View source for Four-dimensional space

{{short description|Geometric space with four dimensions}}
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[[File:8-cell-simple.gif|frame|The 4D equivalent of a [[cube]], known as a [[tesseract]]. The tesseract is rotating in four-dimensional space, and it is displayed here projected into two dimensions.|alt=Animation of a transforming tesseract or 4-cube]]
{{General geometry}}
A '''four-dimensional space''' or '''4D space''' is a mathematical extension of the concept of three-dimensional or 3D space. [[Three-dimensional space]] is the simplest possible abstraction of the observation that one only needs three numbers, called ''[[dimension]]s'', 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<ref>{{Citation|author=Cajori, Florian|year=1926|title=Origins of Fourth Dimension Concepts
|journal=The American Mathematical Monthly|volume=10|pages=403–407|title-link=s:de:T A D (Cajori)}}</ref> followed by [[Joseph-Louis Lagrange]] in the mid-1700s 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 "[[s:What is the Fourth Dimension?|What is the Fourth Dimension?]]", which explained the concept of a four-dimensional cube with a step-by-step 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. [[Albert Einstein|Einstein's]] concept of [[spacetime]] uses such a 4D space, though it has a [[Minkowski space|Minkowski]] structure that is a bit more complicated than [[Euclidean geometry|Euclidean]] 4D space.

Single locations in 4D space can be given as [[Vector space|vectors]] or ''[[n-tuples]]'', 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 the simplest possible 4D objects, the 4D cube or [[tesseract]].

==History==
[[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 &mdash; three dimensions of space, and one of time.<ref>{{cite book|last1=Bell|first1=E.T.|title=Men of Mathematics|date=1965|publisher=Simon and Schuster|location=New York|isbn=978-0-671-62818-5|page=154|edition=1st}}</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 name="Coxeter">{{cite book|last1=Coxeter|first1=H.S.M.|title=Regular Polytopes|date=1973|publisher=Dover Publishing|location=New York|isbn=978-0-486-61480-9|edition=3rd}}</ref>{{rp|141}} and by 1853 [[Ludwig Schläfli]] had discovered many [[polytope]]s in higher dimensions, although his work was not published until after his death.<ref name="Coxeter"/>{{rp|142–143}} Higher dimensions were soon put on firm footing by [[Bernhard Riemann]]'s 1854 [[Habilitationsschrift|thesis]], ''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.

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]]''. Soon after [[tessarine]]s and [[coquaternion]]s were introduced as other four-dimensional [[algebra over a field|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.<ref>{{cite book|last1=Hinton|first1=Charles Howard|authorlink1=Charles Howard Hinton|title=Speculations on the Fourth Dimension: Selected writings of Charles H. Hinton|date=1980|editor-first=Rudolf v. B.|editor-last=Rucker|publisher=Dover|location=New York|isbn=978-0-486-23916-3|page=vii}}</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 name="Hinton">{{cite book|last1=Hinton|first1=Charles Howard|title=The Fourth Dimension|orig-year=1904|date=1993|publisher=Health Research|location=Pomeroy, Washington|isbn=978-0-7873-0410-2|page=14|url=https://books.google.com/books?id=_ZG3MA1wvjIC&pg=PA14|accessdate=17 February 2017|language=en}}</ref><ref>{{cite book|last1=Gardner|first1=Martin|title=Mathematical Carnival: From Penny Puzzles. Card Shuffles and Tricks of Lightning Calculators to Roller Coaster Rides into the Fourth Dimension|date=1975|publisher=[[Knopf Publishing|Knopf]]|location=New York|isbn=978-0-394-49406-7|pages=42, 52–53|edition=1st}}</ref>

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<ref>[[Victor Schlegel]] (1886) ''Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper'', Waren</ref> his method of visualizing four-dimensional objects with [[Schlegel diagram]]s.

