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[[Image:lighterstill.jpg]]
[[Image:lighterstill.jpg]]
[[Image:8-cell-simple.png]]
[[Image:Hypercube diagram.svg|thumb|200px|right|Cube with fourth-dimensional directions creating a hypercube.]]
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.
===Vectors===
===Vectors===
The fourth spatial dimension can be thought of in terms of [[vector (spatial)|vectors]], analogous to arrows, fixed from some single place in space which we call the ''origin'', that point to other places. These are called geometric vectors.
The fourth spatial dimension can be thought of in terms of [[vector (spatial)|vectors]], analogous to arrows, fixed from some single place in space which we call the ''origin'', that point to other places. These are called geometric vectors.
===Geometry with four spatial dimensions===
===Geometry with four spatial dimensions===
[[Image:24-cell.gif|right|thumb|A 3D projection of a rotating [[24-cell]]. It rotates simultaneously about two orthogonal planes.]]
[[Image:24-cell.png]]
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]].
===Dimensional analogy===
===Dimensional analogy===
To make the leap from three spatial dimensions into four, a device called ''dimensional analogy'' is commonly employed. '''Dimensional analogy''' is studying how (''n'' – 1) dimensions relate to ''n'' dimensions, and then inferring how ''n'' dimensions would relate to (''n'' + 1) dimensions.
To make the leap from three spatial dimensions into four, a device called ''dimensional analogy'' is commonly employed. '''Dimensional analogy''' is studying how (''n'' – 1) dimensions relate to ''n'' dimensions, and then inferring how ''n'' dimensions would relate to (''n'' + 1) dimensions.
==The "fourth dimension" in popular culture==
==The "fourth dimension" in popular culture==
*[[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—"]]<!-- Quote marks are part of the title --> by [[Robert A. Heinlein]]
