all directions, in this case the set of all translations forms a lattice. The absolute value of the determinant of the matrix formed by a set of vectors is the hypervolume of the n-dimensional parallelepiped the set subtends. This parallelepiped is a region of the symmetry, any pattern on or in it is possible. E. g. in 2D, instead of a and b we can take a. In general in 2D, we can take pa + qb and ra + sb for integers p, q, r and this ensures that a and b themselves are integer linear combinations of the other two vectors. If not, not all translations are possible with the other pair, each pair a, b defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fully defines the whole object, without further symmetry, this parallelogram is a fundamental domain. The vectors a and b can be represented by complex numbers, for two given lattice points, equivalence of choices of a third point to generate a lattice shape is represented by the modular group, see lattice. With rotational symmetry of order two of the pattern on the tile we have p2, the rectangle is a more convenient unit to consider as fundamental domain than a parallelogram consisting of part of a tile and part of another one
21.
Translation
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Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. While interpreting—the facilitating of oral or sign-language communication between users of different languages—antedates writing, translation began only after the appearance of written literature, there exist partial translations of the Sumerian Epic of Gilgamesh into Southwest Asian languages of the second millennium BCE. Translators always risk inappropriate spill-over of source-language idiom and usage into the target-language translation, on the other hand, spill-overs have imported useful source-language calques and loanwords that have enriched the target languages. Indeed, translators have helped substantially to shape the languages into which they have translated, because of the laboriousness of translation, since the 1940s engineers have sought to automate translation or to mechanically aid the human translator. The rise of the Internet has fostered a world-wide market for services and has facilitated language localization. Translation studies systematically study the theory and practice of translation, the English word translation derives from the Latin word translatio, which comes from trans, across + ferre, to carry or to bring. Thus translatio is a carrying across or a bringing across, in this case, the Germanic languages and some Slavic languages have calqued their words for the concept of translation on translatio. The Romance languages and the remaining Slavic languages have derived their words for the concept of translation from an alternative Latin word, traductio, the Ancient Greek term for translation, μετάφρασις, has supplied English with metaphrase — as contrasted with paraphrase. Metaphrase corresponds, in one of the more recent terminologies, to formal equivalence, nevertheless, metaphrase and paraphrase may be useful as ideal concepts that mark the extremes in the spectrum of possible approaches to translation. Discussions of the theory and practice of translation reach back into antiquity, the ancient Greeks distinguished between metaphrase and paraphrase. Literally graceful, it were an injury to the author that they should be changed, Dryden cautioned, however, against the license of imitation, i. e. of adapted translation, When a painter copies from the life. He has no privilege to alter features and lineaments, despite occasional theoretical diversity, the actual practice of translation has hardly changed since antiquity. The grammatical differences between languages and free-word-order languages have been no impediment in this regard. The particular syntax characteristics of a source language are adjusted to the syntactic requirements of the target language. When a target language has lacked terms that are found in a language, translators have borrowed those terms. However, due to shifts in ecological niches of words, an etymology is sometimes misleading as a guide to current meaning in one or the other language. For example, the English actual should not be confused with the cognate French actuel, the Polish aktualny, the Swedish aktuell, the translators role as a bridge for carrying across values between cultures has been discussed at least since Terence, the 2nd-century-BCE Roman adapter of Greek comedies. The translators role is, however, by no means a passive, mechanical one, the main ground seems to be the concept of parallel creation found in critics such as Cicero
22.
