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There are three conventional spatial [[dimension]]s: length (or depth), width, and height, often expressed as x, y and z.  x and y axes appear on a plane Cartesian graph and z is found in functions such as a "z-buffer" in computer graphics, for processing "depth" in imagery. The '''fourth dimension''' is often identified with [[time]], and as such is used to explain [[space-time]] in Einstein's theories of [[special relativity]] and [[general relativity]]. When a reference is used to four-dimensional co-ordinates, it is likely that what is referred to is the three spatial dimensions plus a time-line. If four (or more) spatial dimensions are referred to, this should be stated at the outset, to avoid confusion with the more common notion that time is the Einsteinian fourth dimension.

There are three conventional spatial [[dimension]]s: length (or depth), width, and height, often expressed as x, y and z.  x and y axes appear on a plane Cartesian graph and z is found in functions such as a "z-buffer" in computer graphics, for processing "depth" in imagery. The '''fourth dimension''' is often identified with [[time]], and as such is used to explain [[space-time]] in Einstein's theories of [[special relativity]] and [[general relativity]]. When a reference is used to four-dimensional co-ordinates, it is likely that what is referred to is the three spatial dimensions plus a time-line. If four (or more) spatial dimensions are referred to, this should be stated at the outset, to avoid confusion with the more common notion that time is the Einsteinian fourth dimension.

===Geometry with four spatial dimensions===

===Geometry with four spatial dimensions===

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In four spatial dimensions, Euclidean geometry provides for a greater variety of shapes to exist than in three dimensions. Just as three-dimensional [[polyhedron]]s are spatial enclosures made out of connected two-dimensional faces, the four-dimensional [[polychoron]]s are enclosures of four-dimensional space made out of three-dimensional ''cells''. Where in three dimensions there are exactly five regular polyhedrons, or [[Platonic solid]]s, that can exist, six [[convex regular 4-polytope|regular polychoron]]s exist in four dimensions. Five of the six can be interpreted as natural extensions of the Platonic solids, just as the [[cube]], itself a Platonic solid, is a natural extension of the two-dimensional [[square (geometry)|square]].

In four spatial dimensions, Euclidean geometry provides for a greater variety of shapes to exist than in three dimensions. Just as three-dimensional [[polyhedron]]s are spatial enclosures made out of connected two-dimensional faces, the four-dimensional [[polychoron]]s are enclosures of four-dimensional space made out of three-dimensional ''cells''. Where in three dimensions there are exactly five regular polyhedrons, or [[Platonic solid]]s, that can exist, six [[convex regular 4-polytope|regular polychoron]]s exist in four dimensions. Five of the six can be interpreted as natural extensions of the Platonic solids, just as the [[cube]], itself a Platonic solid, is a natural extension of the two-dimensional [[square (geometry)|square]].

*[[Fourth Dimension (Stratovarius album)|Fourth Dimension]] is an album by [[Power Metal]] band [[Stratovarius]]

*[[Fourth Dimension (Stratovarius album)|Fourth Dimension]] is an album by [[Power Metal]] band [[Stratovarius]]

*The fourth dimension has been a subject of popular fascination since at least the 1920s. See ''Into the Fourth Dimension'' (1926) by [[Ray Cummings]], the comic [[Eugene the Jeep]] or [["—And He Built a Crooked House—"]]<!-- Quote marks are part of the title --> by [[Robert A. Heinlein]]

*The fourth dimension has been a subject of popular fascination since at least the 1920s. See ''Into the Fourth Dimension'' (1926) by [[Ray Cummings]], the comic [[Eugene the Jeep]] or [["—And He Built a Crooked House—"]] by [[Robert A. Heinlein]]

*''[[Donnie Darko]]'' uses the Fourth Dimension as a plot for [[Time Travel]]ing.  The reference is related to water being a fourth dimensional tool for time traveling.

