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Re: My Ingenius Intelect

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devilscheese1 is not online. devilscheese1
Joined: 01 Dec 2012
Total Posts: 105
21 Mar 2013 09:19 PM
Check it here is my science essay.
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it.[1] [2] thus a line has a dimension of one because only one coordinate is needed to specify a point on it (for example, the point at 5 on a number line). A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and its longitude). The inside of a cube, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces.
In physical terms, dimension refers to the constituent structure of all space (cf. volume) and its position in time (perceived as a scalar dimension along the t-axis), as well as the spatial constitution of objects within—structures that correlate with both particle and field conceptions, interact according to relative properties of mass—and are fundamentally mathematical in description. These, or other axes, may be referenced to uniquely identify a point or structure in its attitude and relationship to other objects and occurrences. Physical theories that incorporate time, such as general relativity, are said to work in 4-dimensional "spacetime", (defined as a Minkowski space). Modern theories tend to be "higher-dimensional" including quantum field and string theories. The state-space of quantum mechanics is an infinite-dimensional function space.
The concept of dimension is not restricted to physical objects. High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.
Contents
• 1 In mathematics
o 1.1 Dimension of a vector space
o 1.2 Manifolds
o 1.3 Varieties
o 1.4 Krull dimension
o 1.5 Lebesgue covering dimension
o 1.6 Inductive dimension
o 1.7 Hausdorff dimension
o 1.8 Hilbert spaces
• 2 In physics
o 2.1 Spatial dimensions
o 2.2 Time
o 2.3 Additional dimensions
• 3 Networks and dimension
• 4 Literature
• 5 Philosophy
• 6 More dimensions
• 7 See also
o 7.1 A list of topics indexed by dimension
• 8 References
• 9 Further reading
In mathematics
In mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. For example, a point on the unit circle in the plane can be specified by two Cartesian coordinates, but one can make do with a single coordinate (the polar coordinate angle), so the circle is 1-dimensional even though it exists in the 2-dimensional plane. This intrinsic notion of dimension is one of the chief ways the mathematical notion of dimension differs from its common usages.
The dimension of Euclidean n-space En is n. When trying to generalize to other types of spaces, one is faced with the question “what makes En n-dimensional?" One answer is that to cover a fixed ball in En by small balls of radius ε; one needs on the order of ε-n such small balls. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, but there are also other answers to that question. For example, one may observe that the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces.
A tesseract is an example of a four-dimensional object. Whereas outside of mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions", mathematicians usually express this as: "The tesseract has dimension 4", or: "The dimension of the tesseract is 4".
Although the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann. Riemann's 1854 Habilitationsschrift, Schlafi's 1852 Theorie der vielfachen Kontinuität, Hamilton's 1843 discovery of the quaternions and the construction of the Cayley Algebra marked the beginning of higher-dimensional geometry.
The rest of this section examines some of the more important mathematical definitions of the dimensions.
Dimension of a vector space
Main article: Dimension (vector space)
The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension.
Manifolds
A connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension. One can show that this yields a uniquely defined dimension for every connected topological manifold.
For connected differential manifolds the dimension is also the dimension of the tangent vector space at any point
The theory of manifolds, in the field of geometric topology, is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases n > 4 are simplified by having extra space in which to "work"; and the cases n = 3 and 4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Poincaré conjecture, where four different proof methods are applied.
Varieties
Main article: Dimension of an algebraic variety
The dimension of an algebraic variety may be defined in various equivalent ways. The most intuitive way is probably the dimension of the tangent space at any regular point. Another intuitive way is to define the dimension as the number of hyperplanes that are needed in order to have an intersection with the variety that is reduced to a finite number of points (dimension zero). This definition is based on the fact that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety.
An algebraic set being a finite union of algebraic varieties, its dimension is the maximum of the dimensions of its components. It is equal to the maximal length of the chains of sub varieties of the given variety (the length of such a chain is the number of " ").
Krull dimension
Main article: Krull dimension
The Krull dimension of a commutative ring is the maximal length of prime ideals in it. It is strongly related to the dimension of an algebraic variety, because of a natural correspondence between sub varieties and prime ideals of ring of the polynomials on the variety.
For an algebra over a field, the dimension as vector space is finite if and only if its Krull dimension is 0.
Lebesgue covering dimension
Main article: Lebesgue covering dimension
For any normal topological space X, the Lebesgue covering dimension of X is defined to be n if n is the smallest integer for which the following holds: any open cover has an open refinement (a second open cover where each element is a subset of an element in the first cover) such that no point is included in more than n + 1 elements. In this case dim X = n. For X a manifold, this coincides with the dimension mentioned above. If no such integer n exists, then the dimension of X is said to be infinite, and one writes dim X = ∞. Moreover, X has dimension −1, i.e. dim X = −1 if and only if X is empty. This definition of covering dimension can be extended from the class of normal spaces to all Tychonoff spaces merely by replacing the term "open" in the definition by the term "functionally open".
Inductive dimension
Main article: Inductive dimension
An inductive definition of dimension can be created as follows. Consider a discrete set of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a new direction, one obtains a 2-dimensional object. In general one obtains an (n + 1)-dimensional object by dragging an n dimensional object in a new direction.
The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that (n + 1)-dimensional balls have n dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.
Hausdorff dimension
Main article: Hausdorff dimension
For structurally complicated sets, especially fractals, the Hausdorff dimension is useful. The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values.[3] The box dimension or Minkowski dimension is a variant of the same idea. In general, there exist more definitions of fractal dimensions that work for highly irregular sets and attain non-integer positive real values. Fractals have been found useful to describe many natural objects and phenomena.
Hilbert spaces
Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the same cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the above dimensions coincide.
In physics
Spatial dimensions
Classical physics theories describe three physical dimensions: from a particular point in space, the basic directions in which we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative distance. Moving diagonally upward and forward is just as the name of the direction implies; i.e., moving in a linear combination of up and forward. In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions. (See Space and Cartesian coordinate system.)
Number of dimensions Example co-ordinate systems
1
Number line
Angle

