Orientation (vector space)
In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary finite dimension. In this setting, the orientation of an ordered basis is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple rotation. Thus, in three dimensions, it is impossible to make the left hand of a human figure into the right hand of the figure by applying a rotation alone, but it is possible to do so by reflecting the figure in a mirror; 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 declared to be positively oriented, but the choice is arbitrary, as they may be assigned a negative orientation.
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 finite-dimensional real vector space and let b1 and b2 be two ordered bases for V, it is a standard 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; 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 the other; every ordered basis lives in another. Thus any choice of a privileged ordered basis for V determines an orientation: the orientation class of the privileged basis is declared to be positive. For example, the standard basis on Rn provides a standard orientation on Rn. Any choice of a linear isomorphism between V and Rn will 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. This is because the determinant of a permutation matrix is equal to the signature of the associated permutation. Let A be a nonsingular linear mapping of vector space Rn to Rn; this mapping is orientation-preserving. For instance, in R3 a rotation around the Z Cartesian axis by an angle α is orientation-preserving: A 1 = while a reflection by the XY Cartesian plane is not orientation-preserving: A 2 = The concept of orientation degenerates in the zero-dimensional case. A zero-dimensional vector space has the zero vector; the only basis of a zero-dimensional vector space is the empty set ∅. Therefore, there is a single equivalence class of ordered bases, the class whose sole member is the empty set; this means that an orientation of a zero-dimensional space is a function →. It is therefore possible to orient a point in two different ways and negative.
Because there is only a single ordered basis ∅, a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing ↦ + 1 or ↦ − 1 therefore chooses an orientation of every basis of every zero-dimensional vector space. If all zero-dimensional vector spaces are assigned this orientation because all isomorphisms among zero-dimensional vector spaces preserve the ordered basis, they preserve the orientation; this is unlike the case of higher-dimensional vector spaces where there is no way to choose an orientation so that it is preserved under all isomorphisms. However, there are situations where it is desirable to give different orientations to different points. For example, consider the fundamental theorem of calculus as an instance of Stokes' theorem. A closed interval is a one-dimensional m
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system; the Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute metaphysical, sense.
Today, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, Euclidean space is a good approximation for it only over short distances. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects; this is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas. The Elements is a systematization of earlier knowledge of geometry, its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones, they are now nearly all lost. There are 13 books in the Elements: Books I–IV and VI discuss plane geometry.
Many results about plane figures are proved, for example "In any triangle two angles taken together in any manner are less than two right angles." and the Pythagorean theorem "In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced, it is proved. Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base; the platonic solids are constructed. Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of simple axioms; until the advent of non-Euclidean geometry, these axioms were considered to be true in the physical world, so that all the theorems would be true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.
Near the beginning of the first book of the Elements, Euclid gives five postulates for plane geometry, stated in terms of constructions: Let the following be postulated:To draw a straight line from any point to any point. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance; that all right angles are equal to one another.: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique; the Elements include the following five "common notions": Things that are equal to the same thing are equal to one another. If equals are added to equals the wholes are equal. If equals are subtracted from equals the differences are equal.
Things that coincide with one another are equal to one another. The whole is greater than the part. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than the others, they aspired to create a system of certain propositions, to them it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible, since one can construct consistent systems of geometry in which the parallel postulate is true, others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated. For example, Playfair's axiom states: In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the giv
In geometry, a figure is chiral if it is not identical to its mirror image, or, more if it cannot be mapped to its mirror image by rotations and translations alone. An object, not chiral is said to be achiral. In 3 dimensions, not all achiral objects have a mirror plane. For example, a 3-dimensional object with inversion centre as its only nontrivial symmetry operation is achiral but has no mirror plane. A chiral object and its mirror image are said to be enantiomorphs; the word chirality is derived from the hand, the most familiar chiral object. A non-chiral figure is called amphichiral; some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule. Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral; the J, L, S and Z-shaped tetrominoes of the popular video game Tetris exhibit chirality, but only in a two-dimensional space.
