"stereophonic sound rather than the music. ... The music is better than the liner notes (which spend most of their time describing the sound) lets on but is not all that significant, emphasizing swing standards along with an occasional Latin novelty."" The Fourth Dimension in Sound The Fourth Dimension in Sound (subtitled A Musical Experiment in the Adaptation of Instruments to Modern Electronics) is an album by bandleader and arranger Shorty Rogers recorded in late 1961 and released on the Warner Bros. label. The album was produced principally as a stereo test and demonstration record to be used by hi-fi"
"Four-dimensional space A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible generalization of the observation that one only needs three numbers, called ""dimensions"", to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring its length (often labeled ""x""), width (""y""), and depth (""z""). The idea of adding a fourth dimension began with Joseph-Louis Lagrange in the mid-1700s and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann."
"H. S. M. Coxeter felt compelled to write: Mathematically four-dimensional space is simply a space with four spatial dimensions, that is a space that needs four parameters to specify a point in it. For example, a general point might have position vector a, equal to This can be written in terms of the four standard basis vectors (e, e, e, e), given by so the general vector a is Vectors add, subtract and scale as in three dimensions. The dot product of Euclidean three-dimensional space generalizes to four dimensions as It can be used to calculate the norm or length"
"measured along these axes can be called ""height"", ""width"", and ""depth"". Comparatively, four-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled ""w"". To describe the two additional cardinal directions, Charles Howard Hinton coined the terms ""ana"" and ""kata"", from the Greek words meaning ""up toward"" and ""down from"", respectively. A position along the ""w"" axis can be called ""spissitude"", as coined by Henry More. The geometry of four-dimensional space is much more complex than that of three-dimensional space, due to the extra degree of freedom. Just as in three dimensions there are polyhedra"
"care must be exercised not to accept results that are not more rigorously tested. For example, consider the formulas for the circumference of a circle formula_11 and the surface area of a sphere: formula_12. One might be tempted to suppose that the surface volume of a hypersphere is formula_13, or perhaps formula_14, but either of these would be wrong. The correct formula is formula_15. Four-dimensional space A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible generalization of the observation that one only needs three numbers,"
"fourth spatial dimension is curled up in a circle of a very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set, and construction of this compact dimension is referred to as compactification. In modern geometry, the extra fifth dimension can be understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the"
"everyday life, there are three familiar dimensions of space: height, width and length. Einstein's general theory of relativity treats time as a dimension on par with the three spatial dimensions; in general relativity, space and time are not modeled as separate entities but are instead unified to a four-dimensional spacetime. In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime. In spite of the fact that the universe is well described by four-dimensional spacetime, there are several reasons why physicists consider theories in other dimensions. In some cases, by modeling spacetime in a"
"Four-dimensionalism In philosophy, four-dimensionalism (also known as the ""doctrine of temporal parts"") is an ontological position that an object's persistence through time is like its extension through space. Thus, an object that exists in time has temporal parts in the various subregions of the total region of time it occupies, just like an object that exists in a region of space has at least one part in every subregion of that space. Four-dimensionalists typically argue for treating time as analogous to space, usually leading them to endorse the doctrine of ""eternalism"". This is a philosophical approach to the ontological nature"
"Higher dimensional spaces have since become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without the use of such spaces. Einstein's concept of spacetime uses such a 4D space, though it has a Minkowski structure that is a bit more complicated than Euclidean 4D space. Single locations in 4D space can be given as vectors or ""n-tuples"", i.e. as ordered lists of numbers such as ""(t,x,y,z)"". It is only when such locations are linked together into more complicated shapes that the full richness and geometric"
"a sphere, at least on the length-scale of the observable Universe. However, present observations cannot exclude the possibilities that the Universe has more dimensions (which is postulated by theories such as the String theory) and that its spacetime may have a multiply connected global topology, in analogy with the cylindrical or toroidal topologies of two-dimensional spaces. The spacetime of the Universe is usually interpreted from a Euclidean perspective, with space as consisting of three dimensions, and time as consisting of one dimension, the ""fourth dimension"". By combining space and time into a single manifold called Minkowski space, physicists have simplified"
"the phenomenon that separates dimension 4 from others."" 4-manifold In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are of importance in physics because, in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. The homotopy type of a"
"to focus instead on ""mysticism of the all-is-one-one-is-all thinking of an Ouspensky."" The ""Quarterly Review of Biology"" declared it to be ""nice"", and ""at times (...) enchanting"", comparing it to ""The Tao of Physics"". The Fourth Dimension (book) The Fourth Dimension: Toward a Geometry of Higher Reality (1984) is a popular mathematics book by Rudy Rucker, a Silicon Valley professor of mathematics and computer science. It provides a popular presentation of set theory and four dimensional geometry as well as some mystical implications. A foreword is provided by Martin Gardner and the 200+ illustrations are by David Povilaitis. ""The Fourth"
