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61-521: An acceleron is a hypothetical subatomic particle postulated to relate the mass of the neutrino to the dark energy conjectured to be responsible for the accelerating expansion of the universe . The acceleron was postulated by researchers at the University of Washington in 2004. This particle physics –related article is a stub . You can help Misplaced Pages by expanding it . Subatomic particle Experiments show that light could behave like

122-414: A = 0 {\displaystyle a=0} . While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements i , j , k {\displaystyle i,j,k} , as well as the dot product and cross product , which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions. It

183-491: A solid figure . Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n -dimensional Euclidean space. The set of these n -tuples is commonly denoted R n , {\displaystyle \mathbb {R} ^{n},} and can be identified to the pair formed by a n -dimensional Euclidean space and a Cartesian coordinate system . When n = 3 , this space

244-487: A parallelogram , and hence are coplanar. A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P . The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball ). The volume of the ball is given by V = 4 3 π r 3 , {\displaystyle V={\frac {4}{3}}\pi r^{3},} and

305-606: A stream of particles (called photons ) as well as exhibiting wave-like properties. This led to the concept of wave–particle duality to reflect that quantum-scale particles behave both like particles and like waves ; they are sometimes called wavicles to reflect this. Another concept, the uncertainty principle , states that some of their properties taken together, such as their simultaneous position and momentum , cannot be measured exactly. The wave–particle duality has been shown to apply not only to photons but to more massive particles as well. Interactions of particles in

366-522: A choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space. Computationally, it is necessary to work with the more concrete description R 3 {\displaystyle \mathbb {R} ^{3}} in order to do concrete computations. A more abstract description still

427-525: A field , which is not commutative nor associative , but is a Lie algebra with the cross product being the Lie bracket. Specifically, the space together with the product, ( R 3 , × ) {\displaystyle (\mathbb {R} ^{3},\times )} is isomorphic to the Lie algebra of three-dimensional rotations, denoted s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} . In order to satisfy

488-412: A given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line. A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of

549-400: A hyperplane satisfy a single linear equation , so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in R 3 {\displaystyle \mathbb {R} ^{3}} form

610-529: A lighter particle having magnitude of electric charge ≤   e exists (which is unlikely). Its charge is not shown yet. All observable subatomic particles have their electric charge an integer multiple of the elementary charge . The Standard Model's quarks have "non-integer" electric charges, namely, multiple of ⁠ 1 / 3 ⁠   e , but quarks (and other combinations with non-integer electric charge) cannot be isolated due to color confinement . For baryons, mesons, and their antiparticles

671-419: A plane curve about a fixed line in its plane as an axis is called a surface of revolution . The plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle. Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution

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732-442: A subtle way. By definition, there exists a basis B = { e 1 , e 2 , e 3 } {\displaystyle {\mathcal {B}}=\{e_{1},e_{2},e_{3}\}} for V {\displaystyle V} . This corresponds to an isomorphism between V {\displaystyle V} and R 3 {\displaystyle \mathbb {R} ^{3}} :

793-445: A unique plane, so skew lines are lines that do not meet and do not lie in a common plane. Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can lie in

854-472: A vector A is denoted by || A || . The dot product of a vector A = [ A 1 , A 2 , A 3 ] with itself is which gives the formula for the Euclidean length of the vector. Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by where θ is the angle between A and B . The cross product or vector product

915-627: Is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product A × B of the vectors A and B is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics , and engineering . In function language, the cross product is a function × : R 3 × R 3 → R 3 {\displaystyle \times :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}} . The components of

976-439: Is a mathematical space in which three values ( coordinates ) are required to determine the position of a point . Most commonly, it is the three-dimensional Euclidean space , that is, the Euclidean space of dimension three, which models physical space . More general three-dimensional spaces are called 3-manifolds . The term may also refer colloquially to a subset of space, a three-dimensional region (or 3D domain ),

1037-758: Is a right circular cone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder . In analogy with the conic sections , the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0 , {\displaystyle Ax^{2}+By^{2}+Cz^{2}+Fxy+Gyz+Hxz+Jx+Ky+Lz+M=0,} where A , B , C , F , G , H , J , K , L and M are real numbers and not all of A , B , C , F , G and H are zero,

1098-406: Is called particle physics . The term high-energy physics is nearly synonymous to "particle physics" since creation of particles requires high energies: it occurs only as a result of cosmic rays , or in particle accelerators . Particle phenomenology systematizes the knowledge about subatomic particles obtained from these experiments. The term " subatomic particle" is largely a retronym of