In 1908, [[Hermann Minkowski]] presented a paper<ref>{{Citation|author=Minkowski, Hermann|year=1909|title=Raum und Zeit|journal=Physikalische Zeitschrift|volume=10|pages=75–88|title-link=s:de:Raum und Zeit (Minkowski)}}
*Various English translations on Wikisource: [[s:Space and Time|Space and Time]]
</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">{{cite book|last1=Møller|first1=C.|title=The Theory of Relativity|url=https://archive.org/details/theoryofrelativi0000mlle|url-access=registration|date=1972|publisher=Clarendon Press|location=Oxford|isbn=978-0-19-851256-1|page=[https://archive.org/details/theoryofrelativi0000mlle/page/93 93]|edition=2nd}}</ref> But the geometry of spacetime, being [[non-Euclidean]], is profoundly different from that popularised by Hinton. The study of [[Minkowski space]] 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:
{{quote|Little, if anything, is gained by representing the fourth Euclidean dimension as ''time''. In fact, this idea, so attractively developed by [[The Time Machine|H. G. Wells in ''The Time Machine'']], has led such authors as [[An Experiment with Time|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 Polytopes''<ref name="Coxeter"/>{{rp|119}}}}

==Vectors==
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
: <math>\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{pmatrix}.</math>
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
:<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>
so the general vector '''a''' is
: <math> \mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 + a_4\mathbf{e}_4.</math>
Vectors add, subtract and scale as in three dimensions.

The [[dot product]] of Euclidean three-dimensional space generalizes to four dimensions as
: <math>\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4.</math>
It can be used to calculate the [[norm (mathematics)|norm]] or [[Euclidean distance|length]] of a vector,
:<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>
and calculate or define the [[angle]] between two non-zero vectors as
:<math> \theta = \arccos{\frac{\mathbf{a} \cdot \mathbf{b}}{\left|\mathbf{a}\right| \left|\mathbf{b}\right|}}.</math>

Minkowski spacetime is four-dimensional space with geometry defined by a non-degenerate [[pairing]] different from the dot product:
: <math>\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 - a_4 b_4.</math>
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.

The [[cross product]] is not defined in four dimensions. Instead the [[exterior product]] is used for some applications, and is defined as follows:
: <math> \begin{align}
\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} \\
+ (a_2b_4 - a_4b_2)\mathbf{e}_{24} + (a_3b_4 - a_4b_3)\mathbf{e}_{34}. \end{align}</math>
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.

==Orthogonality and vocabulary==
In the familiar three-dimensional space in which we live, 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''.

Comparatively, four-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 position along the ''w'' axis can be called ''spissitude'', as coined by [[Henry More]].

As mentioned above, Herman Minkowski exploited the idea of four dimensions to discuss cosmology including the finite [[velocity of light]]. In appending a time dimension to three dimensional space, he specified an alternative perpendicularity, [[hyperbolic orthogonality]]. This notion provides his four-dimensional space with a modified [[relativity of simultaneity|simultaneity]] appropriate to electromagnetic relations in his cosmos. Minkowski's world overcame problems associated with the traditional [[absolute space and time]] cosmology previously used in a universe of three space dimensions and one time dimension.

==Geometry==
{{See also|Rotations in 4-dimensional Euclidean space}}
The geometry of four-dimensional space is much more complex than that of three-dimensional space, due to the extra degree of freedom.

Just as in three dimensions there are [[polyhedron|polyhedra]] made of two dimensional [[polygon]]s, in four dimensions there are [[4-polytope]]s made of polyhedra. In three dimensions, there are 5 regular polyhedra known as the [[Platonic solid]]s. In four dimensions, there are 6 [[convex regular 4-polytope]]s, the analogues of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex [[uniform 4-polytope]]s, analogous to the 13 semi-regular [[Archimedean solid]]s in three dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4-polytopes.

{| class=wikitable
|+ Regular polytopes in four dimensions<br>(Displayed as orthogonal projections in each [[Coxeter plane]] of symmetry)
|-
!A<sub>4</sub>, [3,3,3]
!colspan=2|B<sub>4</sub>, [4,3,3]
!F<sub>4</sub>, [3,4,3]
!colspan=2|H<sub>4</sub>, [5,3,3]
|- align=center
|[[File:4-simplex t0.svg|altN=4-simplex|120px]]<br>[[5-cell]]<br>{{CDD|node_1|3|node|3|node|3|node}}<BR>{3,3,3}
|[[File:4-cube t0.svg|altN=4-cube|120px]]<br>[[tesseract]]<br>{{CDD|node_1|4|node|3|node|3|node}}<BR>{4,3,3}
|[[File:4-cube t3.svg|altN=4-orthoplex|120px]]<br>[[16-cell]]<br>{{CDD|node_1|3|node|3|node|4|node}}<BR>{3,3,4}
|[[File:24-cell graph.svg|altN=24-cell|120px]]<br>[[24-cell]]<br>{{CDD|node_1|3|node|4|node|3|node}}<BR>{3,4,3}
|[[File:120-cell graph H4.svg|altN=120-cell|120px]]<br>[[120-cell]]<br>{{CDD|node_1|5|node|3|node|3|node}}<BR>{5,3,3}
|[[File:600-cell graph H4.svg|altN=600-cell|120px]]<br>[[600-cell]]<br>{{CDD|node_1|3|node|3|node|5|node}}<BR>{3,3,5}
|}

In three dimensions, a circle may be [[extrude]]d to form a [[cylinder (geometry)|cylinder]]. In four dimensions, there are several different cylinder-like 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 four-dimensional space, each with its own properties.

