
Review of Short Phrases and Links 
This Review contains major "Dimensions" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 Dimensions are independent components of a coordinate grid needed to locate a point in a certain defined "space".
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 Dimensions are components of a coordinate grid typically used to locate a point in space, or on the globe, such as by latitude, longitude and planet (Earth).
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 Dimensions are different for Periodicals automation flatsize mail.
 Such dimensions are predicted in theories with supersymmetry or superstrings.
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 Extra dimensions are integral to several theoretical models of the universe; string theory, for example, suggests as many as seven extra dimensions of space.
 The dimensions of a cube are the lengths of the three edges which meet at any vertex.
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 Wythoff's construction in three dimensions is by placing a vertex in the triangle, and dropping perpendiculars to each of the edges.
 Regular polyhedra generalize the notion of a regular polygon to three dimensions.
 This is why physicists take the notion of hyperspace (10 dimensions of spacetime) seriously.
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 The notion of extra dimensions also helps to resolve the hierarchy problem which is the question of why gravity is so much weaker than any other force.
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 Briefly; a simplex is a generalization of the concept of a triangle into forms with more, or fewer, than two dimensions.
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 It is a generalization of the concept of a plane into a different number of dimensions.
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 Dimension X, an adventure game setting based on the concept of parallel universes, which are referred to as dimensions.
 Furthermore, all string theories predict the existence of degrees of freedom which are usually described as extra dimensions.
 Once this space has been quantized, only half of the dimensions simultaneously commute and so the number of degrees of freedom has been halved.
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 Counting layers gives 10 degrees of freedom (5+1+1+1+1+1 = 10), which equals the dimensions postulated by string theory.
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 A gas in a room have trillions of degrees of freedom if you consider all the particles individually, but only moves in 3+1 dimensions.
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 The Big Bang theory states that it is the point in which all dimensions came into being, the start of both space and time.
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 The point groups in two dimensions with respect to any point leave that point fixed.
 The simplex in four dimensions (the pentatope) is a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that.
 First, it is not true in other dimensions than 3 (in a plane, a linear map that is an isometry and that keeps an axis unchanged is a reflection).
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 Most perceptual maps show only two dimensions at a time, for example price on one axis and quality on the other.
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 If you want to generalize rotations to higher dimensions, it's often much easier to think of rotations "over" a plane, instead of around a "hyper" axis.
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 You can scale the dimensions up or down for the size and stiffness of the rope you are using.
 Changes might occur in size, the overall dimensions of a bone, or shape alone, such that if one dimension increases, another must decrease.
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 So understanding the mass of the electron is essential to understanding the size and dimensions of everything around us.
 Know the soul to be free of any gender and not bound by any dimensions of shape and size.
 In addition, in four dimensions and higher, open sets can be very different in terms of shape, size, measure, and topology, and still have the same capacity.
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 A Linear Collider would determine the number, size and shape of extra dimensions through their small effects on particle masses and interactions.
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 The concept of a vector can be extended to three or more dimensions.
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 The method is currently limited to potential flow in two dimensions, although the procedure can be extended to three.
 Therefore, vectors can be extended to three dimensions by simply adding the 'z' value.
 Torque has dimensions of force times distance and the SI units of torque are stated as "newtonmetres".
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 Torque has dimensions of distance × force and the SI units of torque are stated as " newton  metres ".
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 Torque has dimensions of force times distance.
 Polytope The equivalent of a polyhedron, but in any number of dimensions.
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 In three dimensions, stellation consists of "solidifying" parts of the extended face of a polyhedron, then repeating the same pattern for each face.
 Consider the other extreme case of the complete isometric graph with n vertices, this is equivalent to the regular simplex in (n1) dimensions.
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 FIG. 5A shows in two dimensions the points (vertices) 507 scanned during a typical range scan.
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 For example in three dimensions the vectors (v 0, v 1, v 2, v 3) are the vertices of a 3simplex or tetrahedron.
 Examples in two dimensions include the square, the regular pentagon and hexagon, and other regular polygons, including star polygons.
 The method the sphere gives to the square can be generalized so that the form of fourdimensional objects can be seen in three dimensions.
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 In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions,, and the volume of a solid to the cube,.
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 Due to some quantum quirks, the sum of these configurations could yield a spacetime with any number of dimensions.
 As expected then, the dimensionality of the cross product is the sum of the dimensions.