1159-466: Is called a quadric surface . There are six types of non-degenerate quadric surfaces: The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane π and all the lines of R through that conic that are normal to π ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well. Both

1220-416: Is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics , it serves as a model of the physical universe , in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of space-time . While this space remains the most compelling and useful way to model the world as it is experienced, it

1281-406: Is found in linear algebra , where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors . A vector can be pictured as an arrow. The vector's magnitude

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1342-425: Is its length, and its direction is the direction the arrow points. A vector in R 3 {\displaystyle \mathbb {R} ^{3}} can be represented by an ordered triple of real numbers. These numbers are called the components of the vector. The dot product of two vectors A = [ A 1 , A 2 , A 3 ] and B = [ B 1 , B 2 , B 3 ] is defined as: The magnitude of

1403-427: Is made of two up quarks and one down quark , while the neutron is made of two down quarks and one up quark. These commonly bind together into an atomic nucleus, e.g. a helium-4 nucleus is composed of two protons and two neutrons. Most hadrons do not live long enough to bind into nucleus-like composites; those that do (other than the proton and neutron) form exotic nuclei . Any subatomic particle, like any particle in

1464-753: Is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions ( coordinates ), any three directions can be chosen, provided that these directions do not lie in the same plane . Furthermore, if these directions are pairwise perpendicular , the three values are often labeled by the terms width /breadth , height /depth , and length . Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes

1525-664: Is the Kronecker delta . Written out in full, the standard basis is E 1 = ( 1 0 0 ) , E 2 = ( 0 1 0 ) , E 3 = ( 0 0 1 ) . {\displaystyle E_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},E_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},E_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}.} Therefore R 3 {\displaystyle \mathbb {R} ^{3}} can be viewed as

1586-528: Is the Levi-Civita symbol . It has the property that A × B = − B × A {\displaystyle \mathbf {A} \times \mathbf {B} =-\mathbf {B} \times \mathbf {A} } . Its magnitude is related to the angle θ {\displaystyle \theta } between A {\displaystyle \mathbf {A} } and B {\displaystyle \mathbf {B} } by

1647-507: Is the number of protons in its nucleus. Neutrons are neutral particles having a mass slightly greater than that of the proton. Different isotopes of the same element contain the same number of protons but different numbers of neutrons. The mass number of an isotope is the total number of nucleons (neutrons and protons collectively). Chemistry concerns itself with how electron sharing binds atoms into structures such as crystals and molecules . The subatomic particles considered important in

1708-418: Is to model physical space as a three-dimensional affine space E ( 3 ) {\displaystyle E(3)} over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of R 3 {\displaystyle \mathbb {R} ^{3}} ,

1769-406: Is unknown, as some very important Grand Unified Theories (GUTs) actually require it. The μ and τ muons, as well as their antiparticles, decay by the weak force. Neutrinos (and antineutrinos) do not decay, but a related phenomenon of neutrino oscillations is thought to exist even in vacuums. The electron and its antiparticle, the positron , are theoretically stable due to charge conservation unless

1830-513: The Standard Model are: All of these have now been discovered through experiments, with the latest being the top quark (1995), tau neutrino (2000), and Higgs boson (2012). Various extensions of the Standard Model predict the existence of an elementary graviton particle and many other elementary particles , but none have been discovered as of 2021. The word hadron comes from Greek and

1891-438: The baryons containing an odd number of quarks (almost always 3), of which the proton and neutron (the two nucleons ) are by far the best known; and the mesons containing an even number of quarks (almost always 2, one quark and one antiquark), of which the pions and kaons are the best known. Except for the proton and neutron, all other hadrons are unstable and decay into other particles in microseconds or less. A proton

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1952-457: The three-dimensional space that obeys the laws of quantum mechanics , can be either a boson (with integer spin ) or a fermion (with odd half-integer spin). In the Standard Model, all the elementary fermions have spin 1/2, and are divided into the quarks which carry color charge and therefore feel the strong interaction, and the leptons which do not. The elementary bosons comprise the gauge bosons (photon, W and Z, gluons) with spin 1, while

2013-440: The 1960s, used to distinguish a large number of baryons and mesons (which comprise hadrons ) from particles that are now thought to be truly elementary . Before that hadrons were usually classified as "elementary" because their composition was unknown. A list of important discoveries follows: Three-dimensional space In geometry , a three-dimensional space ( 3D space , 3-space or, rarely, tri-dimensional space )

2074-513: The 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton 's development of the quaternions . In fact, it was Hamilton who coined the terms scalar and vector , and they were first defined within his geometric framework for quaternions . Three dimensional space could then be described by quaternions q = a + u i + v j + w k {\displaystyle q=a+ui+vj+wk} which had vanishing scalar component, that is,