In three dimensions, curves can form [[knot (mathematics)|knot]]s but surfaces cannot (unless they are self-intersecting). In four dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction—but 2D surfaces can form non-trivial, non-self-intersecting knots in 4D space.<ref>{{cite book|last1=Carter|first1=J.Scott|last2=Saito|first2=Masahico|title=Knotted Surfaces and Their Diagrams|publisher=American Mathematical Society|isbn=978-0-8218-7491-2|url=https://books.google.com/books?id=TIGVq4GeEM4C}}</ref>{{Page needed|date=December 2016}} Because these surfaces are two-dimensional, 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|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 https://books.google.com/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? -->

===Hypersphere===
[[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]].]]
{{Main article|Hypersphere}}
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:

: <math> \mathbf V = \begin{matrix} \frac{1}{2} \end{matrix} \pi^2 R^4</math>

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>{{cite book|last1=D'Inverno|first1=Ray|title=Introducing Einstein's Relativity|date=1998|publisher=Clarendon Press|location=Oxford|isbn=978-0-19-859653-0|page=319|edition=Reprint}}</ref>

==Cognition==
Research using [[virtual reality]] finds that humans, in spite of living in a three-dimensional world, can, without special practice, make spatial judgments about line segments, embedded in four-dimensional space, based on their length (one dimensional) and the angle (two dimensional) between them.<ref name="Ambinder">{{cite journal|last1=Ambinder|first1=Michael S.|last2=Wang|first2=Ranxiao Frances|last3=Crowell|first3=James A.|last4=Francis|first4=George K.|last5=Brinkmann|first5=Peter|title=Human four-dimensional spatial intuition in virtual reality|journal=Psychonomic Bulletin & Review|date=October 2009|volume=16|issue=5|pages=818–823|doi=10.3758/PBR.16.5.818|pmid=19815783|url=http://pbr.psychonomic-journals.org/content/16/5/818/suppl/DC1|accessdate=17 February 2017}}</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">{{cite journal|last1=Aflalo|first1=T. N.|last2=Graziano|first2=M. S. A.|title=Four-dimensional spatial reasoning in humans|journal=Journal of Experimental Psychology: Human Perception and Performance|date=2008|volume=34|issue=5|pages=1066–1077|doi=10.1037/0096-1523.34.5.1066|pmid=18823195|url=http://www.princeton.edu/~graziano/Aflalo_08.pdf|accessdate=17 February 2017|citeseerx=10.1.1.505.5736}}</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>{{cite web|url=http://www.urticator.net/maze/ |title=4D Maze Game |publisher=urticator.net |date= |accessdate=2016-12-16}}</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).

==Dimensional analogy==
[[File:Tesseract net.svg|thumb|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.<ref>{{cite book|last1=Kaku|first1=Michio|authorlink=Michio Kaku|title=Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension|date=1995|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-286189-4|pages=Part I, Chapter 3|edition=reissued|title-link=Hyperspace (book)}}</ref>

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.

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.

===Cross-sections===
As a three-dimensional object passes through a two-dimensional plane, two-dimensional beings in this plane would only observe a [[Cross section (geometry)|cross-section]] of the three-dimensional 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 four-dimensional object passed through a three dimensional (hyper)surface, one could observe a three-dimensional cross-section of the four-dimensional object&mdash;for example, a 4-sphere would appear first as a point, then as a growing sphere, with the sphere then shrinking to a single point and then disappearing.<ref>{{cite book|last1=Rucker|first1=Rudy|title=The Fourth Dimension: A Guided Tour of the Higher Universe|date=1996|publisher=Houghton Mifflin|location=Boston|isbn=978-0-395-39388-8|page=18}}</ref> This means of visualizing aspects of the fourth dimension was used in the novel ''Flatland'' and also in several works of [[Charles Howard Hinton]].<ref name="Hinton"/>{{rp|11–14}}

===Projections===
A useful application of dimensional analogy in visualizing higher dimensions is in [[Graphical projection|projection]]. A projection is a way for representing an ''n''-dimensional object in {{math|''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. By doing this, the dimension orthogonal to the screen (''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. 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 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 three-dimensional [[cube]] with analogous projections of the four-dimensional tesseract into three-dimensional space.

