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| {{Refimprove|date=February 2009}}
| | Nice to meet you, my title is Refugia. Minnesota has always been his house but his wife wants them to transfer. For many years I've been operating as a payroll clerk. What I love doing is to collect badges but I've been taking on new things recently.<br><br>Also visit my web site: [http://mmservice.dk/weightlossfoodprograms67803 mmservice.dk] |
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| [[File:8-cell-simple.gif|frame|right|3D projection of a [[tesseract]] undergoing a [[Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in four dimensional space.]]
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| 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 over two centuries, both for its own interest and for the insights it offered into mathematics and related fields.
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| Algebraically it is generated by applying the rules of [[vector (mathematics)|vector]]s and [[coordinate geometry]] to a space with four dimensions. In particular a [[Euclidean vector|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 [[Euclidean distance|metric]] and [[Euclidean norm|norm]], and so all directions are treated as the same: the additional dimension is indistinguishable from the other three.
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| In modern [[physics]], [[space (mathematics)|space]] and [[time]] are unified in a four-dimensional [[Minkowski space|Minkowski continuum]] called [[spacetime]], whose [[Metric space|metric]] treats the time dimension differently from the three spatial dimensions (see [[#Vectors|below]] for the definition of the Minkowski metric/pairing). Spacetime is thus ''not'' a Euclidean space.
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| ==History==
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| {{see also|n-dimensional space#History}}
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| [[Joseph-Louis Lagrange|Lagrange]] wrote in his ''Mécanique analytique'' (published 1788, based on work done around 1755) that [[mechanics]] can be viewed as operating in a four-dimensional space — three of dimensions of space, and one of time.<ref>Bell, E.T. (1937). ''Men of Mathematics'', Simon and Schuster, p. 154.</ref> In 1827 [[August Ferdinand Möbius|Möbius]] realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image,<ref>Coxeter, H. S. M. (1973). ''Regular Polytopes'', Dover Publications, Inc., p. 141.</ref> and by 1853 [[Ludwig Schläfli]] had discovered many [[polytope]]s in higher dimensions, although his work was not published until after his death.<ref>Coxeter, H. S. M. (1973). ''Regular Polytopes'', Dover Publications, Inc., pp. 142–143.</ref> 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''<sub>1</sub>, ..., ''x''<sub>''n''</sub>). The possibility of geometry in [[higher dimension]]s, including four dimensions in particular, was thus established.
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| An arithmetic of four dimensions called [[quaternion]]s 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]]''.
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| 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.<ref>Rudolf v.B. Rucker, editor ''Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton'', p. vii, Dover Publications Inc., 1980 ISBN 0-486-23916-0</ref> 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''.<ref>{{cite book
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| | title = Fourth Dimension
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| | last = Hinton
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| | url = http://www.archive.org/details/fourthdimension00hintarch
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| | first = Charles Howard
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| | year = 1904
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| | isbn = 1-5645-9708-3}}</ref><ref>{{cite book
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| | title = Mathematical Carnival
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| | year = 1975
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| | last = Gardner
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| | first = Martin
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| | authorlink = Martin Gardner
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| | publisher = [[Knopf Publishing]]
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| | isbn = 0-394-49406-7
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| | pages = 42, 52–53}}</ref>
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| In 1908, [[Hermann Minkowski]] presented a paper<ref>{{Citation|author=Minkowski, Hermann|year=1909|title=[[s:de:Raum und Zeit (Minkowski)|Raum und Zeit]]|journal=Physikalische Zeitschrift|volume=10|pages=75–88}}
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| *Various English translations on Wikisource: [[s:Space and Time|Space and Time]]
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| </ref> consolidating the role of time as the fourth dimension of [[spacetime]], the basis for [[Albert Einstein|Einstein's]] theories of [[Special relativity|special]] and [[general relativity]].<ref name=Møller>
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| {{cite book
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| |author=C Møller
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| |title=The Theory of Relativity
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| |year= 1952
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| |page=93
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| |publisher=Clarendon Press
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| |location=Oxford UK
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| |isbn=0-19-851256-2}}
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| </ref> But the geometry of spacetime, being [[non-Euclidean]], is profoundly different from that popularised by Hinton. The study of such [[Minkowski space]]s 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:
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| {{quote|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.