Euclidean plane isometry
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In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types, translations, rotations, reflections, the set of Euclidean plane isometries forms a group under composition, the Euclidean group in two dimensions. It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections, informally, a Euclidean plane isometry is any way of transforming the plane without deforming it. For example, suppose that the Euclidean plane is represented by a sheet of transparent plastic sitting on a desk, examples of isometries include, Shifting the sheet one inch to the right. Rotating the sheet by ten degrees around some marked point, turning the sheet over to look at it from behind. Notice that if a picture is drawn on one side of the sheet, then turning the sheet over. These are examples of translations, rotations, and reflections respectively, there is one further type of isometry, called a glide reflection. However, folding, cutting, or melting the sheet are not considered isometries, neither are less drastic alterations like bending, stretching, or twisting. An isometry of the Euclidean plane is a transformation of the plane. That is, it is a map M, R2 → R2 such that for any points p and q in the plane, d = d and it can be shown that there are four types of Euclidean plane isometries. The line L is called the axis or the associated mirror. The combination of rotations about the origin and reflections about a line through the origin is obtained with all orthogonal matrices forming orthogonal group O, in the case of a determinant of −1 we have, R0, θ =. Which is a reflection in the x-axis followed by a rotation by an angle θ, or equivalently, reflection in a parallel line corresponds to adding a vector perpendicular to it. A translation can be seen as a composite of two parallel reflections, rotations, denoted by Rc, θ, where c is a point in the plane, and θ is the angle of rotation. In terms of coordinates, rotations are most easily expressed by breaking them up into two operations, first, a rotation around the origin is given by R0, θ =. These matrices are the matrices, with determinant 1. They form the orthogonal group SO. A rotation around c can be accomplished by first translating c to the origin, then performing the rotation around the origin, and finally translating the origin back to c
23.
Rotation
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A rotation is a circular movement of an object around a center of rotation. A three-dimensional object always rotates around a line called a rotation axis. If the axis passes through the center of mass, the body is said to rotate upon itself. A rotation about a point, e. g. the Earth about the Sun, is called a revolution or orbital revolution. The axis is called a pole, mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two, a rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion, the axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question then the body is said to orbit, there is no fundamental difference between a “rotation” and an “orbit” and or spin. The key distinction is simply where the axis of the rotation lies and this distinction can be demonstrated for both “rigid” and “non rigid” bodies. If a rotation around a point or axis is followed by a rotation around the same point/axis. The reverse of a rotation is also a rotation, thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis and that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the rotations are known as yaw, pitch. This terminology is used in computer graphics. In astronomy, rotation is an observed phenomenon. Stars, planets and similar bodies all spin around on their axes, the rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features and this rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the equator
24.
Reflection (mathematics)
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In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points, this set is called the axis or plane of reflection. The image of a figure by a reflection is its image in the axis or plane of reflection. For example the image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b, a reflection is an involution, when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term reflection is used for a larger class of mappings from a Euclidean space to itself. Such isometries have a set of fixed points that is an affine subspace, for instance a reflection through a point is an involutive isometry with just one fixed point, the image of the letter p under it would look like a d. This operation is known as a central inversion, and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation, other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term reflection means reflection in a hyperplane, a figure that does not change upon undergoing a reflection is said to have reflectional symmetry. Some mathematicians use flip as a synonym for reflection, in a plane geometry, to find the reflection of a point drop a perpendicular from the point to the line used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure, step 2, construct circles centered at A′ and B′ having radius r. P and Q will be the points of intersection of two circles. Point Q is then the reflection of point P through line AB, the matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1,1,1. The product of two matrices is a special orthogonal matrix that represents a rotation. Every rotation is the result of reflecting in an number of reflections in hyperplanes through the origin. Thus reflections generate the group, and this result is known as the Cartan–Dieudonné theorem. Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes, in general, a group generated by reflections in affine hyperplanes is known as a reflection group. The finite groups generated in this way are examples of Coxeter groups, note that the second term in the above equation is just twice the vector projection of v onto a
25.
Linear independence
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These concepts are central to the definition of dimension. A vector space can be of finite-dimension or infinite-dimension depending on the number of independent basis vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a space is linearly dependent are central to determining a basis for a vector space. Thus, v1 is shown to be a combination of the remaining vectors. This implies that no vector in the set can be represented as a combination of the remaining vectors in the set. In other words, a set of vectors is independent if the only representations of 0 → as a linear combination of its vectors is the trivial representation in which all the scalars ai are zero. In order to allow the number of linearly independent vectors in a space to be countably infinite. More generally, let V be a space over a field K. A set X of elements of V is linearly independent if the corresponding family x∈X is linearly independent. Equivalently, a family is dependent if a member is in the span of the rest of the family. The trivial case of the empty family must be regarded as independent for theorems to apply. A set of vectors which is independent and spans some vector space. For example, the space of all polynomials in x over the reals has the subset as a basis. A geographic example may help to clarify the concept of linear independence, a person describing the location of a certain place might say, It is 3 miles north and 4 miles east of here. This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space, the person might add, The place is 5 miles northeast of here. Although this last statement is true, it is not necessary, in this example the 3 miles north vector and the 4 miles east vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third 5 miles northeast vector is a combination of the other two vectors, and it makes the set of vectors linearly dependent, that is, one of the three vectors is unnecessary
26.