"Five-dimensional space A five-dimensional space is a space with five dimensions. If interpreted physically, that is one more than the usual three spatial dimensions and the fourth dimension of time used in relativitistic physics. It is an abstraction which occurs frequently in mathematics, where it is a legitimate construct. In physics and mathematics, a sequence of ""N"" numbers can be understood to represent a location in an ""N""-dimensional space. Whether or not the universe is five-dimensional is a topic of debate. Much of the early work on five dimensional space was in an attempt to develop a theory that unifies"
"Space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Debates concerning the nature, essence and the mode of existence of space date back to"
"but are instead unified to a four-dimensional spacetime, three spatial dimensions and one time dimension. In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime. In spite of the fact that the universe is well described by four-dimensional spacetime, there are several reasons why physicists consider theories in other dimensions. In some cases, by modeling spacetime in a different number of dimensions, a theory becomes more mathematically tractable, and one can perform calculations and gain general insights more easily. There are also situations where theories in two or three spacetime dimensions are useful"
"width and depth"" have been promulgated as three distinct dimensions within the Euclidean context, each with its own independence, Fuller considered the tetrahedron a minimal starting point for spatial cognition. His use of ""4D"" was in many passages close to synonymous with the ordinary meaning of ""3D,"" with the dimensions of physicality (time, mass) considered additional dimensions. Geometers and ""schooled"" people speak of length, breadth, and height as constituting a hierarchy of three independent dimensional states -- ""one-dimensional,"" ""two-dimensional,"" and ""three-dimensional"" -- which can be conjoined like building blocks. But length, breadth, and height simply do not exist independently of"
"normally understood to be due to concentrations of matter in four-dimensional space, the above result is interpreted as saying that four-dimensional matter is induced from geometry in five-dimensional space. In particular, the soliton solutions of formula_74 can be shown to contain the Friedmann–Lemaître–Robertson–Walker metric in both radiation-dominated (early universe) and matter-dominated (later universe) forms. The general equations can be shown to be sufficiently consistent with classical tests of general relativity to be acceptable on physical principles, while still leaving considerable freedom to also provide interesting cosmological models. The Kaluza–Klein theory has a particularly elegant presentation in terms of geometry. In"
"4 dimensions in Hilbert space and dynamics described by 4-dimensional space-time. As a consequence of the four-dimensional nature of space-time in relativity, relativistic quantum mechanics uses 4×4 matrices to describe spin operators and observables. When physicist Paul Dirac tried to modify the Schrödinger equation so that it was consistent with Einstein's theory of relativity, he found it was only possible by including matrices in the resulting Dirac Equation, implying the wave must have multiple components leading to spin. Spin-½ In quantum mechanics, spin is an intrinsic property of all elementary particles. Fermions, the particles that constitute ordinary matter, have half-integer"
"complexity of 4D and higher dimensional spaces emerge. A hint to that complexity can be seen in the accompanying animation of one of simplest possible 4D objects, the 4D cube or tesseract. Lagrange wrote in his ""Mécanique analytique"" (published 1788, based on work done around 1755) that mechanics can be viewed as operating in a four-dimensional space — three dimensions of space, and one of time. In 1827 Möbius realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image, and by 1853 Ludwig Schläfli had discovered many polytopes in higher dimensions, although his work"
"Four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space. Physical events correspond to mathematical points in time and space, the set of all of them together forming a mathematical model of physical four-dimensional spacetime. The history of an object traces a curve in spacetime, called its world line. If the object is massive, so that its speed is less than the speed of light, the world line may be parametrized by the proper time of the"
"dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer. General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field. In ordinary space, a position is specified by three numbers, known as dimensions. In the Cartesian coordinate system, these are called x, y, and z. A position in spacetime is called an ""event"", and requires four numbers to be specified: the three-dimensional location in space, plus the position in"
"by four numbers: a time coordinate and three space coordinates; thus spacetime is a four-dimensional space. The mathematical term for spacetime is a four-dimensional manifold. The concept may be applied as well to a higher-dimensional space. For easy visualizations of four dimensions, two space coordinates are often suppressed. The event is then represented by a point in a Minkowski diagram, which is a plane usually plotted with the time coordinate, say formula_1, upwards and the space coordinate, say formula_2 horizontally. As expressed by F.R. Harvey A world line traces out the path of a single point in spacetime. A world"
"LHC, then we can say it is unlikely to discover supersymmetry at CERN in the foreseeable future. Our physical space is observed to have three large spatial dimensions and, along with time, is a boundless 4-dimensional continuum known as spacetime. However, nothing prevents a theory from including more than 4 dimensions. In the case of string theory, consistency requires spacetime to have 10 dimensions (3D regular space + 1 time + 6D hyperspace). The fact that we see only 3 dimensions of space can be explained by one of two mechanisms: either the extra dimensions are compactified on a very"
