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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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[[Image:24-cell.png|right]]

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]].