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 The dimension of the space is the sum of the dimensions of the two subspaces, minus the dimension of their intersection.
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 Conventional gravity does not place any limits on the possible dimensions of spacetime: its equations can, in principle, be formulated in any dimension.
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 We will introduce the notion of semiconformally flat surfaces and establish a complete classification of the possible dimensions for this family.
 Supersymmetry severely restricts the possible dimensions of a pbrane.
 According to certain Euclidean space perceptions, the universe has three dimensions of space and one dimension of time.
 The universe appears to have a smooth spacetime continuum consisting of three spatial dimensions and one temporal (time) dimension.
 Also, when thinking of space and time it is natural to ask why the universe appears to have exactly one time and three space dimensions.
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 If the M i are actually vector spaces, then the dimension of the direct sum is equal to the sum of the dimensions of the M i.
 The height and width dimensions of the foil layer are substantially equal to the dimensions of the flexible roofing material.
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 The divergence of the curl of any vector field (in three dimensions) is constant and equal to zero.
 Attachment to visualizing the dimensions precludes understanding the many different dimensions that can be measured (time, mass, color, cost, etc.).
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 Recall that in 3+1 dimensions, the radius of a black hole is proportional to its mass, "R=2M", in "c=hbar=G=1" Planck units.
 This work could reveal the secrets of dark matter, the existence of extra dimensions and explain mass and gravity.
 Physical quantities in these scalar field theories may have dimensions of length, time or mass, or some combination of the three.
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 Some quantities which are physically different, and have different unit names, may have the same dimensions, for example, torque and work.
 Physical quantities having different dimensions cannot be compared to one another either.
 The dimensions of a physical quantity is associated with symbols, such as M, L, T which represent mass, length and time, each raised to rational powers.
 The quantity A Â· B is the scalar valued interior product, while A âˆ§ B is the grade 4 exterior product that arises in four or more dimensions.
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 In electromagnetism, for example, it may be useful to use dimensions of M, L, T, and Q, where Q represents quantity of electric charge.
 Its principal ruins consisted of the propylon and two columns of a temple, which was apparently of small dimensions, but of elegant proportions.
 If you want to combine rotations, in 2D you can just add their angles, but in higher dimensions you must multiply their matrices.
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 Properties of the conjugate transpose (A + B) * = A * + B * for any two matrices A and B of the same dimensions.
 Two matrices can only be equal if they have the same dimensions and same elements.
 Because all of the particles feel the gravitational force, G is universal, so G can be used to form quantities with dimensions, giving the Planck scale.
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 Quantum Mechanics is a description that gives the best answers possible using the mathematics of only four dimensions and time and the theory of particles.
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 Particles are then supposed to be modes of vibration in the geometry of these extra dimensions.
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 This theory is nothing but an extension of Einstein Spacetime and, contrary to other theories, it requires only four dimensions: x, y, z and t.
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 For example, many theories of quantum gravity can create universes with arbitrary numbers of dimensions or with arbitrary cosmological constants.
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 It asserts that all theories that attribute more than three spatial dimensions and one temporal dimension to the world of experience are unstable.
 The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions.
 By "world line" we mean a curve traced out in the 4 dimensions of spacetime which could be the history of a particle or a point on a shadow.
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 A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane.
 The Euler characteristic of other surfaces is a useful topological invariant, which can be extended to higher dimensions using Betti numbers.
 The Euler characteristic of other surfaces is a useful topological invariant, which can be extended to higher dimensions using Betti number s.
 The hypothesis that the unconscious is structured like a language, that is, in two dimensions, led Lacan to the topology of surfaces.
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 In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface.
 Similar operations may be extended to calculate the curvature and length of a curve and to analogous properties of surfaces in any number of dimensions.
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 Gaussian curvature, but only after introducing the full Riemann tensor), which is good for building intuition about curvature in higher dimensions.
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 Likewise spacetime is a manifold, of four (or maybe ten or more) dimensions.
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 The notion of shapes like these can be generalized to higher dimensions, and such a shape is called a manifold.
 In dimensions two and higher, a simple but important invariant criterion is the question of whether a manifold admits a meaningful orientation.
 For rotations in three dimensions, this is the axis of the rotation (a concept that has no meaning in any other dimension).
 Any rotation in three dimensions can be described by a rotation by some angle about some axis.
