The concept of the Fourth Dimension has been used as an explanation for the manifestation of claimed psyhic, and mystical phenomena. Unfortunately, like Quantum Mechanics its understanding is incomplete without the development of MultiDimensional Science..
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Algebraically it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular a vector with four elements (a 4tuple) can be used to represent a position in fourdimensional space. The space is a Euclidean space, so has a metric and norm, and so all directions are treated as the same: the additional dimension is indistinguishable from the other three.
In modern physics, space and time are unified in a fourdimensional Minkowski continuum called spacetime, whose metric treats the time dimension differently from the three spatial dimensions (see below for the definition of the Minkowski metric/pairing). Spacetime is thus not a Euclidean space.
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[edit] History
See also: ndimensional space#History
The possibility of spaces with dimensions higher than three was first studied by mathematicians in the 19th century. In 1827 Möbius realized that a fourth dimension would allow a threedimensional form to be rotated onto its mirrorimage,^{[1]} and by 1853 Ludwig Schläfli had discovered many polytopes in higher dimensions, although his work was not published until after his death.^{[2]} Higher dimensions were soon put on firm footing by Bernhard Riemann's 1854 Habilitationsschrift, Über die Hypothesen welche der Geometrie zu Grunde liegen, in which he considered a "point" to be any sequence of coordinates (x_{1}, ..., x_{n}). The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established.An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843. This associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis.
One of the first major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay What is the Fourth Dimension?; published in the Dublin University magazine.^{[3]} He coined the terms tesseract, ana and kata in his book A New Era of Thought, and introduced a method for visualising the fourth dimension using cubes in the book Fourth Dimension.^{[4]}^{[5]}
In 1908, Hermann Minkowski presented a paper^{[6]} consolidating the role of time as the fourth dimension of spacetime, the basis for Einstein's theories of special and general relativity.^{[7]} But the geometry of spacetime, being nonEuclidean, is profoundly different from that popularised by Hinton. The study of such Minkowski spaces required new mathematics quite different from that of fourdimensional Euclidean space, and so developed along quite different lines. This separation was less clear in the popular imagination, with works of fiction and philosophy blurring the distinction, so in 1973 H. S. M. Coxeter felt compelled to write:
Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H. G. Wells in The Time Machine, has led such authors as John William Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski's geometry of spacetime is not Euclidean, and consequently has no connection with the present investigation.—H. S. M. Coxeter, Regular Polytopes^{[8]}
[edit] Vectors
Mathematically fourdimensional 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 toThe dot product of Euclidean threedimensional space generalizes to four dimensions as
The cross product is not defined in four dimensions. Instead the exterior product is used for some applications, and is defined as follows:
[edit] Orthogonality and vocabulary
In the familiar 3dimensional space that we live in there are three coordinate axes — usually labeled x, y, and z — with each axis orthogonal (i.e. perpendicular) to the other two. The six cardinal directions in this space can be called up, down, east, west, north, and south. Positions along these axes can be called altitude, longitude, and latitude. Lengths measured along these axes can be called height, width, and depth.Comparatively, 4dimensional 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 length measured along the w axis can be called spissitude, as coined by Henry More.
[edit] Geometry
The geometry of 4dimensional space is much more complex than that of 3dimensional space, due to the extra degree of freedom.Just as in 3 dimensions there are polyhedra made of two dimensional polygons, in 4 dimensions there are polychora (4polytopes) made of polyhedra. In 3 dimensions there are 5 regular polyhedra known as the Platonic solids. In 4 dimensions there are 6 convex regular polychora, the analogues of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform polychora, analogous to the 13 semiregular Archimedean solids in three dimensions.
A_{4}  BC_{4}  F_{4}  H_{4}  

5cell  tesseract  16cell  24cell  120cell  600cell 
In 3 dimensions, curves can form knots but surfaces cannot (unless they are selfintersecting). In 4 dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction, but 2dimensional surfaces can form nontrivial, nonselfintersecting knots in 4dimensional space ^{[9]}. Because these surfaces are 2dimensional, they can form much more complex knots than strings in 3dimensional space can. The Klein bottle is an example of such a knotted surface^{[citation needed]}. Another such surface is the real projective plane^{[citation needed]}.
