Shape of the universe

{{Short description|Local and global geometry of the universe}} {{Redirect|Edge of the universe|the Bee Gees song|Edge of the Universe (song)|the documentary|Journey to the Edge of the Universe}} {{Cosmology|comp/struct}}

In physical [[cosmology]], the shape of the [[universe]] refers to both its local and global geometry. Local geometry is defined primarily by its [[curvature]], while the global geometry is characterised by its [[topology]] (which itself is constrained by curvature). [[General relativity]] explains how spatial curvature (local geometry) is constrained by [[gravity]]. The global topology of the universe cannot be deduced from measurements of curvature inferred from observations within the family of homogeneous general relativistic models alone, due to the existence of locally indistinguishable spaces with varying global topological characteristics.{{sfn|Luminet|2016}} For example; a [[Simply connected space|multiply connected]] space like a [[3-torus|3 torus]] has everywhere zero curvature but is finite in extent, whereas a flat [[Simply connected space|simply connected]] space is infinite in extent (such as [[Euclidean space]]).{{sfn|Luminet|2016}}

Observational evidence ([[WMAP]], [[BOOMERanG experiment|BOOMERanG]], and [[Planck (spacecraft)|Planck]], for example) indicates that the observable universe is spatially flat to within a 0.4% [[margin of error]] of the [[Friedmann equations#Density parameter|curvature density parameter]] with an unknown global topology.<ref name="NASA_Shape">{{cite web |date=24 January 2014 |title=Will the Universe expand forever? |url=http://map.gsfc.nasa.gov/universe/uni_shape.html |access-date=16 March 2015 |publisher=[[NASA]]}}</ref><ref name="Fermi_Flat">{{cite web |last=Biron |first=Lauren |date=7 April 2015 |title=Our universe is Flat |url=http://www.symmetrymagazine.org/article/april-2015/our-flat-universe?email_issue=725 |work=Symmetry Magazine |publisher=[[FermiLab]]/[[SLAC National Accelerator Laboratory|SLAC]]}}</ref> It is unknown whether the universe is simply connected like euclidean space or multiply connected like a torus.<ref name="Akrami2024_Promise_future">{{cite Q|Q136902920|article-number=171501|doi-access=free|quote=While unambiguous indicators of topology have yet to be detected, ... Much more can be done to discover, or constrain, the topology of space.}}</ref>

== Shape of the observable universe == {{Main|Observable universe|}} {{See also|Distance measures (cosmology)}}

The universe's structure can be examined from two angles:{{sfn|Luminet|2016}} # '''Local''' geometry: This relates to the curvature of the universe, primarily concerning what we can observe. # '''Global''' geometry: This pertains to the universe's overall shape and structure. The observable universe (of a given current observer) is a roughly spherical region extending about 46 billion light-years in every direction (from that observer, the observer being the current Earth, unless specified otherwise).<ref>{{Cite news |last=Crane |first=Leah |date=29 June 2024 |editor-last=de Lange |editor-first=Catherine |title=How big is the universe, really? |url=https://www.newscientist.com/article/mg26234970-500-how-big-is-the-universe-the-shape-of-space-time-could-tell-us/ |url-access=subscription |work=New Scientist |page=31}}</ref> It appears older and more [[redshift]]ed the deeper we look into space.<ref name="NASA_WMAP_Overview">{{cite web |title=WMAP Overview |publisher=NASA |website=NASA Science |url=https://science.nasa.gov/mission/wmap/wmap-overview/ |access-date=25 May 2026}}</ref> In theory, we could look all the way back to the [[Big Bang]], but in practice, we can only see up to the [[cosmic microwave background]] (CMB) (roughly {{val|370,000}} years after the Big Bang) as anything [[Recombination (cosmology)|beyond that is opaque]]. Studies show that the observable universe is [[isotropic]] and [[homogeneous]] on the largest scales.<ref name="PrimackStructure">{{cite web |last=Primack |first=Joel R. |title=Formation of Structure in the Universe |website=NASA/IPAC Extragalactic Database |publisher=Caltech |url=https://ned.ipac.caltech.edu/level5/Primack/Primack1_2.html |access-date=25 May 2026}}</ref>

If the observable universe encompasses the entire universe, we might determine its structure through observation. However, if the observable universe is smaller, we can only grasp a portion of it, making it impossible to deduce the global geometry through observation.{{sfn|Luminet|2016}} Different mathematical models of the universe's global geometry can be constructed, all consistent with observations and general relativity.<ref name="Akrami2024">{{cite journal |last1=Akrami |first1=Yashar |last2=Anselmi |first2=Stefano |last3=Copi |first3=Craig J. |display-authors=et al. |date=2024 |title=Promise of Future Searches for Cosmic Topology |journal=Physical Review Letters |volume=132 |issue=17 |article-number=171501 |arxiv=2210.11426 |doi=10.1103/PhysRevLett.132.171501 |doi-access=free}}</ref> Hence, it is unclear whether the observable universe matches the entire universe or is significantly smaller, though it is generally accepted that the universe is larger than the observable universe.