Translation (geometry)
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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv. If T is a translation, then the image of a subset A under the function T is the translate of A by T, the translate of A by Tv is often written A + v. In a Euclidean space, any translation is an isometry, the set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. When considering spacetime, a change of time coordinate is considered to be a translation, for example, the Galilean group and the Poincaré group include translations with respect to time
27.
Glide reflection
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In 2-dimensional geometry, a glide reflection is a type of opposite isometry of the Euclidean plane, the composition of a reflection in a line and a translation along that line. A single glide is represented as frieze group p11g, a glide reflection can be seen as a limiting rotoreflection, where the rotation becomes a translation. It can also be given a Schoenflies notation as S2∞, Coxeter notation as, the combination of a reflection in a line and a translation in a perpendicular direction is a reflection in a parallel line. However, a reflection cannot be reduced like that. Thus the effect of a combined with any translation is a glide reflection. These are the two kinds of indirect isometries in 2D, for example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it. It fixes a system of parallel lines, the isometry group generated by just a glide reflection is an infinite cyclic group. In the case of reflection symmetry, the symmetry group of an object contains a glide reflection. If that is all it contains, this type is frieze group p11g, example pattern with this symmetry group, Frieze group nr.6 is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a product of Z and C2. Example pattern with this group, A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach. For any symmetry group containing some glide reflection symmetry, the vector of any glide reflection is one half of an element of the translation group. This corresponds to wallpaper group pg, with additional symmetry it occurs also in pmg, pgg, if there are also true reflection lines in the same direction then they are evenly spaced between the glide reflection lines. A glide reflection line parallel to a true reflection line already implies this situation and this corresponds to wallpaper group cm. With additional symmetry it occurs also in cmm, p3m1, p31m, p4m, in 3D the glide reflection is called a glide plane. It is a reflection in a combined with a translation parallel to the plane. In the Euclidean plane 3 of 17 wallpaper groups require glide reflection generators, p2gg has orthogonal glide reflections and 2-fold rotations. Cm has parallel mirrors and glides, and pg has parallel glides, Glide symmetry can be observed in nature among certain fossils of the Ediacara biota, the machaeridians, and certain palaeoscolecid worms
28.
Rotational symmetry
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Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. With the modified notion of symmetry for vector fields the symmetry group can also be E+, for symmetry with respect to rotations about a point we can take that point as origin. These rotations form the orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m =3 this is the rotation group SO, for chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space, because of Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. Note that 1-fold symmetry is no symmetry, the notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
29.
Orientation (vector space)
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In linear algebra, the notion of orientation makes sense in arbitrary finite dimension. In this setting, the orientation of a basis is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple rotation. As a result, in the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed, the orientation on a real vector space is the arbitrary choice of which ordered bases are positively oriented and which are negatively oriented. In the three-dimensional Euclidean space, right-handed bases are typically declared to be positively oriented, a vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called unoriented. Let V be a real vector space and let b1. It is a result in linear algebra that there exists a unique linear transformation A, V → V that takes b1 to b2. The bases b1 and b2 are said to have the same orientation if A has positive determinant, the property of having the same orientation defines an equivalence relation on the set of all ordered bases for V. If V is non-zero, there are two equivalence classes determined by this relation. An orientation on V is an assignment of +1 to one equivalence class, every ordered basis lives in one equivalence class or another. Thus any choice of an ordered basis for V determines an orientation. For example, the basis on Rn provides a standard orientation on Rn. Any choice of an isomorphism between V and Rn will then provide an orientation on V. The ordering of elements in a basis is crucial, two bases with a different ordering will differ by some permutation. They will have the same/opposite orientations according to whether the signature of this permutation is ±1 and this is because the determinant of a permutation matrix is equal to the signature of the associated permutation. Similarly, let A be a linear mapping of vector space Rn to Rn. This mapping is orientation-preserving if its determinant is positive, a zero-dimensional vector space has only a single point, the zero vector. Consequently, the basis of a zero-dimensional vector space is the empty set ∅. Therefore, there is an equivalence class of ordered bases, namely
30.