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| |[[H. S. M. Coxeter]]|''Regular Polytopes''<ref name="Coxeter, H. S. M. 1973 p. 119">Coxeter, H. S. M. (1973). ''Regular Polytopes'', Dover Publications, Inc., p. 119.</ref>}}
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| ==Vectors==
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| Mathematically four-dimensional space is simply a space with four spatial dimensions, that is a [[space (mathematics)|space]] that needs four parameters to specify a [[point (geometry)|point]] in it. For example, a general point might have position [[Euclidean vector|vector]] '''a''', equal to
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| : <math>\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{pmatrix}.</math>
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| This can be written in terms of the four [[standard basis]] vectors ('''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>, '''e'''<sub>4</sub>), given by
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| :<math>\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}; \mathbf{e}_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}, </math>
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| so the general vector '''a''' is
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| : <math> \mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 + a_4\mathbf{e}_4.</math>
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| Vectors add, subtract and scale as in three dimensions.
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| The [[dot product]] of Euclidean three-dimensional space generalizes to four dimensions as
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| : <math>\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4.</math>
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| It can be used to calculate the [[norm (mathematics)|norm]] or [[Euclidean distance|length]] of a vector,
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| :<math> \left| \mathbf{a} \right| = \sqrt{\mathbf{a} \cdot \mathbf{a} } = \sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2},</math>
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| and calculate or define the [[angle]] between two vectors as
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| :<math> \theta = \arccos{\frac{\mathbf{a} \cdot \mathbf{b}}{\left|\mathbf{a}\right| \left|\mathbf{b}\right|}}.</math>
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| Minkowski spacetime is four-dimensional space with geometry defined by a nondegenerate [[pairing]] different from the dot product:
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| : <math>\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 - a_4 b_4.</math>
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| 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 <math>b_4</math> actually decreases the metric distance. This leads to many of the well known apparent "paradoxes" of relativity.
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| The [[cross product]] is not defined in four dimensions. Instead the [[exterior product]] is used for some applications, and is defined as follows:
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| : <math> \begin{align}
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| \mathbf{a} \wedge \mathbf{b} = (a_1b_2 - a_2b_1)\mathbf{e}_{12} + (a_1b_3 - a_3b_1)\mathbf{e}_{13} + (a_1b_4 - a_4b_1)\mathbf{e}_{14} + (a_2b_3 - a_3b_2)\mathbf{e}_{23} \\
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| + (a_2b_4 - a_4b_2)\mathbf{e}_{24} + (a_3b_4 - a_4b_3)\mathbf{e}_{34}. \end{align}</math>
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| This is [[bivector]] valued, with bivectors in four dimensions forming a [[six-dimensional space|six-dimensional]] linear space with basis ('''e'''<sub>12</sub>, '''e'''<sub>13</sub>, '''e'''<sub>14</sub>, '''e'''<sub>23</sub>, '''e'''<sub>24</sub>, '''e'''<sub>34</sub>). They can be used to generate rotations in four dimensions.
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| ==Orthogonality and vocabulary==
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| In the familiar 3-dimensional space that we live in there are three [[coordinate system|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''.
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| 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]].
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| ==Geometry==
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| {{See also|Rotations in 4-dimensional Euclidean space}}
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| The geometry of 4-dimensional space is much more complex than that of 3-dimensional space, due to the extra degree of freedom.
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| Just as in 3 dimensions there are [[polyhedron|polyhedra]] made of two dimensional [[polygon]]s, in 4 dimensions there are [[polychoron|polychora]] (4-[[polytope]]s) made of polyhedra. In 3 dimensions there are 5 regular polyhedra known as the [[Platonic solid]]s. In 4 dimensions there are 6 [[convex regular polychoron|convex regular polychora]], the analogues of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex [[uniform polychoron|uniform polychora]], analogous to the 13 semi-regular [[Archimedean solid]]s in three dimensions.
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| {| class=wikitable
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| |+ Regular polytopes in four dimensions<BR>(Displayed as orthogonal projections in each [[Coxeter plane]] of symmetry)
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| |-
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| !A<sub>4</sub>
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| !colspan=2|BC<sub>4</sub>
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| !F<sub>4</sub>
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| !colspan=2|H<sub>4</sub>
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| |- align=center
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| |[[File:4-simplex t0.svg|altN=4-simplex|120px]]<BR>[[5-cell]]<BR>{{CDD|node_1|3|node|3|node|3|node}}
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| |[[File:4-cube t0.svg|altN=4-cube|120px]]<BR>[[tesseract]]<BR>{{CDD|node_1|4|node|3|node|3|node}}
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| |[[File:4-cube t3.svg|altN=4-orthoplex|120px]]<BR>[[16-cell]]<BR>{{CDD|node|4|node|3|node|3|node_1}}
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| |[[File:24-cell graph.svg|altN=24-cell|120px]]<BR>[[24-cell]]<BR>{{CDD|node_1|3|node|4|node|3|node}}
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| |[[File:120-cell graph H4.svg|altN=120-cell|120px]]<BR>[[120-cell]]<BR>{{CDD|node_1|5|node|3|node|3|node}}
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| |[[File:600-cell graph H4.svg|altN=600-cell|120px]]<BR>[[600-cell]]<BR>{{CDD|node|5|node|3|node|3|node_1}}
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| |}
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| In 3 dimensions, a circle may be [[extrude]]d to form a [[cylinder (geometry)|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.
