Fourdimensional space
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A fourdimensional space or 4D space is a mathematical extension of the concept of threedimensional or 3D space. Threedimensional 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, width, and height (often labeled x, y, and z).
The idea of adding a fourth dimension began with Jean le Rond d'Alembert with his "Dimensions" published in 1754^{[1]} followed by JosephLouis Lagrange in the mid1700s and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann. In 1880 Charles Howard Hinton popularized these insights in an essay titled "What is the Fourth Dimension?", which explained the concept of a fourdimensional cube with a stepbystep generalization of the properties of lines, squares, and cubes. The simplest form of Hinton's method is to draw two ordinary cubes separated by an "unseen" distance, and then draw lines between their equivalent vertices. This can be seen in the accompanying animation, whenever it shows a smaller inner cube inside a larger outer cube. The eight lines connecting the vertices of the two cubes in that case represent a single direction in the "unseen" fourth dimension.
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 ntuples, 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 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.
Contents
HistoryEdit
Lagrange wrote in his Mécanique analytique (published 1788, based on work done around 1755) that mechanics can be viewed as operating in a fourdimensional space — three dimensions of space, and one of time.^{[2]} In 1827 Möbius realized that a fourth dimension would allow a threedimensional form to be rotated onto its mirrorimage,^{[3]}^{:141} and by 1853 Ludwig Schläfli had discovered many polytopes in higher dimensions, although his work was not published until after his death.^{[3]}^{:142–143} Higher dimensions were soon put on firm footing by Bernhard Riemann's 1854 thesis, 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. Soon after tessarines and coquaternions were introduced as other fourdimensional algebras over R.
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.^{[4]} 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.^{[5]}^{[6]}
Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by Martin Gardner in his January 1962 "Mathematical Games column" in Scientific American. In 1886 Victor Schlegel described^{[7]} his method of visualizing fourdimensional objects with Schlegel diagrams.
In 1908, Hermann Minkowski presented a paper^{[8]} consolidating the role of time as the fourth dimension of spacetime, the basis for Einstein's theories of special and general relativity.^{[9]} But the geometry of spacetime, being nonEuclidean, is profoundly different from that popularised by Hinton. The study of Minkowski space 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^{[3]}^{:119}
VectorsEdit
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 to
This can be written in terms of the four standard basis vectors (e_{1}, e_{2}, e_{3}, e_{4}), given by
so the general vector a is
Vectors add, subtract and scale as in three dimensions.
The dot product of Euclidean threedimensional 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 nonzero vectors as
Minkowski spacetime is fourdimensional 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 4spaces, 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 wellknown 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 sixdimensional linear space with basis (e_{12}, e_{13}, e_{14}, e_{23}, e_{24}, e_{34}). They can be used to generate rotations in four dimensions.
Orthogonality and vocabularyEdit
In the familiar threedimensional space in which we live, 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, fourdimensional 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.
GeometryEdit
The geometry of fourdimensional space is much more complex than that of threedimensional space, due to the extra degree of freedom.
Just as in three dimensions there are polyhedra made of two dimensional polygons, in four dimensions there are 4polytopes made of polyhedra. In three dimensions, there are 5 regular polyhedra known as the Platonic solids. In four dimensions, there are 6 convex regular 4polytopes, the analogues of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform 4polytopes, analogous to the 13 semiregular Archimedean solids in three dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4polytopes.
A_{4}, [3,3,3]  B_{4}, [4,3,3]  F_{4}, [3,4,3]  H_{4}, [5,3,3]  

5cell {3,3,3} 
tesseract {4,3,3} 
16cell {3,3,4} 
24cell {3,4,3} 
120cell {5,3,3} 
600cell {3,3,5} 
In three dimensions, a circle may be extruded to form a cylinder. In four dimensions, there are several different cylinderlike objects. A sphere may be extruded to obtain a spherical cylinder (a cylinder with spherical "caps", known as a spherinder), and a cylinder may be extruded to obtain a cylindrical prism (a cubinder). The Cartesian product of two circles may be taken to obtain a duocylinder. All three can "roll" in fourdimensional space, each with its own properties.
In three dimensions, curves can form knots but surfaces cannot (unless they are selfintersecting). In four dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction—but 2D surfaces can form nontrivial, nonselfintersecting knots in 4D space.^{[10]}^{[page needed]} Because these surfaces are twodimensional, they can form much more complex knots than strings in 3D 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]}
HypersphereEdit
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:
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.^{[11]}
CognitionEdit
Research using virtual reality finds that humans, in spite of living in a threedimensional world, can, without special practice, make spatial judgments about line segments, embedded in fourdimensional space, based on their length (one dimensional) and the angle (two dimensional) between them.^{[12]} 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."^{[12]} In another study,^{[13]} 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.^{[14]} 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 lowerdimensional cases were for comparison and for the participants to learn the method).
Dimensional analogyEdit
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.^{[15]}
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.
CrosssectionsEdit
As a threedimensional object passes through a twodimensional plane, twodimensional beings in this plane would only observe a crosssection of the threedimensional object within this plane. For example, if a spherical balloon passed through a sheet of paper, beings in the paper would see first a single point, then a circle gradually growing larger, until it reaches the diameter of the balloon, and then getting smaller again, until it shrank to a point and then disappeared. Similarly, if a fourdimensional object passed through a three dimensional (hyper)surface, one could observe a threedimensional crosssection of the fourdimensional object—for example, a 4sphere would appear first as a point, then as a growing sphere, with the sphere then shrinking to a single point and then disappearing.^{[16]} This means of visualizing aspects of the fourth dimension was used in the novel Flatland and also in several works of Charles Howard Hinton.^{[5]}^{:11–14}
ProjectionsEdit
A useful application of dimensional analogy in visualizing higher dimensions 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. By doing this, the dimension orthogonal to the screen (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. The shadow, cast by a fictitious grid model of a rotating tesseract on a plane surface, as shown in the figures, is also the result of projections.
