View source for Five-dimensional space

{{short description|Geometric space with five dimensions}}
{{About|mathematical spaces having five dimensions|the musical group|The 5th Dimension|alternate planes of existence in fiction|fourth dimension in literature}}{{More citations needed|date=May 2024}}[[File:5-cube t0.svg|thumb|A 2D [[orthogonal projection]] of a [[5-cube]]|240px]]
A '''five-dimensional space''' is a [[space (mathematics)|space]] with five [[dimension]]s. In [[mathematics]], a [[tuple|sequence]] of ''N'' [[real numbers|numbers]] can represent a [[coordinate system|location]] in an ''N''-dimensional [[Euclidean space|space]]. If interpreted physically, that is one more than the usual three [[space|spatial]] dimensions and the fourth dimension of [[time]] used in [[theory of relativity|relativistic physics]].<ref name="pbs">{{cite web |title=How Many Dimensions Does the Universe Really Have |publisher=Public Broadcasting Service |author=Paul Halpern |date=April 3, 2014 |url=https://www.pbs.org/wgbh/nova/article/how-many-dimensions-does-the-universe-really-have/ |access-date=September 12, 2015}}</ref>

==Physics==
Much of the early work on five-dimensional space was in an attempt to develop a [[unified field theory|theory that unifies]] the four [[fundamental interactions]] in nature: [[strong interaction|strong]] and [[weak interaction|weak]] nuclear forces, [[gravity]], and [[electromagnetism]]. [[Germany|German]] mathematician [[Theodor Kaluza]] and [[Sweden|Swedish]] physicist [[Oskar Klein]] independently developed the [[Kaluza–Klein theory]] in 1921, which used the fifth dimension to unify [[gravity]] with [[electromagnetic force]]. Although their approaches were later found to be at least partially inaccurate, the concept provided a basis for further research over the past century.<ref name="pbs"/>

To explain why this dimension would not be directly observable, Klein suggested that the fifth dimension would be rolled up into a tiny, compact loop on the order of 10{{sup|-33}} centimeters.<ref name="pbs"/> Under his reasoning, he envisioned light as a disturbance caused by rippling in the higher dimension just beyond human perception, similar to how fish in a pond can only see shadows of ripples across the surface of the water caused by raindrops.<ref name="discovery">{{cite web |title=Black Holes on a String in the Fifth Dimension |first=Jennifer |last=Oulette |date=March 6, 2011 |publisher=Discovery News |url=http://news.discovery.com/space/black-holes-on-a-string-in-the-fifth-dimension.htm |access-date=September 12, 2015 |url-status=dead |archive-url=https://web.archive.org/web/20151101013219/http://news.discovery.com/space/black-holes-on-a-string-in-the-fifth-dimension.htm |archive-date=November 1, 2015}}</ref> While not detectable, it would indirectly imply a connection between seemingly unrelated forces. The Kaluza–Klein theory experienced a revival in the 1970s due to the emergence of [[string theory|superstring theory]] and [[supergravity]]: the concept that reality is composed of vibrating strands of energy, a postulate only mathematically viable in ten dimensions or more.{{Clarify|reason=unclear what is meant by statement that supergravity requires ten-dimensional space; or what is meant by ten-dimensional space|date=April 2023}} Superstring theory then evolved into a more generalized approach known as [[M-theory]]. M-theory suggested a potentially observable extra dimension in addition to the ten essential dimensions which would allow for the existence of superstrings. The other 10 dimensions are compacted, or "rolled up", to a size below the subatomic level.<ref name="pbs"/><ref name="discovery"/> The Kaluza–Klein theory today is seen as essentially a [[gauge theory]], with the [[gauge group|gauge]] being the [[circle group]].{{citation needed|date=September 2015}}

