Editing Four-dimensional space (section)

==Geometry==
{{See also|Rotations in 4-dimensional Euclidean space}}{{Citations needed|section|date=November 2022}}
The geometry of four-dimensional space is much more complex than that of three-dimensional space, due to the extra degree of freedom.

Just as in three dimensions there are [[polyhedron|polyhedra]] made of two dimensional [[polygon]]s, in four dimensions there are polychora made of polyhedra. In three dimensions, there are 5 regular polyhedra known as the [[Platonic solid]]s. In four dimensions, there are 6 [[convex regular 4-polytope]]s, the analogs of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex [[uniform 4-polytope]]s, analogous to the 13 semi-regular [[Archimedean solid]]s in three dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4-polytopes.

{| class=wikitable
|+ Regular polytopes in four dimensions<br>(Displayed as orthogonal projections in each [[Coxeter plane]] of symmetry)
|-
!A<sub>4</sub>, [3,3,3]
!colspan=2|B<sub>4</sub>, [4,3,3]
!F<sub>4</sub>, [3,4,3]
!colspan=2|H<sub>4</sub>, [5,3,3]
|- align=center
|[[File:4-simplex t0.svg|altN=4-simplex|120px]]<br>[[5-cell]]<br>{{CDD|node_1|3|node|3|node|3|node}}<BR>{3,3,3}
|[[File:4-cube t0.svg|altN=4-cube|120px]]<br>[[tesseract]]<br>{{CDD|node_1|4|node|3|node|3|node}}<BR>{4,3,3}
|[[File:4-cube t3.svg|altN=4-orthoplex|120px]]<br>[[16-cell]]<br>{{CDD|node_1|3|node|3|node|4|node}}<BR>{3,3,4}
|[[File:24-cell graph.svg|altN=24-cell|120px]]<br>[[24-cell]]<br>{{CDD|node_1|3|node|4|node|3|node}}<BR>{3,4,3}
|[[File:600-cell graph H4.svg|altN=600-cell|120px]]<br>[[600-cell]]<br>{{CDD|node_1|3|node|3|node|5|node}}<BR>{3,3,5}
|[[File:120-cell graph H4.svg|altN=120-cell|120px]]<br>[[120-cell]]<br>{{CDD|node_1|5|node|3|node|3|node}}<BR>{5,3,3}
|}

In three dimensions, a circle may be [[extrude]]d to form a [[cylinder (geometry)|cylinder]]. In four dimensions, there are several different cylinder-like objects. A sphere may be extruded to obtain a spherical cylinder (a cylinder with spherical "caps", known as a [[spherinder]]), and a cylinder may be extruded to obtain a cylindrical prism (a cubinder).{{citation needed|date=March 2022}} The [[Cartesian product]] of two circles may be taken to obtain a [[duocylinder]]. All three can "roll" in four-dimensional space, each with its properties.

In three dimensions, curves can form [[knot (mathematics)|knot]]s but surfaces cannot (unless they are self-intersecting). In four dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction—but 2D surfaces can form non-trivial, non-self-intersecting knots in 4D space.<ref>{{cite book |last1=Carter |first1=J. Scott |url=https://books.google.com/books?id=TIGVq4GeEM4C |title=Knotted Surfaces and Their Diagrams |last2=Saito |first2=Masahico |publisher=[[American Mathematical Society]] |isbn=978-0-8218-7491-2}}</ref> Because these surfaces are two-dimensional, they can form much more complex knots than strings in 3D space can. The [[Klein bottle]] is an example of such a knotted surface.{{citation needed|date=January 2013}} Another such surface is the [[real projective plane]].{{citation needed|date=January 2013}} <!-- did a google search and can't find anything on these as examples of knotted surfaces.  This book https://books.google.com/books?id=TIGVq4GeEM4C seems to talk about the Klein bottle as an unknotted surface. Presumably if knotted has to be knotted relative to some other surface it is homeomorphic to which is unknotted, which surface is that? -->

===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>