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 For instance, an eigenvector of a rotation in three dimensions is a vector located within the axis about which the rotation is performed.
 In SI units, angular frequency is measured in radians per second, with dimensions T −1 since radians are dimensionless.
 Put another way, there is no such thing as absolute motion, either in the three dimensions of space, or in the fourth dimension identified by Einstein, time.
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 Before Einstein 's work on relativistic physics, time and space were viewed as independent dimensions.
 Einstein's equation says that mass literally is energy and momentum  it is the length of that vector in four dimensions.
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 Ehrenfest also showed that if space has an even number of dimensions, then the different parts of a wave impulse will travel at different speeds.
 In three dimensions all bivectors can be generated by the exterior product of two vectors.
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 While bivectors are isomorphic to vectors (via the dual) in three dimensions they can be represented by skewsymmetric matrices in any dimension.
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 All bivectors in two dimensions are of this form, that is multiples of the bivector e 1 e 2, written e 12 to emphasise it is a bivector rather than a vector.
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 When he returns to Flatland, his preaching on the existence and significance of higher dimensions gets him thrown into prison.
 For example, there's a marvelous lattice in 24 dimensions called the Leech lattice, which gives the densest lattice packing of spheres in that dimension.
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 The 24 vertices of the 24cell correspond to the 24 dimensions of the Leech Lattice.
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 Therefore, the photon in flat spacetime will be massless—and the theory consistent—only for a particular number of dimensions.
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 These new phenomena include effects from new dimensions of space, quantum gravity, and the vibrational modes of the superstring.
 The bosons are again interpreted as spacetime dimensions and so the critical dimension for the superstring is 10.
 Then, in 1971, string theory incorporated supersymmetry, and this merger resulted in the creation of superstrings that can exist in ten dimensions.
 Many scientists, however, do not believe superstrings are the answers, because they have not detected the additional dimensions required by string theory.
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 The elegant universe: superstrings, hidden dimensions, and the quest for the ultimate theory.
 In two dimensions, potential flow reduces to a very simple system that is analysed using complex numbers (see below).
 The state organisation of the Ottoman Empire was a very simple system that had two main dimensions: the military administration and the civil administration.
 Magnitude: Magnitude in three dimensions is the same as in two dimensions, with the addition of a 'z' term in the radicand.
 In two dimensions the prefactor (âˆ’ 1) 2 s can be replaced by any complex number of magnitude 1 (see Anyon).
 This misses anyon s and braid statistics in lower dimensions.
 Schläfli showed that there are 6 regular convex polytopes in 4 dimensions, and exactly three in each higher dimension.
 The calculation is the same in seven dimensions, except it is more complicated because of the higher dimension.
 For a square matrix A of order n to be diagonalizable, the sum of the dimensions of the eigenspaces must be equal to n.
 Let R = real numbers C = complex numbers H = quaternions O = octonions Let SO(n,1) denote the Lorentz group in n+1 dimensions.
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 The restricted Lorentz group is a connected normal subgroup of the full Lorentz group with the same dimension (in this case, 6 dimensions).
 The Lorentz group and the Poincare group of isometries of spacetime are Lie groups of dimensions 6 and 10 that are used in special relativity.
 In Euclidean space of dimensions 0, 1, 2, and 3, the simplexes are the point, line segment, triangle and tetrahedron, respectively.
 More general Heisenberg groups H n may be defined for higher dimensions in Euclidean space, and more generally on symplectic vector spaces.
 It may be embedded in Euclidean space of dimensions 4 and higher.
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Categories
 Topology > Topological Spaces > Manifolds > Dimension
 Science > Astronomy > Universe > Space
 Information > Science > Mathematics > Geometry
 Mathematics > Algebra > Linear Algebra > Vectors
 Science > Physics > Volume > Cube
Related Keywords
* Arbitrary Dimension
* Arbitrary Dimensions
* Complex
* Conformal Geometry
* Conidia
* CrossPolytope
* Cube
* Dimension
* Dimensions Higher
* Extra Dimensions
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* Four Dimensions
* Geometry
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* Homotopy Groups
* Hypercube
* Infinite Dimensions
* Lqg
* MTheory
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* Mathematics
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* Microns
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* Physics
* Plane
* Polyhedra
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* Seven Dimensions
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* Space
* SpaceTime
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* Supergravity
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* Time Dimension
* Various Dimensions
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