[edit] Hypersphere
The set of points in Euclidean 4space having the same distance R from a fixed point P_{0} forms a hypersurface known as a 3sphere. The hypervolume of the enclosed space is:[edit] Cognition
Research using virtual reality finds that humans in spite of living in a threedimensional world can without special practice make spatial judgments based on the length of, and angle between, line segments embedded in fourdimensional space.^{[11]} The researchers noted that "the participants in our study had minimal practice in these tasks, and it remains an open question whether it is possible to obtain more sustainable, definitive, and richer 4D representations with increased perceptual experience in 4D virtual environments."^{[11]} In another study,^{[12]} the ability of humans to orient themselves in 2D, 3D and 4D mazes has been tested. Each maze consisted of four path segments of random length and connected with orthogonal random bends, but without branches or loops (i.e. actually labyrinths). The graphical interface was based on John McIntosh's free 4D Maze game.^{[13]} The participating persons had to navigate through the path and finally estimate the linear direction back to the starting point. The researchers found that some of the participants were able to mentally integrate their path after some practice in 4D (the lower dimensional cases were for comparison and for the participants to learn the method).[edit] Dimensional analogy
To understand the nature of fourdimensional space, a device called dimensional analogy is commonly employed. Dimensional analogy is the study of how (n − 1) dimensions relate to n dimensions, and then inferring how n dimensions would relate to (n + 1) dimensions.^{[14]}Dimensional analogy was used by Edwin Abbott Abbott in the book Flatland, which narrates a story about a square that lives in a twodimensional world, like the surface of a piece of paper. From the perspective of this square, a threedimensional being has seemingly godlike powers, such as ability to remove objects from a safe without breaking it open (by moving them across the third dimension), to see everything that from the twodimensional perspective is enclosed behind walls, and to remain completely invisible by standing a few inches away in the third dimension.
By applying dimensional analogy, one can infer that a fourdimensional being would be capable of similar feats from our threedimensional perspective. Rudy Rucker illustrates this in his novel Spaceland, in which the protagonist encounters fourdimensional beings who demonstrate such powers.
[edit] Crosssections
As a threedimensional object passes through a twodimensional plane, a twodimensional being would only see a crosssection of the threedimensional object. For example, if a balloon passed through a sheet of paper, a being on the paper would see a circle gradually grow larger, then smaller again. Similarly, if a fourdimensional object passed through threedimensions, we would see a threedimensional crosssection of the fourdimensional object–for example, a sphere.^{[15]}[edit] Projections
A useful application of dimensional analogy in visualizing the fourth dimension is in projection. A projection is a way for representing an ndimensional object in n − 1 dimensions. For instance, computer screens are twodimensional, and all the photographs of threedimensional people, places and things are represented in two dimensions by projecting the objects onto a flat surface. When this is done, depth is removed and replaced with indirect information. The retina of the eye is also a twodimensional array of receptors but the brain is able to perceive the nature of threedimensional objects by inference from indirect information (such as shading, foreshortening, binocular vision, etc.). Artists often use perspective to give an illusion of threedimensional depth to twodimensional pictures.Similarly, objects in the fourth dimension can be mathematically projected to the familiar 3 dimensions, where they can be more conveniently examined. In this case, the 'retina' of the fourdimensional eye is a threedimensional array of receptors. A hypothetical being with such an eye would perceive the nature of fourdimensional objects by inferring fourdimensional depth from indirect information in the threedimensional images in its retina.
The perspective projection of threedimensional objects into the retina of the eye introduces artifacts such as foreshortening, which the brain interprets as depth in the third dimension. In the same way, perspective projection from four dimensions produces similar foreshortening effects. By applying dimensional analogy, one may infer fourdimensional "depth" from these effects.
As an illustration of this principle, the following sequence of images compares various views of the 3dimensional cube with analogous projections of the 4dimensional tesseract into threedimensional space.
Cube  Tesseract  Description 

The image on the left is a cube viewed faceon. The analogous viewpoint of the tesseract in 4 dimensions is the cellfirst perspective projection, shown on the right. One may draw an analogy between the two: just as the cube projects to a square, the tesseract projects to a cube.Note that the other 5 faces of the cube are not seen here. They are obscured by the visible face. Similarly, the other 7 cells of the tesseract are not seen here because they are obscured by the visible cell.  