"The Fourth Dimension (book) The Fourth Dimension: Toward a Geometry of Higher Reality (1984) is a popular mathematics book by Rudy Rucker, a Silicon Valley professor of mathematics and computer science. It provides a popular presentation of set theory and four dimensional geometry as well as some mystical implications. A foreword is provided by Martin Gardner and the 200+ illustrations are by David Povilaitis. ""The Fourth Dimension: Toward a Geometry of Higher Reality"" was reprinted in 1985 as the paperback ""The Fourth Dimension: A Guided Tour of the Higher Universes"". It was again reprinted in paperback in 2014 by Dover"
"gravitation, is a real 4-dimensional ""Pseudo-Riemannian manifold"". In QM, wave functions describing particles are complex-valued functions of real space and time variables. The set of all wavefunctions for a given system is an infinite-dimensional complex Hilbert space. The notion of spacetime having more than four dimensions is of interest in its own mathematical right. Its appearance in physics can be rooted to attempts of unifying the fundamental interactions, originally gravity and electromagnetism. These ideas prevail in string theory and beyond. The idea of ""complex"" spacetime has received considerably less attention, but it has been considered in conjunction with the Lorentz–Dirac"
"forward the holographic principle, which explains that the ""information about an extra dimension is visible as a curvature in a spacetime with one fewer dimension"". For example, holograms are three-dimensional pictures placed on a two-dimensional surface, which gives the image a curvature when the observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal (test) particle. 'T Hooft has speculated that the fifth dimension is really the ""spacetime fabric"". According to Klein’s definition, ""a geometry is the study of the invariant properties of a spacetime, under transformations"
"are characterized by fractal dimensions. The concept of dimension can be generalized to include networks embedded in space. The dimension characterize their spatial constraints. Science fiction texts often mention the concept of ""dimension"" when referring to parallel or alternate universes or other imagined planes of existence. This usage is derived from the idea that to travel to parallel/alternate universes/planes of existence one must travel in a direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth (or higher) spatial (or non-spatial) dimension, not the"
"four dimensions. This premise was not unheard of. Abstract five-dimensional space was already a legitimate mathematical construct, and the boson-exchange theories of Theodor Kaluza and Oskar Klein required a fifth dimension for the unification of gravitation with electromagnetism (Kaluza–Klein theory, 1921). Similarly, in the 1930s and 1940s, Albert Einstein had considered fifth-dimensional unification before turning to other approaches. But Lovelace did not stop with the fifth or sixth dimension. Instead, he continued to increase the number of dimensions until, strangely, at ""D"" = 26 the problem with tachyons vanished and unitarity was restored. Despite the earlier models that worked in"
"""The Philadelphia Inquirer"", about the influence of the fourth dimension on avant-garde painting; describing how the artists' employed ""harmonic use of forms"" distinguishing between the ""representation or rendering of space and the designing in space"": If we still further add to design in the third dimension, a consideration of weight, pressure, resistance, movement, as distinguished from motion, we arrive at what may legitimately be called design in the fourth dimension, or the harmonic use of what may arbitrarily be called volume. It is only at this point that we can appreciate the masterly productions of such a man as Cézanne."
"Four-vector In special relativity, a four-vector (also known as a 4-vector) is an object with four components, which transform in a specific way under Lorentz transformation. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (½,½) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). Four-vectors describe, for instance, position in spacetime modeled"
"three spatial coordinates plus one temporal. This is usually associated with theory of relativity, although four dimensions were used for such models since Galilei. The choice of theory leads to different structure, though: in Galilean relativity the coordinate is privileged, but in Einsteinian relativity it is not. Special relativity is set in Minkowski space. General relativity uses curved spaces, which may be thought of as with a curved metric for most practical purposes. None of these structures provide a (positive-definite) metric on . Euclidean also attracts the attention of mathematicians, for example due to its relation to quaternions, a 4-dimensional"
"Dynamical dimensional reduction Dynamical dimensional reduction or spontaneous dimensional reduction is the apparent reduction in the number of spacetime dimensions as a function of the distance scale, or conversely the energy scale, with which spacetime is probed. At least within the current level of experimental precision, our universe has three dimensions of space and one of time. However, the idea that the number of dimensions may increase at extremely small length scales was first proposed more than a century ago, and is now fairly commonplace in theoretical physics. Contrary to this, a number of recent results in quantum gravity suggest"
"flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy). The best-known treatment of time as a dimension is Poincaré and Einstein's special relativity (and extended to general relativity), which treats perceived space and time as components of a four-dimensional manifold, known as spacetime, and in the special, flat case as Minkowski space. In physics, three dimensions of space and one of time is the accepted norm. However, there are theories that attempt to unify the four fundamental forces by introducing extra dimensions. Most notably, superstring theory"