2
Cartesian (2-dimensional)
Polar
Latitude and longitude

3
Cartesian (3-dimensional)
Cylindrical
Spherical

Time
A temporal dimension is a dimension of time. Time is often referred to as the "fourth dimension" for this reason, but that is not to imply that it is a spatial dimension. A temporal dimension is one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that we cannot move freely in time but subjectively move in one direction.
The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it. The equations of classical mechanics are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the perception of time 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.
Additional dimensions
In physics, three dimensions of space and one of time is the accepted norm. There are theories that try to unify different forces and such—these theories require more dimensions. Superstring theory, M-theory and Bosonic string theory respectively posit that physical space has 10, 11 and 26 dimensions. These extra dimensions are said to be spatial. However, we perceive only three spatial dimensions and, to date, no experimental or observational evidence is available to confirm the existence of these extra dimensions. A possible explanation that has been suggested is that space acts as if it were "curled up" in the extra dimensions on a subatomic scale, possibly at the quark/string level of scale or below.
An analysis of results from the Large Hadron Collider in December 2010 severely constrains theories with large extra dimensions.[6]
Networks and dimension
Some complex networks are characterized by fractal dimensions.[7] The concept of dimension can be generalized to include networks embedded in space.[8] The dimension characterizes their spatial constraints.
Literature
Perhaps the most basic way the word dimension is used in literature is as a hyperbolic synonym for feature, attribute, aspect, or magnitude. Frequently the hyperbole is quite literal as in he's so 2-dimensional, meaning that one can see at a glance what he is. This contrasts with 3-dimensional objects, which have an interior that is hidden from view, and a back that can only be seen with further examination.
Science fiction texts often mention the concept of dimension, when really referring to parallel universes, alternate universes, or other 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 standard ones.
One of the most heralded science fiction novellas regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novel Flatland by Edwin A. Abbott. Isaac Asimov, in his foreword to the Signet Classics 1984 edition, described Flatland as "The best introduction one can find into the manner of perceiving dimensions."
The idea of other dimensions was incorporated into many early science fiction stories, appearing prominently, for example, in Miles J. Breuer's The Appendix and the Spectacles (1928) and Murray Leinster's The Fifth-Dimension Catapult (1931); and appeared irregularly in science fiction by the 1940s. Some of the classic stories involving other dimensions include Robert A. Heinlein's 1941 —And He Built a Crooked House, in which a California architect designs a house based on a three-dimensional projection of a tesseract, and Alan E. Nourse's Tiger by the Tail and The Universe Between, both from 1951. Another reference is Madeleine L'Engle's novel A Wrinkle in Time (1962), which uses the 5th Dimension as a way for "tesseracting the universe" or in a better sense, "folding" space in half to move across it quickly. The fourth and fifth dimensions were also a key component of the book The Boy Who Reversed Himself, by William Sleator.
Philosophy
In 1783, Kant wrote: "That everywhere space (which is not itself the boundary of another space) has three dimensions and that space in general cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests immediately on intuition and indeed on pure intuition a priori because it is apodictically (demonstrably) certain."[9]