Euclidean vector
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In mathematics, physics, and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra, a Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by A B →. A vector is what is needed to carry the point A to the point B and it was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics, the velocity and acceleration of a moving object, many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their magnitude and direction can still be represented by the length, the mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the system include pseudovectors and tensors. The concept of vector, as we know it today, evolved gradually over a period of more than 200 years, about a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence, working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially he realized an equivalence relation on the pairs of points in the plane, the term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s and a 3-dimensional vector. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments, grassmanns work was largely neglected until the 1870s. Peter Guthrie Tait carried the standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇, in 1878 Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product and this approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwells Treatise on Electricity and Magnetism, the first half of Gibbss Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibbs lectures, in physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a line segment, or arrow
31.
Displacement (vector)
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A displacement is a vector that is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a line from the initial position to the final position of the point. The velocity then is distinct from the speed which is the time rate of change of the distance traveled along a specific path. The velocity may be defined as the time rate of change of the position vector. For motion over an interval of time, the displacement divided by the length of the time interval defines the average velocity. In dealing with the motion of a body, the term displacement may also include the rotations of the body. In this case, the displacement of a particle of the body is called linear displacement, for a position vector s that is a function of time t, the derivatives can be computed with respect to t. These derivatives have common utility in the study of kinematics, control theory, vibration sensing and other sciences, by extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the displacement function. Such higher-order terms are required in order to represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering. The fourth order derivative is called jounce
32.
Rotation (mathematics)
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Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a space that preserves at least one point. It can describe, for example, the motion of a body around a fixed point. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude, mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group, for example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations, the rotation group is a Lie group of rotations about a fixed point. This fixed point is called the center of rotation and is identified with the origin. The rotation group is a point stabilizer in a group of motions. For a particular rotation, The axis of rotation is a line of its fixed points and they exist only in n >2. The plane of rotation is a plane that is invariant under the rotation, unlike the axis, its points are not fixed themselves. The axis and the plane of a rotation are orthogonal, a representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory, rotations of spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as affine rotations, whereas the latter are vector rotations, see the article below for details. A motion of a Euclidean space is the same as its isometry, but a rotation also has to preserve the orientation structure. The improper rotation term refers to isometries that reverse the orientation, in the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise the identity component. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point, there are no non-trivial rotations in one dimension. In two dimensions, only a single angle is needed to specify a rotation about the origin – the angle of rotation that specifies an element of the circle group. The rotation is acting to rotate an object counterclockwise through an angle θ about the origin, composition of rotations sums their angles modulo 1 turn, which implies that all two-dimensional rotations about the same point commute
33.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0
34.
Crystallographic restriction theorem
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The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold, crystals are modeled as discrete lattices, generated by a list of independent finite translations. Because discreteness requires that the spacings between lattice points have a bound, the group of rotational symmetries of the lattice at any point must be a finite group. The strength of the theorem is not all finite groups are compatible with a discrete lattice, in any dimension. The special cases of 2D and 3D are most heavily used in applications, a rotation symmetry in dimension 2 or 3 must move a lattice point to a succession of other lattice points in the same plane, generating a regular polygon of coplanar lattice points. We now confine our attention to the plane in which the symmetry acts, now consider an 8-fold rotation, and the displacement vectors between adjacent points of the polygon. If a displacement exists between any two points, then that same displacement is repeated everywhere in the lattice. So collect all the edge displacements to begin at a lattice point. The edge vectors become radial vectors, and their 8-fold symmetry implies a regular octagon of lattice points around the collection point, but this is impossible, because the new octagon is about 80% as large as the original. The significance of the shrinking is that it is unlimited, the same construction can be repeated with the new octagon, and again and again until the distance between lattice points is as small as we like, thus no discrete lattice can have 8-fold symmetry. The same argument applies to any rotation, for k greater than 6. A shrinking argument also eliminates 5-fold symmetry, Consider a regular pentagon of lattice points. If it exists, then we can take every other edge displacement and assemble a 5-point star, the vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, but about 60% smaller than the original. The existence of quasicrystals and Penrose tilings shows that the assumption of a translation is necessary. And without the discrete lattice assumption, the construction not only fails to reach a contradiction. Thus 5-fold rotational symmetry cannot be eliminated by an argument missing either of those assumptions, Consider two lattice points A and B separated by a translation vector r. Consider an angle α such that a rotation of angle α about any point is a symmetry of the lattice. Rotating about point B by α maps point A to a new point A. Similarly, since both rotations mentioned are symmetry operations, A and B must both be lattice points
35.