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| In 3 dimensions, curves can form [[knot (mathematics)|knot]]s 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.<ref>J. Scott Carter, Masahico Saito [http://books.google.co.uk/books?id=TIGVq4GeEM4C Knotted Surfaces and Their Diagrams]</ref> 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 {{citation needed|date=January 2013}}. Another such surface is the [[real projective plane]]{{citation needed|date=January 2013}}. <!-- did a google search and can't find anything on these as examples of knotted surfaces. This book http://books.google.co.uk/books?id=TIGVq4GeEM4C seems to talk about the Klein bottle as an unknotted surface. Presumably if knotted has to be knotted relative to some other surface it is homeomorphic to which is unknotted, which surface is that? -->
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| ===Hypersphere===
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| [[File:Clifford-torus.gif|thumb|right|256px|[[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]].]]
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| The set of points in [[Euclidean space|Euclidean 4-space]] having the same distance R from a fixed point P<sub>0</sub> forms a [[hypersurface]] known as a [[3-sphere]]. The hyper-volume of the enclosed space is:
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| : <math> \mathbf V = \begin{matrix} \frac{1}{2} \end{matrix} \pi^2 R^4</math>
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| 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.<ref>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</ref>
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| ==Cognition==
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| 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.<ref name="Ambinder">Ambinder MS, Wang RF, Crowell JA, Francis GK, Brinkmann P. (2009). Human four-dimensional spatial intuition in virtual reality. Psychon Bull Rev. 16(5):818-23. {{doi|10.3758/PBR.16.5.818}} PMID 19815783 [http://pbr.psychonomic-journals.org/content/16/5/818/suppl/DC1 online supplementary material]</ref> 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."<ref name="Ambinder"/> In another study,<ref name="Aflalo">Aflalo TN, Graziano MS (2008). Four-Dimensional Spatial Reasoning in Humans. ''Journal of Experimental Psychology'': Human Perception and Performance 34(5):1066-1077. {{doi|10.1037/0096-1523.34.5.1066}} [http://www.princeton.edu/~graziano/Aflalo_08.pdf Preprint]</ref> 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 [[labyrinth]]s). The graphical interface was based on John McIntosh's free 4D Maze game.<ref name=McIntosh>John McIntosh's four dimensional maze game. [http://www.urticator.net/maze/ Free software]</ref> 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).
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| ==Dimensional analogy==
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| [[File:Tesseract net.svg|thumb|A net of a tesseract]]
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| 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.<ref>[[Michio Kaku]] (1994). ''[[Hyperspace (book)|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.</ref>
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| Dimensional analogy was used by [[Edwin Abbott Abbott]] <!--yes, his middle name is the same as his surname--> 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.
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| 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 (novel)|Spaceland]]'', in which the protagonist encounters four-dimensional beings who demonstrate such powers.
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| ==Cross-sections==
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| 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 through 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.<ref>{{cite book |title=One Two Three . . . Infinity: Facts and Speculations of Science |edition=3rd |first1=George |last1=Gamow |publisher=Courier Dover Publications |year=1988 |isbn=0-486-25664-2 |page=68 |url=http://books.google.com/books?id=EZbcwk6SkhcC}}, [http://books.google.com/books?id=EZbcwk6SkhcC&pg=PA68 Extract of page 68]</ref>
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| ===Projections===
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| A useful application of dimensional analogy in visualizing the fourth dimension is in [[Graphical projection|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 [[human eye|eye]] is also a two-dimensional [[Array data structure|array]] of [[Sensory receptor|receptor]]s 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.). [[Artist]]s often use [[perspective (graphical)|perspective]] to give an illusion of three-dimensional depth to two-dimensional pictures.
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| 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.
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| 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.
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| 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.
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| {|class="wikitable"
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| !Cube
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| !Tesseract
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| !Description
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| |-
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| |[[File:Cube-face-first.png|160px]]
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| |[[File:Tesseract-perspective-cell-first.png|160px]]
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| |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.
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| 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.
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| |-
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| |[[File:Cube-edge-first.png|160px]]
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| |[[File:Tesseract-perspective-face-first.png|160px]]
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| |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 [[trapezoid]]s, the face-first projection of the tesseract consists of two [[frustum]]s.