In mathematics, four-dimensional space ("4D") is an abstract concept derived by generalizing the rules of three-dimensional space. It has been studied by mathematicians and philosophers for almost three hundred years, both for its own interest and for the insights it offered into mathematics and related fields.
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 4-tuple) can be used to represent a position in four-dimensional 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 four-dimensional Minkowski continuum called space-time, whose metric treats the time dimension differently from the three spatial dimensions (see below for the definition of the Minkowski metric/pairing). Space-time is thus not a Euclidean space.
Contents
• 1 History
• 2 Vectors
• 3 Orthogonality and vocabulary
• 4 Geometry
o 4.1 Hypersphere
• 5 Cognition
• 6 Dimensional analogy
• 7 Cross-sections
o 7.1 Projections
o 7.2 Shadows
o 7.3 Bounding volumes
o 7.4 Visual scope
o 7.5 Limitations
• 8 See also
• 9 References
• 10 External links
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 three-dimensional form to be rotated onto its mirror-image,[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 (x1, ..., xn). 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 non-Euclidean, is profoundly different from that popularised by Hinton. The study of such Minkowski spaces required new mathematics quite different from that of four-dimensional 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 space-time is not Euclidean, and consequently has no connection with the present investigation.
—H. S. M. Coxeter, Regular Polytope]
Vectors
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 (e1, e2, e3, e4), 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 of a vector,

And calculate or define the angle between two vectors as

Minkowski spacetime is four-dimensional space with geometry defined by a nondegenerate pairing different from the dot product:

As an example, the distance squared between the points (0,0,0,0) and (1,1,1,0) is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between (0,0,0,0) and (1,1,1,1) is 4 in Euclidean space and 2 in Minkowski space; increasing actually decreases the metric distance. This leads to many of the well known apparent "paradoxes" of relativity.
The cross product is not defined in four dimensions. Instead the exterior product is used for some applications, and is defined as follows:

This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (e12, e13, e14, e23, e24, e34). They can be used to generate rotations in four dimensions.
Orthogonality and vocabulary
In the familiar 3-dimensional 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, 4-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 length measured along the w axis can be called spissitude, as coined by Henry More.
Geometry
See also: Rotations in 4-dimensional Euclidean space
The geometry of 4-dimensional space is much more complex than that of 3-dimensional 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 (4-polytopes) 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 semi-regular Archimedean solids in three dimensions.
Regular polytopes in four dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)
A4 BC4 F4 H4