Primitive cell
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The concept is used particularly in describing crystal structure in two and three dimensions, though it makes sense in all dimensions. A lattice can be characterized by the geometry of its primitive cell, the primitive cell is a primitive unit. A primitive unit is a section of the tiling that generates the whole tiling using only translations, the primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller and that is, for a point in the lattice r, the arrangement of points appears the same from r′ = r + T→ as from r. Since the primitive cell is defined by the primitive axes a→1, a→2, a→3, a primitive cell is considered to contain exactly one lattice point. A Wigner–Seitz cell is a primitive cell centered on the lattice point it contains. This is a type of Voronoi cell, the Wigner–Seitz cell of the reciprocal lattice in momentum space is called the Brillouin zone. A 2-dimensional primitive cell is a parallelogram, which in special cases may have orthogonal angles, or equal lengths, a crystal can be categorized by its lattice and the atoms that lie in a primitive cell. A cell will fill all the space without leaving gaps by repetition of crystal translation operations. A 3-dimensional primitive cell is a parallelepiped, which in special cases may have orthogonal angles, or equal lengths, Wigner–Seitz cell Bravais lattice Wallpaper group Space group
36.
John Horton Conway
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John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey, Conway was born in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at an early age, his mother has recalled that he could recite the powers of two when he was four years old. By the age of eleven his ambition was to become a mathematician, after leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, who was a terribly introverted adolescent in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person and he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport. Having solved the problem posed by Davenport on writing numbers as the sums of fifth powers. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos and he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University, Conway is especially known for the invention of the Game of Life, one of the early examples of a cellular automaton. His initial experiments in that field were done with pen and paper, since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, and articles. It is a staple of recreational mathematics, there is an extensive wiki devoted to curating and cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, at times Conway has said he hates the game of life–largely because it has come to overshadow some of the other deeper and more important things he has done. Nevertheless, the game did help launch a new branch of mathematics, the Game of Life is now known to be Turing complete. Conways career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner, when Gardner featured Conways Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, for instance, he discussed Conways game of Sprouts, Hackenbush, and his angel and devil problem. In the September 1976 column he reviewed Conways book On Numbers and Games, Conway is widely known for his contributions to combinatorial game theory, a theory of partisan games. This he developed with Elwyn Berlekamp and Richard Guy, and with them also co-authored the book Winning Ways for your Mathematical Plays and he also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, as well as philosophers football and he developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, and Conways soldiers
37.
