Fourth dimension

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3D Shadow of a tesseract rotating around a plane in 4D.

In physics and mathematics, a sequence of n numbers can be understood as a location in an n-dimensional space. When n = 4, the set of all such locations is called 4-dimensional Euclidean space.

Such a space differs from our more familiar three-dimensional space in that it has an additional dimension, a new direction in which movement is possible. This fourth spatial dimension is a concept distinct from the time dimension in spacetime.

Contents

[edit] History

The possibility of spaces with dimensions higher than three was first studied by mathematicians in the 19th century. In 1827 Möbius realised that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image,[1] and by 1853 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, \ldots, x_n). The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established.

In 1880, Charles H. Hinton published his essay What is the Fourth Dimension? in the Dublin University magazine.[3] He also coined the term tesseract, referring to a four-dimensional cube.

In 1908, Hermann Minkowski presented a paper[4] which introduced the idea of time being the fourth dimension of spacetime, which would be the basis for Einstein's theories of Special and General Relativity.[5] This association of the fourth dimension with time rather than space has become the popular understanding of the term, even though it is applicable only to Einstein's theories of relativity. Nevertheless, mathematicians today continue to study the rich geometry of four-dimensional space regarding the fourth dimension as a spatial, and not temporal, dimension.

[edit] The fourth spatial dimension

In the spatial sense, the fourth dimension is a space with literally 4 spatial dimensions, or four mutually orthogonal directions of movement. This space, known as 4-dimensional Euclidean space, is the space used by mathematicians when studying geometric objects such as 4-dimensional polytopes. It is not to be confused with the Minkowskian notion of time being the fourth dimension. Regarding this, Coxeter writes:

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 J. W. 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 Polytopes[6]

Mathematically, the 4-dimensional spatial equivalent of conventional 3-dimensional geometry is the Euclidean 4-space, a 4-dimensional normed vector space with the Euclidean norm. The "length" of a vector

\mathbf{x} = (p, q, r, s)

expressed in the standard basis is given by

\| \mathbf{x} \| = \sqrt{p^{2} + q^{2} + r^{2} + s^{2}}

which is the natural generalization of the Pythagorean Theorem to 4 dimensions. This allows for the definition of distance between two points and the angle between two vectors (see Euclidean space for more information).

[edit] Orthogonality

In the familiar 3-dimensional space that we live in, there are three pairs of cardinal directions: up/down (altitude), north/south (latitude), and east/west (longitude). These pairs of directions are mutually orthogonal: they are at right angles to each other. Mathematically, they lie on three coordinate axes, usually labelled x, y, and z. The z-buffer in computer graphics refers to this z-axis, representing depth in the 2-dimensional imagery displayed on the computer screen.

A space of four spatial dimensions has an additional pair of cardinal directions which is orthogonal to the other three. This additional pair of directions lies on a fourth coordinate axis perpendicular to the x, y, and z axes, usually labelled w. Attested terms for these extra directions include ana/kata.

[edit] 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[7].

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 being able to remove objects from a safe without breaking it open (by moving them across the third dimension), being able to see everything that from the two-dimensional perspective is enclosed behind walls, and remaining 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 demonstrates this in his novel Spaceland, in which the protagonist encounters four-dimensional beings who demonstrate such powers.

[edit] 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 3-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 trapezoids 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 lie behind the 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 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 each of the corners connected. Similarly, if the wireframe of a four-dimensional cube were lit from “above” (in the fourth direction), its shadow would be that of a three-dimensional cube within another three-dimensional cube.

[edit] 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 squares. 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. This helps one understand features of such projections that may otherwise be very puzzling.

[edit] 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.

[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 C = 2πr and the surface area of a sphere: A = 4πr2. One might be tempted to suppose that the surface volume of a hypersphere is V = 6πr3, or perhaps V = 8πr3, but either of these would be wrong. The correct formula is V = 2π2r3.

[edit] Geometry

The geometry of 4-dimensional space is much richer than that of 3-dimensional space, due to the extra degree of freedom.

Just as in 3 dimensions, one may construct polyhedra from polygons, in 4 dimensions one may construct polychora (4-polytopes) from 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. In 3 dimensions, there are 13 Archimedean solids, whereas in 4 dimensions, there are 58 convex uniform polychora (64 including the regular polychora).

In 3 dimensions, one may extrude a circle to form a cylinder. In 4 dimensions, there are several different cylinder-like objects. One may extrude a sphere to obtain a spherical cylinder (a cylinder with spherical "caps"), or one may extrude a cylinder to obtain a cylindrical prism. One may also take the Cartesian product of two circles 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.

[edit] Hypersphere

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:

\mathbf V = 2 \pi^2 R^3

This is part of the 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.[8]

[edit] See also

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[edit] References

  1. ^ Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., p. 141.
  2. ^ Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., p.p. 142-143.
  3. ^ Rudolf v.B. Rucker, editor Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton, p. vii, Dover Publications Inc., 1980 ISBN 0486239160
  4. ^ Hermann Minkowski, "Raum und Zeit", 80. Versammlung Deutscher Naturforscher (Köln, 1908). Published in Physikalische Zeitschrift 10 104-111 (1909) and Jahresbericht der Deutschen Mathematiker-Vereinigung 18 75-88 (1909). For an English translation, see Lorentz et al. (1952).
  5. ^ C Møller (1952). The Theory of Relativity. Oxford UK: Clarendon Press. p. 93. ISBN 0198512562. http://books.google.com/books?id=Qu9SAAAAMAAJ&dq=intitle:the+intitle:theory+intitle:of+intitle:relativity+inauthor:moller&lr=&as_brr=0. 
  6. ^ Coxeter, H. S. M. (1973). Regular Polytopes, Dover Publications, Inc., p. 119.
  7. ^ 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 0-19-286189-1.
  8. ^ Ray d'Inverno (1992), Introducing Einstein's Relativity, Clarendon Press, chp. 22.8 Geometry of 3-spaces of constant curvature, p.319ff, ISBN 0-19-859653-7

[edit] External links

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