The image on the left shows the same cube viewed edgeon. The analogous viewpoint of a tesseract is the facefirst perspective projection, shown on the right. Just as the edgefirst projection of the cube consists of two trapezoids, the facefirst projection of the tesseract consists of two frustums.The nearest edge of the cube in this viewpoint is the one lying between the red and green faces. Likewise, the nearest face of the tesseract is the one lying between the red and green cells.  
On the left is the cube viewed cornerfirst. This is analogous to the edgefirst perspective projection of the tesseract, shown on the right. Just as the cube's vertexfirst projection consists of 3 deltoids surrounding a vertex, the tesseract's edgefirst projection consists of 3 hexahedral volumes surrounding an edge. Just as the nearest vertex of the cube is the one where the three faces meet, so the nearest edge of the tesseract is the one in the center of the projection volume, where the three cells meet.  
A different analogy may be drawn between the edgefirst projection of the tesseract and the edgefirst projection of the cube. The cube's edgefirst projection has two trapezoids surrounding an edge, while the tesseract has three hexahedral volumes surrounding an edge.  
On the left is the cube viewed cornerfirst. The vertexfirst perspective projection of the tesseract is shown on the right. The cube's vertexfirst projection has three tetragons surrounding a vertex, but the tesseract's vertexfirst projection has four hexahedral volumes surrounding a vertex. Just as the nearest corner of the cube is the one lying at the center of the image, so the nearest vertex of the tesseract lies not on boundary of the projected volume, but at its center inside, where all four cells meet.Note that only three faces of the cube's 6 faces can be seen here, because the other 3 lie behind these three faces, on the opposite side of the cube. Similarly, only 4 of the tesseract's 8 cells can be seen here; the remaining 4 lie behind these 4 in the fourth direction, on the far side of the tesseract. 
[edit] Shadows
A concept closely related to projection is the casting of shadows.If a light is shone on a three dimensional object, a twodimensional shadow is cast. By dimensional analogy, light shone on a twodimensional object in a twodimensional world would cast a onedimensional shadow, and light on a onedimensional object in a onedimensional world would cast a zerodimensional shadow, that is, a point of nonlight. Going the other way, one may infer that light shone on a fourdimensional object in a fourdimensional world would cast a threedimensional shadow.
If the wireframe of a cube is lit from above, the resulting shadow is a square within a square with the corresponding corners connected. Similarly, if the wireframe of a tesseract were lit from “above” (in the fourth direction), its shadow would be that of a threedimensional cube within another threedimensional cube. (Note that, technically, the visual representation shown here is actually a twodimensional shadow of the threedimensional shadow of the fourdimensional wireframe figure.)
[edit] Bounding volumes
Dimensional analogy also helps in inferring basic properties of objects in higher dimensions. For example, twodimensional objects are bounded by onedimensional boundaries: a square is bounded by four edges. Threedimensional objects are bounded by twodimensional surfaces: a cube is bounded by 6 square faces. By applying dimensional analogy, one may infer that a fourdimensional cube, known as a tesseract, is bounded by threedimensional volumes. And indeed, this is the case: mathematics shows that the tesseract is bounded by 8 cubes. Knowing this is key to understanding how to interpret a threedimensional projection of the tesseract. The boundaries of the tesseract project to volumes in the image, not merely twodimensional surfaces.[edit] Visual scope
Being threedimensional, we are only able to see the world with our eyes in two dimensions. A fourdimensional being would be able to see the world in three dimensions. For example, it would be able to see all six sides of an opaque box simultaneously, and in fact, what is inside the box at the same time, just as we can see the interior of a square on a piece of paper. It would be able to see all points in 3dimensional space simultaneously, including the inner structure of solid objects and things obscured from our threedimensional viewpoint. Our brains receive images in the second dimension and use reasoning to help us "picture" threedimensional objects. Just as a fourdimensional creature would probably receive multiple threedimensional pictures.^{[citation needed]}[edit] Limitations
Reasoning by analogy from familiar lower dimensions can be an excellent intuitive guide, but 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 and the surface area of a sphere: . One might be tempted to suppose that the surface volume of a hypersphere is , or perhaps , but either of these would be wrong. The correct formula is .^{[8]}[edit] See also
Wikisource has original text related to this article: 
 Euclidean space
 Euclidean geometry
 4manifold
 Exotic R^{4}
 Dimension
 Fourdimensionalism
 Fifth dimension
 Sixth dimension
 Polychoron
 Polytope
 List of geometry topics
 Block Theory of the Universe
 Flatland, a book by Edwin A. Abbott about two and threedimensional spaces, to understand the concept of four dimensions
 Sphereland, an unofficial sequel to Flatland
 Charles Howard Hinton
 Dimensions, a set of films about two, three and fourdimensional polytopes
 List of fourdimensional games
[edit] References
 ^ Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., p. 141.