Orbifold
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In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold. It is a space with an orbifold structure. The underlying space locally looks like the quotient space of a Euclidean space under the action of a finite group. The definition of Thurston will be described here, it is the most widely used and is applicable in all cases, mathematically, orbifolds arose first as surfaces with singular points long before they were formally defined. In 3-manifold theory, the theory of Seifert fiber spaces, initiated by Seifert, in geometric group theory, post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces. In string theory, the orbifold has a slightly different meaning. The main example of underlying space is a quotient space of a manifold under the properly discontinuous action of an infinite group of diffeomorphisms with finite isotropy subgroups. In particular this applies to any action of a group, thus a manifold with boundary carries a natural orbifold structure. Similarly the quotient space of a manifold by a proper action of S1 carries the structure of an orbifold. Orbifold structure gives a natural stratification by open manifolds on its underlying space and it should be noted that one topological space can carry many different orbifold structures. For example, consider the orbifold O associated with a space of the 2-sphere along a rotation by π, it is homeomorphic to the 2-sphere. It is possible to adopt most of the characteristics of manifolds to orbifolds, in the above example, the orbifold fundamental group of O is Z2 and its orbifold Euler characteristic is 1. The structure of an orbifold encodes not only that of the quotient space, which need not be a manifold. An n-dimensional orbifold is a Hausdorff topological space X, called the space, with a covering by a collection of open sets Ui. φj·ψij = φi the gluing maps are unique up to composition with group elements, note that the orbifold structure determines the isotropy subgroup of any point of the orbifold up to isomorphism, it can be computed as the stabilizer of the point in any orbifold chart. More generally, attached to a covering of an orbifold by orbifold charts. Exactly as in the case of manifolds, differentiability conditions can be imposed on the maps to give a definition of a differentiable orbifold. It will be a Riemannian orbifold if in addition there are invariant Riemannian metrics on the orbifold charts, for applications in geometric group theory, it is often convenient to have a slightly more general notion of orbifold, due to Haefliger
38.
Point group
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In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, point groups can be realized as sets of orthogonal matrices M that transform point x into point y, y = Mx where the origin is the fixed point. Point-group elements can either be rotations or else reflections, or improper rotations and these are the crystallographic point groups. Point groups can be classified into groups and achiral groups. The chiral groups are subgroups of the orthogonal group SO, they contain only orientation-preserving orthogonal transformations. The achiral groups contain also transformations of determinant −1, in an achiral group, the orientation-preserving transformations form a subgroup of index 2. Finite Coxeter groups or reflection groups are those point groups that are generated purely by a set of reflectional mirrors passing through the same point, a rank n Coxeter group has n mirrors and is represented by a Coxeter-Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with symbols for rotational. There are only two one-dimensional point groups, the identity group and the reflection group, point groups in two dimensions, sometimes called rosette groups. The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their Coxeter group and related polygons. The symmetry of the groups can be doubled by an isomorphism. Point groups in three dimensions, sometimes called point groups after their wide use in studying the symmetries of small molecules. They come in 7 infinite families of axial or prismatic groups, the reflection point groups, defined by 1 to 3 mirror planes, can also be given by their Coxeter group and related polyhedra. The group can be doubled, written as, mapping the first and last mirrors onto each other, doubling the symmetry to 48, the four-dimensional point groups are listed in Conway and Smith, Section 4, Tables 4. 1-4.3. The following list gives the four-dimensional reflection groups, each group is specified as a Coxeter group, and like the polyhedral groups of 3D, it can be named by its related convex regular 4-polytope. Front-back symmetric groups like and can be doubled, shown as double brackets in Coxeters notation, the following table gives the five-dimensional reflection groups, by listing them as Coxeter groups. The following table gives the six-dimensional reflection groups, by listing them as Coxeter groups, the following table gives the seven-dimensional reflection groups, by listing them as Coxeter groups. The following table gives the eight-dimensional reflection groups, by listing them as Coxeter groups, S. M. Coxeter, Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C
39.
Dihedral group
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
40.
Harold Scott MacDonald Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
41.