5-cell

tesseract

16-cell

24-cell

120-cell

600-cell

In 3 dimensions, a circle may be extruded to form a cylinder. In 4 dimensions, there are several different cylinder-like objects. A sphere may be extruded to obtain a spherical cylinder (a cylinder with spherical "caps"), and a cylinder may be extruded to obtain a cylindrical prism. The Cartesian product of two circles may be taken to obtain a duocylinder. All three can "roll" in 4-dimensional space, each with its own properties.
In 3 dimensions, curves can form knots but surfaces cannot (unless they are self-intersecting). In 4 dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction, but 2-dimensional surfaces can form non-trivial, non-self-intersecting knots in 4-dimensional space. Because these surfaces are 2-dimensional, they can form much more complex knots than strings in 3-dimensional space can. The Klein bottle is an example of such a knotted surface. Another such surface is the real projective plane.
Hypersphere


Stereographic projection of a Clifford torus: the set of points (cos (a), sin (a), cos (b), sin (b)), which is a subset of the 3-sphere.
The set of points in Euclidean 4-space having the same distance R from a fixed point P0 forms a hypersurface known as a 3-sphere. The hyper-volume of the enclosed space is:

This is part of the Friedmann–Lemaître–Robertson–Walker metric in General relativity where R is substituted by function R(t) with t meaning the cosmological age of the universe. Growing or shrinking R with time means expanding or collapsing universe, depending on the mass density inside.[9]
Cognition
Research using virtual reality finds that humans in spite of living in a three-dimensional world can without special practice make spatial judgments based on the length of, and angle between, line segments embedded in four-dimensional space.[10] 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."[10] In another study,[11] 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.[12] 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).
Dimensional analogy


A net of a tesseract
To understand the nature of four-dimensional 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.[13]
Dimensional analogy was used by Edwin Abbott Abbott in the book Flatland, which narrates a story about a square that lives in a two-dimensional world, like the surface of a piece of paper. From the perspective of this square, a three-dimensional being has seemingly god-like 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 two-dimensional 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 four-dimensional being would be capable of similar feats from our three-dimensional perspective. Rudy Rucker illustrates this in his novel Spaceland, in which the protagonist encounters four-dimensional beings who demonstrate such powers.
Cross-sections
As a three-dimensional object passes through a two-dimensional plane, a two-dimensional being would only see a cross-section of the three-dimensional object. For example, if a balloon passed though a sheet of paper, a being on the paper would see a circle gradually grow larger, then smaller again. Similarly, if a four-dimensional object passed through three-dimensions, we would see a three-dimensional cross-section of the four-dimensional object–for example, a sphere.[14]
Projections
A useful application of dimensional analogy in visualizing the fourth dimension is in projection. A projection is a way for representing an n-dimensional object in n − 1 dimensions. For instance, computer screens are two-dimensional, and all the photographs of three-dimensional 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 two-dimensional array of receptors but the brain is able to perceive the nature of three-dimensional objects by inference from indirect information (such as shading, foreshortening, binocular vision, etc.). Artists often use perspective to give an illusion of three-dimensional depth to two-dimensional 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 four-dimensional eye is a three-dimensional array of receptors. A hypothetical being with such an eye would perceive the nature of four-dimensional objects by inferring four-dimensional depth from indirect information in the three-dimensional images in its retina.
The perspective projection of three-dimensional 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 four-dimensional "depth" from these effects.
As an illustration of this principle, the following sequence of images compares various views of the 3-dimensional cube with analogous projections of the 4-dimensional tesseract into three-dimensional space.
Cube Tesseract Description
The image on the left is a cube viewed face-on. The analogous viewpoint of the tesseract in 4 dimensions is the cell-first 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 edge-on. The analogous viewpoint of a tesseract is the face-first perspective projection, shown on the right. Just as the edge-first projection of the cube consists of two trapezoids, the face-first 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 corner-first. This is analogous to the edge-first perspective projection of the tesseract, shown on the right. Just as the cube's vertex-first projection consists of 3 deltoids surrounding a vertex, the tesseract's edge-first 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 edge-first projection of the tesseract and the edge-first projection of the cube. The cube's edge-first projection has two trapezoids surrounding an edge, while the tesseract has three hexahedral volumes surrounding an edge.
On the left is the cube viewed corner-first. The vertex-first perspective projection of the tesseract is shown on the right. The cube's vertex-first projection has three tetragons surrounding a vertex, but the tesseract's vertex-first 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 lies 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.
Shadows
A concept closely related to projection is the casting of shadows.