2135-487: The Higgs boson is the only elementary particle with spin zero. The hypothetical graviton is required theoretically to have spin 2, but is not part of the Standard Model. Some extensions such as supersymmetry predict additional elementary particles with spin 3/2, but none have been discovered as of 2023. Due to the laws for spin of composite particles, the baryons (3 quarks) have spin either 1/2 or 3/2 and are therefore fermions;

2196-399: The above-mentioned systems. Two distinct points always determine a (straight) line . Three distinct points are either collinear or determine a unique plane . On the other hand, four distinct points can either be collinear, coplanar , or determine the entire space. Two distinct lines can either intersect, be parallel or be skew . Two parallel lines, or two intersecting lines , lie in

2257-490: The abstract vector space, together with the additional structure of a choice of basis. Conversely, V {\displaystyle V} can be obtained by starting with R 3 {\displaystyle \mathbb {R} ^{3}} and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis. As opposed to a general vector space V {\displaystyle V} ,

2318-771: The axioms of a Lie algebra, instead of associativity the cross product satisfies the Jacobi identity . For any three vectors A , B {\displaystyle \mathbf {A} ,\mathbf {B} } and C {\displaystyle \mathbf {C} } A × ( B × C ) + B × ( C × A ) + C × ( A × B ) = 0 {\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )=0} One can in n dimensions take

2379-533: The constituent quarks' charges sum up to an integer multiple of e . Through the work of Albert Einstein , Satyendra Nath Bose , Louis de Broglie , and many others, current scientific theory holds that all particles also have a wave nature. This has been verified not only for elementary particles but also for compound particles like atoms and even molecules. In fact, according to traditional formulations of non-relativistic quantum mechanics, wave–particle duality applies to all objects, even macroscopic ones; although

2440-630: The construction for the isomorphism is found here . However, there is no 'preferred' or 'canonical basis' for V {\displaystyle V} . On the other hand, there is a preferred basis for R 3 {\displaystyle \mathbb {R} ^{3}} , which is due to its description as a Cartesian product of copies of R {\displaystyle \mathbb {R} } , that is, R 3 = R × R × R {\displaystyle \mathbb {R} ^{3}=\mathbb {R} \times \mathbb {R} \times \mathbb {R} } . This allows

2501-491: The construction of the five regular Platonic solids in a sphere. In the 17th century, three-dimensional space was described with Cartesian coordinates , with the advent of analytic geometry developed by René Descartes in his work La Géométrie and Pierre de Fermat in the manuscript Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space. In

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2562-880: The cross product are A × B = [ A 2 B 3 − B 2 A 3 , A 3 B 1 − B 3 A 1 , A 1 B 2 − B 1 A 2 ] {\displaystyle \mathbf {A} \times \mathbf {B} =[A_{2}B_{3}-B_{2}A_{3},A_{3}B_{1}-B_{3}A_{1},A_{1}B_{2}-B_{1}A_{2}]} , and can also be written in components, using Einstein summation convention as ( A × B ) i = ε i j k A j B k {\displaystyle (\mathbf {A} \times \mathbf {B} )_{i}=\varepsilon _{ijk}A_{j}B_{k}} where ε i j k {\displaystyle \varepsilon _{ijk}}

2623-516: The definition of canonical projections, π i : R 3 → R {\displaystyle \pi _{i}:\mathbb {R} ^{3}\rightarrow \mathbb {R} } , where 1 ≤ i ≤ 3 {\displaystyle 1\leq i\leq 3} . For example, π 1 ( x 1 , x 2 , x 3 ) = x {\displaystyle \pi _{1}(x_{1},x_{2},x_{3})=x} . This then allows

2684-493: The definition of the standard basis B Standard = { E 1 , E 2 , E 3 } {\displaystyle {\mathcal {B}}_{\text{Standard}}=\{E_{1},E_{2},E_{3}\}} defined by π i ( E j ) = δ i j {\displaystyle \pi _{i}(E_{j})=\delta _{ij}} where δ i j {\displaystyle \delta _{ij}}

2745-431: The elementary fermions with no color charge. All massless particles (particles whose invariant mass is zero) are elementary. These include the photon and gluon, although the latter cannot be isolated. Most subatomic particles are not stable. All leptons, as well as baryons decay by either the strong force or weak force (except for the proton). Protons are not known to decay , although whether they are "truly" stable

