Editing Four-dimensional space (section)

{{Short description|Geometric space with four dimensions}}
{{More citations needed|date=December 2016}}
{{for|alternate planes of existence in fiction|Fourth dimension in literature}}
<div class="calculator-container" data-calculator-refresh-on-load="true">{{Image frame|width=256|content={{calculator-hideifzero|formula=moving1|[[File:8-cell-simple.gif|alt=Animation of a transforming tesseract or 4-cube]]}}{{calculator-hideifzero|formula=static1|[[File:8-cell-simple_frame.png|alt=Single frame of a transforming tesseract or 4-cube]]|starthidden=1}}|caption=The 4D equivalent of a [[cube]] is known as a [[tesseract]], seen rotating here in four-dimensional space, yet projected into two dimensions for display.<div role="radiogroup" aria-labelledby="animatedgiflabel1" class="calculatorgadget-enabled" style="display:none">{{calculator label|codex=1|label=|id=animatedgiflabel1}}{{Calculator codex radio|id=moving1|name=animatedgifbutton|inline=1|label=Animated|default=1}}{{Calculator codex radio|id=static1|name=animatedgifbutton|inline=1|label=Static}}</div>}}</div>
{{General geometry}}

'''Four-dimensional space''' ('''4D''') is the mathematical extension of the concept of [[three-dimensional space]] (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''[[dimension]]s'', to describe the [[size]]s or [[location]]s of objects in the everyday world.  This concept of ordinary space is called [[Euclidean space]] because it corresponds to [[Euclid]][[Euclidean geometry|'s geometry]], which was originally abstracted from the spatial experiences of everyday life.

Single locations in Euclidean 4D space can be given as [[Vector space|vectors]] or ''[[n-tuples|4-tuples]]'', i.e., as ordered lists of numbers such as {{math|(''x'', ''y'', ''z'', ''w'')}}. For example, the [[volume]] of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled {{mvar|x}}, {{mvar|y}}, and {{mvar|z}}). It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of 4D spaces emerge. A hint of that complexity can be seen in the accompanying 2D animation of one of the simplest possible [[Regular 4-polytope|regular 4D objects]], the [[tesseract]], which is [[Hypercube|analogous]] to the 3D [[cube]].