*''[[Donnie Darko]]'' uses the Fourth Dimension as a plot for [[Time Travel]]ing.  The reference is related to water being a fourth dimensional tool for time traveling.

*[[Alan Moore]]'s [[graphic novel]] ''[[From Hell]]'' uses the fourth dimension as a reference to the insanity of [[Jack the Ripper]]'s character.

*[[Alan Moore]]'s [[graphic novel]] ''[[From Hell]]'' uses the fourth dimension as a reference to the insanity of [[Jack the Ripper]]'s character.

*In [[William Sleator|William Sleator's]] ''The Boy Who Reversed Himself'', the main characters get lost in the fourth spatial dimension, where they encounter highly intelligent beings who are represented by three dimensional cross sections of themselves, which is all that can be perceived of them by the book's three dimensional characters.

*In [[William Sleator|William Sleator's]] ''The Boy Who Reversed Himself'', the main characters get lost in the fourth spatial dimension, where they encounter highly intelligent beings who are represented by three dimensional cross sections of themselves, which is all that can be perceived of them by the book's three dimensional characters.

*In Michael Atkinson's 2006 Science Fiction novella ''Combing Back Through Time'', a history recording probe "combs" space around itself in order to regress in the Fourth Dimension.

*In Michael Atkinson's 2006 Science Fiction novella ''Combing Back Through Time'', a history recording probe "combs" space around itself in order to regress in the Fourth Dimension.

*Most motion simulator attractions use 4-D as a marketing term, referring to the moving seats as the fourth dimension. {{see|Motion Simulator}}

*Most motion simulator attractions use 4-D as a marketing term, referring to the moving seats as the fourth dimension.  

*In the videogame ''[[Mother (video game)|Mother]]'', the main character can use a special PK ability called "Fourth Dimension Slip" to instantly get out of any battle.

*In the videogame ''[[Mother (video game)|Mother]]'', the main character can use a special PK ability called "Fourth Dimension Slip" to instantly get out of any battle.

*In [[Frank Herbert]]'s ''[[Dune (novel)|Dune]]'' the fourth dimension is used when folding space to allow instantaneous travel through space.

*In [[Frank Herbert]]'s ''[[Dune (novel)|Dune]]'' the fourth dimension is used when folding space to allow instantaneous travel through space.

==External links==

==External links==

*[http://etext.lib.virginia.edu/etcbin/toccer-new2?id=ManFour.sgm&images=images/modeng&data=/texts/english/modeng/parsed&tag=public&part=all The Fourth Dimension Simply Explained]

*[https://etext.lib.virginia.edu/etcbin/toccer-new2?id=ManFour.sgm&images=images/modeng&data=/texts/english/modeng/parsed&tag=public&part=all The Fourth Dimension Simply Explained]

*[http://tetraspace.alkaline.org Garrett Jones' tetraspace page]

*[https://tetraspace.alkaline.org Garrett Jones' tetraspace page]

*[http://www.geom.uiuc.edu/~banchoff/Flatland/ Flatland: a Romance of Many Dimensions]

*[https://www.geom.uiuc.edu/~banchoff/Flatland/ Flatland: a Romance of Many Dimensions]

*[http://eusebeia.dyndns.org/4d/vis/vis.html 4D visualization]

*[https://eusebeia.dyndns.org/4d/vis/vis.html 4D visualization]

* [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.

* [https://uk.arxiv.org/abs/hep-ph/0002255 TeV scale gravity, mirror universe, and ... dinosaurs] Article from [https://th-www.if.uj.edu.pl/acta/ Acta Physica Polonica B] by Z.K. Silagadze.

*[http://www.bayarea.net/~kins/thomas_briggs/ Exploring Hyperspace with the Geometric Product]

*[https://www.bayarea.net/~kins/thomas_briggs/ Exploring Hyperspace with the Geometric Product]

[[Category: General Reference]]

[[Category: General Reference]]

[[Category: Mathematics]]

[[Category: Mathematics]]