Similarly, objects in the fourth dimension can be mathematically projected to the familiar three 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 threedimensional cube with analogous projections of the fourdimensional 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. 
ShadowsEdit
A concept closely related to projection is the casting of shadows.
If a light is shone on a threedimensional 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 on a flat twodimensional surface 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 dimension), its shadow would be that of a threedimensional cube within another threedimensional cube suspended in midair (a "flat" surface from a fourdimensional perspective). (Note that, technically, the visual representation shown here is actually a twodimensional image of the threedimensional shadow of the fourdimensional wireframe figure.)
Bounding volumesEdit
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.
Visual scopeEdit
We spatially perceive ourselves as beings in a threedimensional space, but visually we are restricted to one dimension less: we see the world with our eyes as projections to two dimensions, on the surface of the retina. Assuming a fourdimensional being were able to "see" his world in projections to a hypersurface, also just one dimension less, i.e., to three dimensions, it would be able to "see", e.g., 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 all four sides and simultaneously the interior of a rectangle on a piece of paper. The being would be able to discern all points in a 3dimensional subspace simultaneously, including the inner structure of solid 3dimensional objects, things obscured from our viewpoints in three dimensions on twodimensional projections. Our brains receive images in two dimensions and use reasoning to help us "picture" threedimensional objects.
LimitationsEdit
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 .^{[3]}^{:119}
See alsoEdit
ReferencesEdit
 ^ Cajori, Florian (1926), , The American Mathematical Monthly, 10: 403–407
 ^ Bell, E.T. (1965). Men of Mathematics (1st ed.). New York: Simon and Schuster. p. 154. ISBN 9780671628185.
 ^ ^{a} ^{b} ^{c} ^{d} Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover Publishing. ISBN 9780486614809.
 ^ Hinton, Charles Howard (1980). Rucker, Rudolf v. B., ed. Speculations on the Fourth Dimension: Selected writings of Charles H. Hinton. New York: Dover. p. vii. ISBN 9780486239163.
 ^ ^{a} ^{b} Hinton, Charles Howard (1993) [1904]. The Fourth Dimension. Pomeroy, Washington: Health Research. p. 14. ISBN 9780787304102. Retrieved 17 February 2017.
 ^ Gardner, Martin (1975). Mathematical Carnival: From Penny Puzzles. Card Shuffles and Tricks of Lightning Calculators to Roller Coaster Rides into the Fourth Dimension (1st ed.). New York: Knopf. pp. 42, 52–53. ISBN 9780394494067.
 ^ Victor Schlegel (1886) Ueber Projectionsmodelle der regelmässigen vierdimensionalen Körper, Waren
 ^ Minkowski, Hermann (1909),
 Various English translations on Wikisource: Space and Time
 ^ Møller, C. (1972). The Theory of Relativity (2nd ed.). Oxford: Clarendon Press. p. 93. ISBN 9780198512561.
 ^ Carter, J.Scott; Saito, Masahico. Knotted Surfaces and Their Diagrams. American Mathematical Society. ISBN 9780821874912.
 ^ D'Inverno, Ray (1998). Introducing Einstein's Relativity (Reprint ed.). Oxford: Clarendon Press. p. 319. ISBN 9780198596530.
 ^ ^{a} ^{b} Ambinder, Michael S.; Wang, Ranxiao Frances; Crowell, James A.; Francis, George K.; Brinkmann, Peter (October 2009). "Human fourdimensional spatial intuition in virtual reality". Psychonomic Bulletin & Review. 16 (5): 818–823. doi:10.3758/PBR.16.5.818. PMID 19815783. Retrieved 17 February 2017.
 ^ Aflalo, T. N.; Graziano, M. S. A. (2008). "Fourdimensional spatial reasoning in humans" (PDF). Journal of Experimental Psychology: Human Perception and Performance. 34 (5): 1066–1077. CiteSeerX 10.1.1.505.5736. doi:10.1037/00961523.34.5.1066. PMID 18823195. Retrieved 17 February 2017.
 ^ "4D Maze Game". urticator.net. Retrieved 20161216.
 ^ Kaku, Michio (1995). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension (reissued ed.). Oxford: Oxford University Press. pp. Part I, Chapter 3. ISBN 9780192861894.
 ^ Rucker, Rudy (1996). The Fourth Dimension: A Guided Tour of the Higher Universe. Boston: Houghton Mifflin. p. 18. ISBN 9780395393888.
Further readingEdit
 Archibald, R. C (1914). "Time as a Fourth Dimension" (PDF). American Mathematical Society: 409–412.
 Andrew Forsyth (1930) Geometry of Four Dimensions, link from Internet Archive.
 Gamow, George (1988). One Two Three . . . Infinity: Facts and Speculations of Science (3rd ed.). Courier Dover Publications. p. 68. ISBN 9780486256641. Extract of page 68
 E. H. Neville (1921) The Fourth Dimension, Cambridge University Press, link from University of Michigan Historical Math Collection.
External linksEdit
Wikibooks has a book on the topic of: Special Relativity 