Editing Four-dimensional space (section)

===Hypersphere===
<div class="calculator-container" data-calculator-refresh-on-load="true">{{Image frame|width=255|content={{calculator-hideifzero|formula=moving2|[[File:Clifford-torus.gif]]}}{{calculator-hideifzero|formula=static2|[[File:Clifford-torus-frame-50.png]]|starthidden=1}}|caption=[[Stereographic projection]] of a [[Clifford torus]]: the set of points {{math|(cos(''a''), sin(''a''), cos(''b''), sin(''b''))}}, which is a subset of the [[3-sphere]].<div role="radiogroup" aria-labelledby="animatedgiflabel2" class="calculatorgadget-enabled" style="display:none">{{calculator label|codex=1|label=|id=animatedgiflabel2}}{{Calculator codex radio|id=moving2|name=animatedgifbutton|inline=1|label=Animated|default=1}}{{Calculator codex radio|id=static2|name=animatedgifbutton|inline=1|label=Static}}</div>}}</div>
{{Main|Hypersphere}}
The set of points in [[Euclidean space|Euclidean 4-space]] having the same distance {{mvar|R}} from a fixed point {{math|''P''<sub>0</sub>}} forms a [[hypersurface]] known as a [[3-sphere]]. The hyper-volume of the enclosed space is:

: <math> \mathbf V = \begin{matrix} \frac{1}{2} \end{matrix} \pi^2 R^4</math>

This is part of the [[Friedmann–Lemaître–Robertson–Walker metric]] in [[General relativity]] where {{mvar|R}} is substituted by function {{math|''R''(''t'')}} with {{mvar|t}} meaning the cosmological age of the universe. Growing or shrinking {{mvar|R}} with time means expanding or collapsing universe, depending on the mass density inside.<ref>{{cite book|last1=D'Inverno|first1=Ray|title=Introducing Einstein's Relativity|date=1998|publisher=Clarendon Press|location=Oxford|isbn=978-0-19-859653-0|page=319|edition=Reprint}}</ref>