 ^ Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., pp. 142–143.
 ^ Rudolf v.B. Rucker, editor Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton, p. vii, Dover Publications Inc., 1980 ISBN 0486239160
 ^ Hinton, Charles Howard (1904). Fourth Dimension. ISBN 156459708. http://www.archive.org/details/fourthdimension00hintarch.
 ^ Gardner, Martin (1975). Mathematical Carnival. Knopf Publishing. pp. 42, 52–53. ISBN 0 14 02.2041 0.
 ^ Hermann Minkowski, "Raum und Zeit", 80. Versammlung Deutscher Naturforscher (Köln, 1908). Published in Physikalische Zeitschrift 10 104–111 (1909) and Jahresbericht der Deutschen MathematikerVereinigung 18 75–88 (1909). For an English translation, see Lorentz et al. (1952).
 ^ C Møller (1952). The Theory of Relativity. Oxford UK: Clarendon Press. p. 93. ISBN 0198512562.
 ^ ^{a} ^{b} Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., p. 119.
 ^ J. Scott Carter, Masahico Saito Knotted Surfaces and Their Diagrams
 ^ Ray d'Inverno (1992), Introducing Einstein's Relativity, Clarendon Press, chp. 22.8 Geometry of 3spaces of constant curvature, p.319ff, ISBN 0198596537
 ^ ^{a} ^{b} Ambinder MS, Wang RF, Crowell JA, Francis GK, Brinkmann P. (2009). Human fourdimensional spatial intuition in virtual reality. Psychon Bull Rev. 16(5):81823. doi:10.3758/PBR.16.5.818 PMID 19815783 online supplementary material
 ^ Aflalo TN, Graziano MS (2008). FourDimensional Spatial Reasoning in Humans. Journal of Experimental Psychology: Human Perception and Performance 34(5):10661077. doi:10.1037/00961523.34.5.1066 Preprint
 ^ John McIntosh's four dimensional maze game. Free software
 ^ Michio Kaku (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension, Part I, chapter 3, The Man Who "Saw" the Fourth Dimension (about tesseracts in years 1870–1910). ISBN 0192861891.
 ^ Gamow, George (1988). One Two Three . . . Infinity: Facts and Speculations of Science (3rd ed.). Courier Dover Publications. p. 68. ISBN 0486256642. http://books.google.com/books?id=EZbcwk6SkhcC., Extract of page 68
[edit] External links
Wikibooks has a book on the topic of: Special Relativity 
 "Dimensions" videos, showing several different ways to visualize four dimensional objects
 Science News article summarizing the "Dimensions" videos, with clips
 Garrett Jones' tetraspace page
 Flatland: a Romance of Many Dimensions (second edition)
 TeV scale gravity, mirror universe, and ... dinosaurs Article from Acta Physica Polonica B by Z.K. Silagadze.
 Exploring Hyperspace with the Geometric Product
 4D Euclidean space
 4D Building Blocks  Interactive game to explore 4D space
 4DNav  A small tool to view a 4D space as four 3D space uses ADSODA algorithm
 MagicCube 4D A 4dimensional analog of traditional Rubik's Cube.
 Framebyframe animations of 4D  3D analogies

Blogger Reference Link http://www.p2pfoundation.net/MultiDimensional_Science
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