The fifth dimension is difficult to directly observe, though the [[Large Hadron Collider]] provides an opportunity to record indirect evidence of its existence.<ref name="pbs"/> Physicists theorize that collisions of subatomic particles in turn produce new particles as a result of the collision, including a [[graviton]] that escapes from the fourth dimension, or [[brane]], leaking off into a five-dimensional bulk.<ref>{{cite web |title=Physicists probe fifth dimension |last=Boyle |first=Alan |work=NBC news |date=June 6, 2006 |url=https://www.nbcnews.com/id/wbna13070896 |access-date=December 12, 2023}}</ref> M-theory would explain the weakness of gravity relative to the other fundamental forces of nature, as can be seen, for example, when using a magnet to lift a pin off a table—the magnet overcomes the gravitational pull of the entire earth with ease.<ref name="pbs"/>

Mathematical approaches were developed in the early 20th century that viewed the fifth dimension as a theoretical construct. These theories make reference to [[Hilbert space]], a concept that postulates an infinite number of mathematical dimensions to allow for a limitless number of quantum states. [[Albert Einstein|Einstein]], [[Peter Bergmann|Bergmann]], and [[Valentine Bargmann|Bargmann]] later tried to extend the four-dimensional [[spacetime]] of [[theory of relativity|general relativity]] into an extra physical dimension to incorporate electromagnetism, though they were unsuccessful.<ref name="pbs"/> In their 1938 paper, Einstein and Bergmann were among the first to introduce the modern viewpoint that a four-dimensional theory, which coincides with [[Einstein field equations#Einstein–Maxwell equations|Einstein–Maxwell theory]] at long distances, is derived from a five-dimensional theory with complete [[symmetry (physics)|symmetry]] in all five dimensions. They suggested that electromagnetism resulted from a gravitational field that is “polarized” in the fifth dimension.<ref>{{cite journal |last1=Einstein |first1=Albert |last2=Bergmann |first2=Peter |title=On A Generalization Of Kaluza's Theory Of Electricity |journal=[[Annals of Mathematics]] |volume=39 |date=1938 |issue=3 |pages=683–701 |jstor=1968642 |doi=10.2307/1968642}}</ref>

The main novelty of Einstein and Bergmann was to seriously consider the fifth dimension as a physical entity, rather than an excuse to combine the [[metric tensor]] and electromagnetic potential. But they then reneged, modifying the theory to break its five-dimensional symmetry. Their reasoning, as suggested by [[Edward Witten]], was that the more symmetric version of the theory predicted the existence of a new long range field, one that was both [[massless particle|massless]] and [[scalar particle|scalar]], which would have required a fundamental modification to Einstein's [[theory of general relativity]].<ref>{{cite arXiv |title=A Note On Einstein, Bergmann, and the Fifth Dimension |last=Witten |first=Edward |date=January 31, 2014 |class=physics.hist-ph |eprint=1401.8048}}</ref> [[Minkowski space]] and [[Maxwell's equations#In vacuum|Maxwell's equations in vacuum]] can be embedded in a five-dimensional [[Riemann curvature tensor]].{{citation needed|date=September 2015}}

In 1993, the physicist [[Gerard 't Hooft]] put forward the [[holographic principle]], which explains that the ''information about an extra dimension is visible as a curvature in a spacetime with one fewer dimension''. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, which gives the image a curvature when the observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal (test) particle. 'T Hooft has speculated that the fifth dimension is really the "spacetime fabric".<ref>{{cite web |last=Decker |first=Adam |newspaper=[[Scientific American]] |date=February 1, 2022 |title=What Is Spacetime Really Made Of? |url=https://www.scientificamerican.com/article/what-is-spacetime-really-made-of/ |access-date=November 5, 2023 |archive-date=January 18, 2022 |archive-url=https://web.archive.org/web/20220118151328/https://www.scientificamerican.com/article/what-is-spacetime-really-made-of/ |url-status=bot: unknown }}</ref><ref>{{cite web |last=Oulette |first=Jennifer |newspaper=[[Wired(magazine) |Wired]] |date=May 18, 2015 |title=Spooky Quantum Action Might Hold the Universe Together |url=https://www.wired.com/2015/05/spooky-quantum-action-might-hold-universe-together/ |access-date=November 5, 2023}}</ref>