The universe may be compact in some dimensions and not in others.{{sfn|Luminet|2016}} Such models can be tested by searching for topological lensing, such as multiple images of the same distant source or matched patterns in the cosmic microwave background.{{sfn|Luminet|2016}} A small closed universe would produce multiple images of the same object in the sky, though not necessarily of the same age. As of 2024, observational evidence indicates that the observable universe is spatially flat with an unknown global structure.

== Curvature of the universe == {{further|Curvature#Space| Flatness problem}}

The [[curvature of Riemannian manifolds|curvature]] is a quantity describing how the geometry of a space differs locally from flat space. The curvature of any locally [[isotropic space]] (and hence of a locally isotropic universe) falls into one of the three following cases:{{sfn|Luminet|2016}} # Zero curvature (flat){{snd}}a drawn triangle's angles add up to 180° and the [[Pythagorean theorem]] holds; such 3-dimensional space is locally modeled by Euclidean space {{math|'''E'''''3''}}. # Positive curvature{{snd}}a drawn triangle's angles add up to more than 180°; such 3-dimensional space is locally modeled by a region of a [[N-sphere|3-sphere]] {{math|'''S'''''3''}}. # Negative curvature{{snd}}a drawn triangle's angles add up to less than 180°; such 3-dimensional space is locally modeled by a region of a [[hyperbolic space]] {{math|'''H'''''3''}}.

Curved geometries are in the domain of [[non-Euclidean geometry]]. An example of a positively curved space would be the surface of a sphere such as the Earth. A triangle drawn from the equator to a pole will have at least two angles equal 90°, which makes the sum of the 3 angles greater than 180°. An example of a negatively curved surface would be the shape of a [[saddle]] or mountain pass. A triangle drawn on a saddle surface will have the sum of the angles adding up to less than 180°.<ref name="GuardianTopology2024">{{cite news |last=Ball |first=Philip |date=8 June 2024 |title='We're trying to find the shape of space': scientists wonder if the universe is like a doughnut |url=https://www.theguardian.com/science/article/2024/jun/08/shape-universe-topology-doughnut-flat-curvature-dimensions |work=The Guardian |access-date=25 May 2026}}</ref>

[[File:End of universe.jpg|thumb|275px|The local geometry of the universe is determined by whether the [[density parameter#Density parameter|density parameter {{math|Ω}}]] is greater than, less than, or equal to 1. From top to bottom: a [[spherical geometry|spherical universe]] with {{math|Ω > 1}}, a [[hyperbolic geometry|hyperbolic universe]] with {{math|Ω < 1}}, and a [[Euclidean geometry|flat universe]] with {{math|Ω {{=}} 1}}. These depictions of two-dimensional surfaces are merely easily visualizable analogs to the 3-dimensional structure of (local) space.]][[File:Spacetime-diagram-flat-universe-proper-coordinates.png|thumb|275px|[[Comoving and proper distances|Proper distance]] spacetime diagram of our flat [[Lambda-CDM_model|ΛCDM]] universe. [[Particle horizon]]: green, [[Hubble radius]]: blue, [[Event horizon]]: purple, [[Light cone]]: orange.]][[File:Hyperbolic.universe.proper.coordinates.png|thumb|275px|Hyperbolic universe with the same radiation and matter density parameters as ours, but with negative curvature instead of dark energy (ΩΛ→Ωk).]][[File:Big-crunch_spacetime-diagram_matter-dominated_proper-distances.png|thumb|275px|Closed universe without dark energy and with overcritical matter density, which leads to a [[Big Crunch]]. Neither the hyperbolic nor the closed examples have an Event horizon (here the purple curve is the cosmic Antipode).]]

[[General relativity]] explains that mass and energy bend the curvature of spacetime and is used to determine what curvature the universe has by using a value called the [[density parameter]], represented with Omega ({{math|Ω}}). The density parameter is the average density of the universe divided by the critical energy density, that is, the mass energy needed for a universe to be flat. Put another way,{{sfn|Luminet|2016}} * If {{math|Ω {{=}} 1}}, the universe is flat. * If {{math|Ω > 1}}, there is positive curvature. * If {{math|Ω < 1}}, there is negative curvature.