{|class="wikitable"
|-
!Cube
!Tesseract
!Description
|-
|[[File:Cube-face-first.png|160px]]
|[[File:Tesseract-perspective-cell-first.png|160px]]
|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.
|-
|[[File:Cube-edge-first.png|160px]]
|[[File:Tesseract-perspective-face-first.png|160px]]
|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.
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.
|-
|[[File:Cube-vertex-first.png|160px]]
|[[File:Tesseract-perspective-edge-first.png|160px]]
|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.
|-
|[[File:Cube-edge-first.png|160px]]
|[[File:Tesseract-perspective-edge-first.png|160px]]
|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.
|-
|[[File:Cube-vertex-first.png|160px]]
|[[File:Tesseract-perspective-vertex-first.png|160px]]
|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'' 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.

[[File:Schlegel wireframe 8-cell.png|right|200px]]
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 on a flat two-dimensional 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 three-dimensional cube within another three-dimensional cube suspended in midair (a "flat" surface from a four-dimensional perspective). (Note that, technically, the visual representation shown here is actually a two-dimensional image 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===
We spatially perceive ourselves as beings in a three-dimensional 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 four-dimensional 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 3-dimensional subspace simultaneously, including the inner structure of solid 3-dimensional objects, things obscured from our viewpoints in three dimensions on two-dimensional projections. Our brains receive images in two dimensions and use reasoning to help us "picture" three-dimensional objects.

===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
<math>C = 2\pi r</math>
and the surface area of a sphere:
<math>A = 4\pi r^2</math>.
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"/>{{rp|119}}

==See also==
{{div col|colwidth=20em}}
*[[4-manifold]]
*[[Exotic R4|Exotic '''R'''<sup>4</sup>]]
*[[Eugene the Jeep]]
*[[Four-dimensionalism]]
*[[Fourth dimension in art]]
*[[Fourth dimension in literature]]
*[[List of four-dimensional games]]
{{div col end}}

==References==
{{reflist|30em}}

==Further reading==
* {{cite journal|author-last=Archibald|author-first=R. C|author-link=Raymond Clare Archibald|title=Time as a Fourth Dimension | journal = Bulletin of the American Mathematical Society|pages=409–412|year=1914|url=https://www.ams.org/journals/bull/1914-20-08/S0002-9904-1914-02511-X/S0002-9904-1914-02511-X.pdf}}
* [[Andrew Forsyth]] (1930) [https://archive.org/details/geometryoffourdi032760mbp Geometry of Four Dimensions], link from [[Internet Archive]].
* {{cite book |title=One Two Three . . . Infinity: Facts and Speculations of Science |edition=3rd |first1=George |last1=Gamow |authorlink=George Gamow|publisher=Courier Dover Publications |year=1988 |isbn=978-0-486-25664-1 |page=68 |url=https://books.google.com/books?id=EZbcwk6SkhcC}} [https://books.google.com/books?id=EZbcwk6SkhcC&pg=PA68 Extract of page 68]
* [[E. H. Neville]] (1921) [http://quod.lib.umich.edu/u/umhistmath/ABR2619.0001.001?rgn=works;view=toc;rgn1=author;q1=Neville ''The Fourth Dimension''], [[Cambridge University Press]], link from [[University of Michigan]] Historical Math Collection.

==External links==
{{Wikibooks|Special Relativity}}
*[http://www.dimensions-math.org "Dimensions" videos, showing several different ways to visualize four dimensional objects]
*[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]
*[[s:Flatland (second edition)|''Flatland: a Romance of Many Dimensions'' (second edition)]]
*[http://www.math.union.edu/~dpvc/math/4D/welcome.html Frame-by-frame animations of 4D - 3D analogies]

{{Dimension topics}}

{{DEFAULTSORT:Fourth Dimension}}
[[Category:Dimension]]
[[Category:Four-dimensional geometry| ]]
[[Category:Multi-dimensional geometry]]
[[Category:Special relativity]]
[[Category:4 (number)]]