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| [[Image:lighterstill.jpg]] | | [[Image:lighterstill.jpg]] |
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− | [[Image:Hypercube diagram.svg|thumb|200px|right|Cube with fourth-dimensional directions creating a hypercube.]] | |
− | [[Image:glass tesseract animation.gif|thumb|200px|3D projection of a rotating [[tesseract]]]]
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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. |
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| ===Vectors=== | | ===Vectors=== |
− | [[Image:Dice analogy- 1 to 5 dimensions.svg|thumb|Demonstration of objects with 1 to 5 dimensions]]
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| 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. |
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| ===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]]. |
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| ===Dimensional analogy=== | | ===Dimensional analogy=== |
− | [[Image:Tesseract net.svg|thumb|A net of a tesseract]]
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| 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. |
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| ==The "fourth dimension" in popular culture== | | ==The "fourth dimension" in popular culture== |
− | [[Image:4D rubiks cube.jpg|right|thumb|[http://www.superliminal.com/cube/mc4dswing.jar 4 dimensional Rubik's Cube]]]
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| *[[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]] |