Coxeter notation
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The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups defined by pure reflections, there is a correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors, the Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the An group is represented by, to imply n nodes connected by n-1 order-3 branches, example A2 = = or represents diagrams or. Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like, Coxeter allowed for zeros as special cases to fit the An family, like A3 = = = =, like = =. Coxeter groups formed by cyclic diagrams are represented by parenthesese inside of brackets, if the branch orders are equal, they can be grouped as an exponent as the length the cycle in brackets, like =, representing Coxeter diagram or. More complicated looping diagrams can also be expressed with care, the paracompact complete graph diagram or, is represented as with the superscript as the symmetry of its regular tetrahedron coxeter diagram. The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs, so the Coxeter diagram = A2×A2 = 2A2 can be represented by × =2 =. For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, Coxeters notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets which cuts the order of the group in half. This is called a direct subgroup because what remains are only direct isometries without reflective symmetry, + operators can also be applied inside of the brackets, and creates semidirect subgroups that include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches next to it, the subgroup index is 2n for n + operators. So the snub cube, has symmetry +, and the tetrahedron, has symmetry. Johnson extends the + operator to work with a placeholder 1 nodes, in general this operation only applies to mirrors bounded by all even-order branches. The 1 represents a mirror so can be seen as, or, like diagram or, the effect of a mirror removal is to duplicate connecting nodes, which can be seen in the Coxeter diagrams, =, or in bracket notation, = =. Each of these mirrors can be removed so h = = = and this can be shown in a Coxeter diagram by adding a + symbol above the node, = =. If both mirrors are removed, a subgroup is generated, with the branch order becoming a gyration point of half the order, q = = +. For example, = = = ×, order 4. = +, the opposite to halving is doubling which adds a mirror, bisecting a fundamental domain, and doubling the group order
42.
Coxeter group
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups, however, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935, Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the groups of regular polytopes. The condition m i j = ∞ means no relation of the form m should be imposed, the pair where W is a Coxeter group with generators S = is called a Coxeter system. Note that in general S is not uniquely determined by W, for example, the Coxeter groups of type B3 and A1 × A3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition, the relation m i i =1 means that 1 =2 =1 for all i, as such the generators are involutions. If m i j =2, then the r i and r j commute. This follows by observing that x x = y y =1, in order to avoid redundancy among the relations, it is necessary to assume that m i j = m j i. This follows by observing that y y =1, together with m =1 implies that m = m y y = y m y = y y =1. Alternatively, k and k are elements, as y k y −1 = k y y −1 = k. The Coxeter matrix is the n × n, symmetric matrix with entries m i j, indeed, every symmetric matrix with positive integer and ∞ entries and with 1s on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group. The Coxeter matrix can be encoded by a Coxeter diagram. The vertices of the graph are labelled by generator subscripts, vertices i and j are adjacent if and only if m i j ≥3. An edge is labelled with the value of m i j whenever the value is 4 or greater, in particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter graph has two or more connected components, the group is the direct product of the groups associated to the individual components. Thus the disjoint union of Coxeter graphs yields a product of Coxeter groups. The Coxeter matrix, M i j, is related to the n × n Schläfli matrix C with entries C i j = −2 cos , but the elements are modified, being proportional to the dot product of the pairwise generators
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Improper rotation
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In 3D, equivalently it is the combination of a rotation and an inversion in a point on the axis. Therefore it is called a rotoinversion or rotary inversion. A three-dimensional symmetry that has one fixed point is necessarily an improper rotation. In both cases the operations commute, rotoreflection and rotoinversion are the same if they differ in angle of rotation by 180°, and the point of inversion is in the plane of reflection. An improper rotation of an object produces a rotation of its mirror image. The axis is called the rotation-reflection axis and this is called an n-fold improper rotation if the angle of rotation is 360°/n. The notation Sn denotes the group generated by an n-fold improper rotation. The notation n ¯ is used for n-fold rotoinversion, i. e. rotation by an angle of rotation of 360°/n with inversion, the Coxeter notation for S2n is, and orbifold notation is n×, order 2n. The direct subgroup, index 2, is Cn, +, order n, S2n for odd n contain inversion, with S2 = Ci is the group generated by inversion. S2n contain indirect isometries but not inversion for even n, in general, if odd p is a divisor of n, then S2n/p is a subgroup of S2n. For example S4 is a subgroup of S12, in a wider sense, an improper rotation may be defined as any indirect isometry, i. e. an element of E\E+, thus it can also be a pure reflection in a plane, or have a glide plane. An indirect isometry is a transformation with an orthogonal matrix that has a determinant of −1. A proper rotation is an ordinary rotation. In the wider sense, a rotation is defined as a direct isometry, i. e. an element of E+, it can also be the identity. A direct isometry is a transformation with an orthogonal matrix that has a determinant of 1. In either the narrower or the senses, the composition of two improper rotations is a proper rotation, and the composition of an improper and a proper rotation is an improper rotation