Recent research suggests several alternative interpretations of the 5D extension of [[spacetime]], most of them generalizing the earlier [[Kaluza-Klein theory]]. The first approach is [[Paul S. Wesson#Science Books|space-time-matter]], which utilizes an unrestricted group of 5D coordinate transforms to derive new solutions of the Einstein's field equations that agree with the corresponding classical solutions in 4D spacetime.<ref name="Wesson_STM_1999">{{cite book |last1=Wesson |first1=Paul |year=1999 |title=Space-Time-Matter, Modern Kaluza-Klein Theory |publisher=Singapore: World Scientific |ISBN=981-02-3588-7}}</ref> Another 5D representation describes quantum physics from a thermal-space-time ensemble perspective and draws connections with classical field theory as limiting cases.<ref name="thompson_TST_2022">{{cite journal |last1=Thompson |first1=Russell |year=2022 |title=An interpretation of quantum foundations based on density functional theory and polymer self-consistent field theory |journal=Quantum Stud.: Math. Found. |volume=9 |pages=405–416 |arxiv=2111.14236 |doi=10.1007/s40509-022-00276-y}}</ref> Yet another approach, spacekime representation, lifts the ordinary time from an event-ordering positive-real number to complex-time (kime), which effectively transforms longitudinal processes from time-series into 2D manifolds (kime-surfaces).<ref name="dinov_velev_2021">{{cite book |last1=Dinov |first1=Ivo |last2=Velev |first2=Milen |date=2021 |title=Data Science - Time Complexity, Inferential Uncertainty, and Spacekime Analytics |location=Boston/Berlin |publisher=De Gruyter |isbn=9783110697803 |doi=10.1515/9783110697827 |url=https://www.degruyter.com/document/doi/10.1515/9783110697827/html}}</ref>

==Fifth-dimensional geometry==
According to Klein's definition, "a geometry is the study of the invariant properties of a spacetime, under transformations within itself." Therefore, the geometry of the 5th dimension studies the invariant properties of such space-time, as we move within it, expressed in formal equations.<ref>{{cite book |last=Sancho |first=Luis |date=October 4, 2011 |title=Absolute Relativity: The 5th dimension (abridged) |page=442}}</ref> Fifth dimensional geometry is generally represented using 5 [[coordinate system|coordinate values]] (x,y,z,w,v), where moving along the v axis involves moving between different [[hyper-volumes]].<ref>{{Cite web|url=https://e.math.cornell.edu/people/Morgan_Weiler/4540/SB.pdf|title=Hyperbolic Geometry|author=Steven Booth}}</ref>

===Polytopes===
{{Main article|5-polytope}}

In five or more dimensions, only three [[regular polytope]]s exist. In five dimensions, they are:
# The '''[[5-simplex]]''' of the [[simplex]] family, {3,3,3,3}, with 6 vertices, 15 edges, 20 faces (each an [[equilateral triangle]]), 15 cells (each a regular [[tetrahedron]]), and 6 hypercells (each a [[5-cell]]).
# The '''[[5-cube]]''' of the [[hypercube]] family, {4,3,3,3}, with 32 vertices, 80 edges, 80 faces (each a [[square (geometry)|square]]), 40 cells (each a [[cube]]), and 10 hypercells (each a [[tesseract]]).
# The '''[[5-orthoplex]]''' of the [[cross polytope]] family, {3,3,3,4}, with 10 vertices, 40 edges, 80 faces (each a [[triangle]]), 80 cells (each a [[tetrahedron]]), and 32 hypercells (each a [[5-cell]]).

An important uniform 5-polytope is the [[5-demicube]], h{4,3,3,3} has half the vertices of the 5-cube (16), bounded by alternating [[5-cell]] and [[16-cell]] hypercells. The [[expansion (geometry)|expanded]] or [[stericated 5-simplex]] is the vertex figure of the [[A5 lattice|A<sub>5</sub> lattice]], {{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}. It and has a doubled symmetry from its symmetric Coxeter diagram. The kissing number of the lattice, 30, is represented in its vertices.<ref>{{Cite web|url=https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A5.html|title=The Lattice A5|website=www.math.rwth-aachen.de}}</ref> The [[rectified 5-orthoplex]] is the vertex figure of the [[D5 lattice|D<sub>5</sub> lattice]], {{CDD|nodes_10ru|split2|node|3|node|split1|nodes}}. Its 40 vertices represent the [[kissing number]] of the lattice and the highest for dimension 5.<ref>''Sphere packings, lattices, and groups'', by [[John Horton Conway]], Neil James Alexander Sloane, Eiichi Bannai
[https://books.google.com/books?id=upYwZ6cQumoC&dq=Sphere%20Packings%2C%20Lattices%20and%20Groups&pg=PR19]</ref>