Individually they contain no mirror symmetry in the plane. A figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. See for a full mathematical definition of chirality. In three dimensions, every figure that possesses a mirror plane of symmetry S1, an inversion center of symmetry S2, or a higher improper rotation Sn axis of symmetry is achiral. Note, that there are achiral figures lacking both plane and center of symmetry. An example is the figure F 0 =, invariant under the orientation reversing isometry ↦ and thus achiral, but it has neither plane nor center of symmetry; the figure F 1 = is achiral as the origin is a center of symmetry, but it lacks a plane of symmetry. Note that achiral figures can have a center axis. In two dimensions, every figure which possesses an axis of symmetry is achiral
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 finite groups, they play an important role in group theory and chemistry; the notation for the dihedral group differs in abstract algebra. In geometry, Dn or Dihn refers to the symmetries of a group of order 2n. In abstract algebra, D2n refers to this same dihedral group; 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. 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 there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are 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.
The following picture shows the effect of the sixteen elements of D 8 on a stop sign: The first row shows the effect of the eight rotations, the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left. 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, this gives the symmetries of a polygon the algebraic structure of a finite group; the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity. 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, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative. In general, the group Dn has elements r0, …, rn−1 and s0, …, sn−1, with composition given by the following formulae: r i r j = r i + j, r i s j = s i + j, s i r j = s i − j, s i s j = r i − j.
In all cases and subtraction of subscripts are to be performed using modular arithmetic with modulus n. If we center the regular polygon at the origin elements of the dihedral group act as linear transformations of the plane; this lets us represent elements of Dn with composition being matrix multiplication. This is an example of a group representation. For example, the elements of the group D4 can be represented by the following eight matrices: r 0 =, r 1 =, r 2 =, r 3 =, s 0 =, s 1 =, s 2 =
The swastika or sauwastika is a geometrical figure and an ancient religious icon in the cultures of Eurasia. It is used as a symbol of spirituality in Indian religions. In the Western world, it was a symbol of auspiciousness and good luck until the 1930s, when it became a feature of Nazi symbolism as an emblem of Aryan identity and, as a result, was stigmatized by its association with racism and antisemitism; the name swastika comes from Sanskrit meaning'conducive to well being' or'auspicious'. In Hinduism, the symbol with arms pointing clockwise is called swastika, symbolizing surya and good luck, while the counterclockwise symbol is called sauvastika, symbolizing night or tantric aspects of Kali. In Jainism, a swastika is the symbol for Suparshvanatha—the 7th of 24 Tirthankaras, while in Buddhism it symbolizes the auspicious footprints of the Buddha. In several major Indo-European religions, the swastika symbolizes lightning bolts, representing the thunder god and the king of the gods, such as Indra in the religion of the Indus Valley Civilisation, Zeus in the ancient Greek religion, Jupiter in the ancient Roman religion, Thor in the ancient Germanic religion.
The swastika is an icon, found in both human history and the modern world. In various forms, it is otherwise known as the fylfot, tetraskelion, or cross cramponnée. In China it is named. A swastika takes the form of a cross, the arms of which are of equal length and perpendicular to the adjacent arms, each bent midway at a right angle; the symbol is found in the archeological remains of the Indus Valley Civilization and Mesopotamia, as well as in early Byzantine and Christian artwork. The swastika was adopted by several organizations in pre–World War I Europe, by the Nazi Party and Nazi Germany prior to World War II, it was used by the Nazi Party to symbolize German nationalistic pride. To Jews and the enemies of Nazi Germany, it became a symbol of terror. In many Western countries, the swastika is viewed as a symbol of racial supremacism and intimidation because of its association with Nazism. Reverence for the swastika symbol in Asian cultures, in contrast to the West's stigma of the symbol, has led to misinterpretations and misunderstandings.
The word swastika has been used in the Indian subcontinent and the middle east since 500 BC. Its appearance in English dates to the 1870s, replacing gammadion from Greek γαμμάδιον, it is alternatively spelled in contemporary texts as svastika, other spellings were used in the 19th and early 20th century, such as suastika. It was derived from the Sanskrit term, which transliterates to svastika under the used IAST transliteration system, but is pronounced closer to swastika when letters are used with their English values; the first use of the word swastika in a European text is found in 1871 with the publications of Heinrich Schliemann, who discovered more than 1,800 ancient samples of the swastika symbol and its variants while digging the Hisarlik mound near the Aegean Sea coast for the history of Troy. Schliemann linked his findings to the Sanskrit swastika; the word swastika is derived from the Sanskrit root swasti, composed of su and asti. The word swasti occurs in the Vedas as well as in classical literature, meaning'health, success, prosperity', it was used as a greeting.