2806-559: The framework of quantum field theory are understood as creation and annihilation of quanta of corresponding fundamental interactions . This blends particle physics with field theory . Even among particle physicists , the exact definition of a particle has diverse descriptions. These professional attempts at the definition of a particle include: Subatomic particles are either "elementary", i.e. not made of multiple other particles, or "composite" and made of more than one elementary particle bound together. The elementary particles of

2867-506: The heaviest lepton (the tau particle ) is heavier than the two lightest flavours of baryons ( nucleons ). It is also certain that any particle with an electric charge is massive. When originally defined in the 1950s, the terms baryons, mesons and leptons referred to masses; however, after the quark model became accepted in the 1970s, it was recognised that baryons are composites of three quarks, mesons are composites of one quark and one antiquark, while leptons are elementary and are defined as

2928-421: The hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces , meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a regulus . Another way of viewing three-dimensional space

2989-460: The identity ‖ A × B ‖ = ‖ A ‖ ⋅ ‖ B ‖ ⋅ | sin ⁡ θ | . {\displaystyle \left\|\mathbf {A} \times \mathbf {B} \right\|=\left\|\mathbf {A} \right\|\cdot \left\|\mathbf {B} \right\|\cdot \left|\sin \theta \right|.} The space and product form an algebra over

3050-620: The mesons (2 quarks) have integer spin of either 0 or 1 and are therefore bosons. In special relativity , the energy of a particle at rest equals its mass times the speed of light squared , E = mc . That is, mass can be expressed in terms of energy and vice versa. If a particle has a frame of reference in which it lies at rest , then it has a positive rest mass and is referred to as massive . All composite particles are massive. Baryons (meaning "heavy") tend to have greater mass than mesons (meaning "intermediate"), which in turn tend to be heavier than leptons (meaning "lightweight"), but

3111-555: The position of any point in three-dimensional space is given by an ordered triple of real numbers , each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates , though there are an infinite number of possible methods. For more, see Euclidean space . Below are images of

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3172-490: The prerequisite basics of Newtonian mechanics , a series of statements and equations in Philosophiae Naturalis Principia Mathematica , originally published in 1687. The negatively charged electron has a mass of about ⁠ 1 / 1836 ⁠ of that of a hydrogen atom. The remainder of the hydrogen atom's mass comes from the positively charged proton . The atomic number of an element

3233-472: The product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions . It can be useful to describe three-dimensional space as a three-dimensional vector space V {\displaystyle V} over the real numbers. This differs from R 3 {\displaystyle \mathbb {R} ^{3}} in

3294-530: The space R 3 {\displaystyle \mathbb {R} ^{3}} is sometimes referred to as a coordinate space. Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} assumes

3355-418: The surface area of the sphere is A = 4 π r 2 . {\displaystyle A=4\pi r^{2}.} Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere : points equidistant to the origin of the euclidean space R . If a point has coordinates, P ( x , y , z , w ) , then x + y + z + w = 1 characterizes those points on

3416-431: The understanding of chemistry are the electron , the proton , and the neutron . Nuclear physics deals with how protons and neutrons arrange themselves in nuclei. The study of subatomic particles, atoms and molecules, and their structure and interactions, requires quantum mechanics . Analyzing processes that change the numbers and types of particles requires quantum field theory . The study of subatomic particles per se

3477-446: The unit 3-sphere centered at the origin. This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra . A surface generated by revolving

3538-479: The wave properties of macroscopic objects cannot be detected due to their small wavelengths. Interactions between particles have been scrutinized for many centuries, and a few simple laws underpin how particles behave in collisions and interactions. The most fundamental of these are the laws of conservation of energy and conservation of momentum , which let us make calculations of particle interactions on scales of magnitude that range from stars to quarks. These are

3599-491: The work of Hermann Grassmann and Giuseppe Peano , the latter of whom first gave the modern definition of vector spaces as an algebraic structure. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin , the point at which they cross. They are usually labeled x , y , and z . Relative to these axes,

3660-505: Was introduced in 1962 by Lev Okun . Nearly all composite particles contain multiple quarks (and/or antiquarks) bound together by gluons (with a few exceptions with no quarks, such as positronium and muonium ). Those containing few (≤ 5) quarks (including antiquarks) are called hadrons . Due to a property known as color confinement , quarks are never found singly but always occur in hadrons containing multiple quarks. The hadrons are divided by number of quarks (including antiquarks) into

3721-466: Was not until Josiah Willard Gibbs that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbook Vector Analysis written by Edwin Bidwell Wilson based on Gibbs' lectures. Also during the 19th century came developments in the abstract formalism of vector spaces, with

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