{| class=wikitable
|+ Regular and semiregular polytopes in five dimensions<br>(Displayed as orthogonal projections in each [[Coxeter plane]] of symmetry)
|-
!A<sub>5</sub>
!Aut(A<sub>5</sub>)
!colspan=3|B<sub>5</sub>
!D<sub>5</sub>
|- align=center valign=top
|[[File:5-simplex t0.svg|altN=5-simplex|120px]]<br>[[5-simplex]]<br>{{CDD|node_1|3|node|3|node|3|node|3|node}}<br>{3,3,3,3}
|[[File:5-simplex t04 A4.svg|120px]]<br>[[Stericated 5-simplex]]<br>{{CDD|node_1|3|node|3|node|3|node|3|node_1}}
|[[File:5-cube t0.svg|altN=5-cube|120px]]<br>[[5-cube]]<br>{{CDD|node_1|4|node|3|node|3|node|3|node}}<br>{4,3,3,3}
|[[File:5-cube t4.svg|altN=5-orthoplex|120px]]<br>[[5-orthoplex]]<br>{{CDD|node_1|3|node|3|node|3|node|4|node}}<br>{3,3,3,4}
|[[File:5-cube t3.svg|altN=rectified 5-orthoplex|120px]]<br>[[Rectified 5-orthoplex]]<br>{{CDD|node|3|node_1|3|node|3|node|4|node}}<br>r{3,3,3,4}
|[[File:5-demicube t0 D5.svg|120px]]<br>[[5-demicube]]<br>{{CDD|node_h|4|node|3|node|3|node|3|node}}<br>h{4,3,3,3}
|}

===Hypersphere===
A [[hypersphere]] in 5-space (also called a '''4-sphere''' due to its surface being 4-dimensional) consists of the set of all points in 5-space at a fixed distance ''r'' from a central point P, that is [[rotationally symmetrical]]. The hypervolume enclosed by this hypersurface is:
:<math>V=\frac{8\pi ^2r^5}{15}</math>

==See also==
*[[5-manifold]]
*[[Gravity]]
*[[Hypersphere]]
*[[List of regular polytopes#Convex 4|List of regular 5-polytopes]]
*[[Four-dimensional space]]

==References==
{{Reflist|30em}}

==Further reading==
*{{cite book |title=Space-Time-Matter, Modern Kaluza-Klein Theory |last=Wesson |first=Paul S. |year=1999 |publisher=World Scientific |location=Singapore |isbn=981-02-3588-7 |url=https://archive.org/details/spacetimematterm0000wess |url-access=registration}}
*{{cite book |title=Five-Dimensional Physics: Classical and Quantum Consequences of Kaluza-Klein Cosmology |last=Wesson |first=Paul S. |year=2006 |publisher=World Scientific |location=Singapore |isbn=981-256-661-9 |url=https://archive.org/details/fivedimensionalp0000wess |url-access=registration}}
* [[Hermann Weyl|Weyl, Hermann]], ''[https://archive.org/details/raumzeitmateriev00weyl Raum, Zeit, Materie]'', 1918. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 ''[https://archive.org/details/spacetimematter00weyluoft Space Time Matter]'', Methuen, rept. 1952 Dover. {{ISBN|0-486-60267-2}}.

==External links==
* [https://robertinventor.com/software/virtualflower/virtualflower/tesseract.htm Anaglyph of a five dimensional hypercube in hyper perspective]

{{Dimension topics}}

[[Category:Dimension]]
[[Category:Multi-dimensional geometry]]
[[Category:5 (number)]]