The final ka is a common suffix with the same meaning as the English adverbial suffix -ly, so swastika means'associated with well-being'. According to Monier-Williams, a majority of scholars consider it a solar symbol; the sign implies something fortunate, lucky, or auspicious, it denotes auspiciousness or well-being. The earliest known use of the word swastika is in Panini's Ashtadhyayi which uses it to explain one of the Sanskrit grammar rules, in the context of a type of identifying mark on a cow's ear. Most scholarship suggests that Panini lived in or before the 4th-century BC in 6th or 5th century BC. Other names for the symbol include: tetragammadion or cross gammadion, as each arm resembles the Greek letter Γ hooked cross, angled cross, or crooked cross cross cramponned, cramponnée, or cramponny in heraldry, as each arm resembles a crampon or angle-iron fylfot, chiefly in heraldry and architecture tetraskelion meaning'four-legged' when composed of four conjoined legs whirling logs: can denote abundance, prosperity and luck All swastikas are bent crosses based on a chiral symmetry—but they appear with different geometric details: as compact crosses with short legs, as crosses with large arms and as motifs in a pattern of unbroken lines.
One distinct representation of a swastika, as a double swastika or swastika made of squares, appears in a Nepalese silver mohar coin of 1685, kingdom of Patan KM# 337. Chirality describes an absence of reflective symmetry, with the existence of two versions that are mirror images of each other; the mirror-image forms are described as: left-facing and right-facing. The left-facing version is distinguished in some traditions and languages as a distinct symbol from the right-facing and is called the "sauwastika"; the compact swastika can be seen
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and 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. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. 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 an integer. 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
Degrees of freedom (physics and chemistry)
In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, degrees of freedom of the system, are the dimensions of the phase space; the location of a particle in three-dimensional space requires. The direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely determines its past and future position and velocity as a function of time, such a system has six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions, for example, the particle must move along a wire or on a fixed surface the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.
In classical mechanics, the state of a point particle at any given time is described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism. In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system; the specification of all microstates of a system is a point in the system's phase space. In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer, it is useful to specify quadratic degrees of freedom. These are degrees of freedom. In three-dimensional space, three degrees of freedom are associated with the movement of a particle. A diatomic gas molecule has 6 degrees of freedom; this set may be decomposed in terms of translations and vibrations of the molecule. The center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two rotational degrees of one vibrational mode.
The rotations occur around the two axes perpendicular to the line between the two atoms. The rotation around the atom–atom bond is not a physical rotation; this yields, for a diatomic molecule, a decomposition of: N = 6 = 3 + 2 + 1. For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition: 3 N = 3 + 3 + which means that an N-atom molecule has 3N − 6 vibrational degrees of freedom for N > 2. In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one; as defined above one can count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows: For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space, thus its degree of freedom in a 3-D space is 3. For a body consisting of 2 particles in a 3-D space with constant distance between them we can show its degrees of freedom to be 5.
Let's say the other has coordinate with z2 unknown. Application of the formula for distance between two coordinates d = 2 + 2 + 2 results in one equation with one unknown, in which we can solve for z2. One of x1, x2, y1, y2, z1, or z2 can be unknown. Contrary to the classical equipartition theorem, at room temperature, the vibrational motion of molecules makes negligible contributions to the heat capacity; this is because these degrees of freedom are frozen because the spacing between the energy eigenvalues exceeds the energy corresponding to ambient temperatures. In the following table such degrees of freedom are disregarded because of their low effect on total energy. Only the translational and rotational degrees of freedom contribute, in equal amounts, to the heat capacity ratio; this is why γ = 7/5 for diatomic gases at room temperature. However, at high temperatures, on the order of the vibrational temperature, vibrational motion cannot be neglected. Vibrational temperatures are between 103 K and 104 K.
The set of degrees of freedom X1, ... , XN of a system is independent if the energy associated with the set can be written in the following form: E = ∑ i = 1 N E i, where Ei is a function of the sole variable Xi. example: if X1 and X2 are two degrees of freedom, E is the associated energy: If E = X 1 4 + X 2 4 the two degrees of freedom are independent. If E = X 1 4 + X 1