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| 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.
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| |[[File:Cube-vertex-first.png|160px]]
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| |[[File:Tesseract-perspective-edge-first.png|160px]]
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| |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 [[kite (geometry)|deltoids]] surrounding a vertex, the tesseract's edge-first projection consists of 3 [[hexahedron|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.
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| |[[File:Cube-edge-first.png|160px]]
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| |[[File:Tesseract-perspective-edge-first.png|160px]]
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| |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.
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| |[[File:Cube-vertex-first.png|160px]]
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| |[[File:Tesseract-perspective-vertex-first.png|160px]]
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| |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.
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| 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.
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| |}
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| ===Shadows===
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| A concept closely related to projection is the casting of shadows.
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| [[File:Schlegel wireframe 8-cell.png|right|200px]]
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| 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.
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| 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.)
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| ===Bounding volumes===
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| 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.
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| ===Visual scope===
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| 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.{{Citation needed|date=April 2012}}
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| ===Limitations===
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| 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
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| <math>C = 2\pi r</math> | |
| and the surface area of a sphere:
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| <math>A = 4\pi r^2</math>.
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| One might be tempted to suppose that the surface volume of a hypersphere is <math>V=6\pi r^3</math>, or perhaps <math>V=8\pi r^3</math>, but either of these would be wrong. The correct formula is <math>V = 2\pi^2 r^3</math>.<ref name="Coxeter, H. S. M. 1973 p. 119"/>
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| ==See also==
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| {{wikisource|Flatland}}
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| <div style="-moz-column-count:2; column-count:2;">
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| *[[Euclidean space]]
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| *[[Euclidean geometry]]
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| *[[4-manifold]]
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| *[[Exotic R4|Exotic '''R'''<sup>4</sup>]]
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| *[[Fourth dimension in art]]
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| *[[Dimension]]
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| *[[Four-dimensionalism]]
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| *[[Five-dimensional space|Fifth dimension]]
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| *[[Sixth dimension]]
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| *[[Polychoron]]
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| *[[Polytope]]
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| *[[List of geometry topics]]
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| *[[Block Theory of the Universe]]
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| *''[[Flatland]]'', a book by [[Edwin A. Abbott]] about two- and [[three-dimensional space]]s, to understand the concept of four dimensions
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| *''[[Sphereland]]'', an unofficial sequel to ''[[Flatland]]''
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| *[[Charles Howard Hinton]]
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| *''[[Dimensions (animation)|Dimensions]]'', a set of films about two-, three- and four-dimensional [[polytope]]s
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| *[[List of four-dimensional games]]
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| </div>
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| ==References==
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| {{reflist|2}}
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| ==External links==
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| {{Wikibooks|Special Relativity}}
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| *[http://www.dimensions-math.org "Dimensions" videos, showing several different ways to visualize four dimensional objects]
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| *[http://www.sciencenews.org/index/generic/activity/view/id/35740/title/Math_Trek__Seeing_in_four_dimensions Science News article summarizing the "Dimensions" videos, with clips]
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| *[http://tetraspace.alkaline.org Garrett Jones' tetraspace page]
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| *[[s:Flatland (second edition)|Flatland: a Romance of Many Dimensions (second edition)]]
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| * [http://uk.arxiv.org/abs/hep-ph/0002255 TeV scale gravity, mirror universe, and ... dinosaurs] Article from [http://th-www.if.uj.edu.pl/acta/ Acta Physica Polonica B] by Z.K. Silagadze.
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| *[http://www.bayarea.net/~kins/thomas_briggs/ Exploring Hyperspace with the Geometric Product]
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| *[http://eusebeia.dyndns.org/4d/index.html 4D Euclidean space]
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| *[http://math.eretrandre.org/4dbb/ 4D Building Blocks - Interactive game to explore 4D space]
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| *[http://pagesperso-orange.fr/famille_jf_bigot/informatique/factor/4DNav.htm 4DNav - A small tool to view a 4D space as four 3D space] uses [http://www.flowerfire.com/ADSODA/ ADSODA algorithm ]
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| *[http://www.superliminal.com/cube/mc4dswing.jar MagicCube 4D] A 4-dimensional analog of traditional [[Rubik's Cube]].
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| *[http://www.math.union.edu/~dpvc/math/4D/welcome.html Frame-by-frame animations of 4D - 3D analogies]
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| {{Dimension topics}}
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| {{DEFAULTSORT:Fourth Dimension}}
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| [[Category:Four-dimensional geometry| ]]
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| [[Category:Multi-dimensional geometry]]
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| [[Category:Dimension]]
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| [[Category:Special relativity]]
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