If a light is shone on a three dimensional object, a two-dimensional shadow is cast. By dimensional analogy, light shone on a two-dimensional object in a two-dimensional world would cast a one-dimensional shadow, and light on a one-dimensional object in a one-dimensional world would cast a zero-dimensional shadow, that is, a point of non-light. Going the other way, one may infer that light shone on a four-dimensional object in a four-dimensional world would cast a three-dimensional 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 three-dimensional cube within another three-dimensional cube. (Note that, technically, the visual representation shown here is actually a two-dimensional shadow of the three-dimensional shadow of the four-dimensional wireframe figure.)
Bounding volumes
Dimensional analogy also helps in inferring basic properties of objects in higher dimensions. For example, two-dimensional objects are bounded by one-dimensional boundaries: a square is bounded by four edges. Three-dimensional objects are bounded by two-dimensional surfaces: a cube is bounded by 6 square faces. By applying dimensional analogy, one may infer that a four-dimensional cube, known as a tesseract, is bounded by three-dimensional 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 three-dimensional projection of the tesseract. The boundaries of the tesseract project to volumes in the image, not merely two-dimensional surfaces.
Visual scope
Being three-dimensional, we are only able to see the world with our eyes in two dimensions. A four-dimensional 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 3-dimensional space simultaneously, including the inner structure of solid objects and things obscured from our three-dimensional viewpoint. Our brains receive images in the second dimension and use reasoning to help us "picture" three-dimensional objects. Just as a four-dimensional creature would probably receive multiple three-dimensional pictures.
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


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teez115 is not online. teez115
Joined: 07 Aug 2008
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21 Mar 2013 09:20 PM
pasted.
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Halogste is not online. Halogste
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21 Mar 2013 09:20 PM
tl;dr

Actually, I read half.
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Cocopelado is not online. Cocopelado
Joined: 14 Jan 2008
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21 Mar 2013 09:20 PM
I don't take too kindly to plagiarism.

I LIVE IN A VAN, DOWN BY THE RIVER.
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pusfrus is not online. pusfrus
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21 Mar 2013 09:21 PM
Copy and paste works wonder's doesn't it?
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ninetailfox73 is not online. ninetailfox73
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21 Mar 2013 09:24 PM
Might want to remove [1][2], obviously pasted.
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devilscheese1 is not online. devilscheese1
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22 May 2014 04:29 AM
It's not copy and pasted, this was submitted to my high school over the internet and required tags.
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Nteorvolri is not online. Nteorvolri
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22 May 2014 04:50 AM
oh great, someone else who claims they are in high school and copy pastes an article to make them look smart


seriously if you can easily claim you're still in elementary then you're fine
but i knew how to copy and paste huge things at a young age
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devilscheese1 is not online. devilscheese1
Joined: 01 Dec 2012
Total Posts: 105
23 May 2014 12:26 AM
Haha, screw it, I'm done lying. I am in high school but this essay isn't mine.
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IrrationalLogic is not online. IrrationalLogic
Joined: 17 Jan 2011
Total Posts: 1455
23 May 2014 12:31 AM
Good job on copying this massive wall of text.
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stealthyninja588 is not online. stealthyninja588
Joined: 30 Oct 2010
Total Posts: 4023
23 May 2014 12:53 AM
"The correct formula is "
i hate cliffhangers
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