Scientists could experimentally calculate {{math|Ω}} to determine the curvature two ways. One is to count all the [[mass–energy equivalence|mass–energy]] in the universe and take its average density, then divide that average by the critical energy density. Data from the [[Wilkinson Microwave Anisotropy Probe]] (WMAP) as well as the [[Planck (spacecraft)|Planck spacecraft]] give values for the three constituents of all the mass–energy in the universe – normal mass ([[baryonic matter]] and [[dark matter]]), relativistic particles (predominantly [[photon]]s and [[neutrino]]s), and [[dark energy]] or the [[cosmological constant]]:<ref>{{cite web|title= Density Parameter, Omega|url= http://hyperphysics.phy-astr.gsu.edu/hbase/astro/denpar.html|website= hyperphysics.phy-astr.gsu.edu|access-date= 2015-06-01}}</ref><ref>{{Cite journal |last1=Ade |first1=P. A. R. |last2=Aghanim |first2=N. |author-link2=Nabila Aghanim |last3=Armitage-Caplan |first3=C. |last4=Arnaud |first4=M. |last5=Ashdown |first5=M. |last6=Atrio-Barandela |first6=F. |last7=Aumont |first7=J. |last8=Baccigalupi |first8=C. |last9=Banday |first9=A. J. |last10=Barreiro |first10=R. B. |last11=Bartlett |first11=J. G. |last12=Battaner |first12=E. |last13=Benabed |first13=K. |last14=Benoît |first14=A. |last15=Bernard |first15=J.-P. |display-authors=3 |date=November 2014 |title=Planck 2013 results. XVI. Cosmological parameters |journal=[[Astronomy & Astrophysics]] |volume=571 |pages=A16 |arxiv=1303.5076 |bibcode=2014A&A...571A..16P |doi=10.1051/0004-6361/201321591 |issn=0004-6361 |s2cid=118349591 |collaboration=[[Planck Collaboration]] |last16=Bersanelli |first16=M. |last17=Bielewicz |first17=P. |last18=Bobin |first18=J. |last19=Bock |first19=J. J. |last20=Bonaldi |first20=A. |last21=Bond |first21=J. R. |last22=Borrill |first22=J. |last23=Bouchet |first23=F. R. |last24=Bridges |first24=M. |last25=Bucher |first25=M. |last26=Burigana |first26=C. |last27=Butler |first27=R. C. |last28=Calabrese |first28=E. |last29=Cappellini |first29=B.}}</ref> : Ωmass ≈ {{val|0.315|0.018}} : Ωrelativistic ≈ {{val|9.24|e=−5}} : ΩΛ ≈ {{val|0.6817|0.0018}} : Ωtotal = Ωmass + Ωrelativistic + ΩΛ = {{val|1.00|0.02}} The actual value for critical density value is measured as ''ρ''critical = {{val|9.47|e=−27|u=kg.m-3}}. From these values, within experimental error, the universe seems to be spatially flat.

Another way to measure Ω is to do so geometrically by measuring an angle across the observable universe. This can be done by using the [[CMB]] and measuring the power spectrum and temperature [[anisotropy]]. For instance, one can imagine finding a gas cloud that is not in thermal equilibrium due to being so large that light speed cannot propagate the thermal information. Knowing this propagation speed, we then know the size of the gas cloud as well as the distance to the gas cloud, we then have two sides of a triangle and can then determine the angles. Using a method similar to this, the [[BOOMERanG experiment]] has determined that the sum of the angles to 180° within experimental error, corresponding to {{nowrap|Ωtotal ≈ {{val|1.00|0.12}}}}.<ref>{{Cite journal |last1=de Bernardis |first1=P. |last2=Ade |first2=P. A. R. |last3=Bock |first3=J. J. |last4=Bond |first4=J. R. |last5=Borrill |first5=J. |last6=Boscaleri |first6=A. |last7=Coble |first7=K. |last8=Crill |first8=B. P. |last9=De Gasperis |first9=G. |last10=Farese |first10=P. C. |last11=Ferreira |first11=P. G. |last12=Ganga |first12=K. |last13=Giacometti |first13=M. |last14=Hivon |first14=E. |last15=Hristov |first15=V. V. |display-authors=3 |date=April 2000 |title=A flat Universe from high-resolution maps of the cosmic microwave background radiation |journal=[[Nature (journal)|Nature]] |language=en |volume=404 |issue=6781 |pages=955–959 |arxiv=astro-ph/0004404 |bibcode=2000Natur.404..955D |doi=10.1038/35010035 |issn=0028-0836 |pmid=10801117 |s2cid=4412370 |last16=Iacoangeli |first16=A. |last17=Jaffe |first17=A. H. |last18=Lange |first18=A. E. |last19=Martinis |first19=L. |last20=Masi |first20=S. |last21=Mason |first21=P. V. |last22=Mauskopf |first22=P. D. |last23=Melchiorri |first23=A. |last24=Miglio |first24=L. |last25=Montroy |first25=T. |last26=Netterfield |first26=C. B. |last27=Pascale |first27=E. |last28=Piacentini |first28=F. |last29=Pogosyan |first29=D. |last30=Prunet |first30=S.}}</ref>

These and other astronomical measurements constrain the spatial curvature to be very close to zero, although they do not constrain its sign. This means that although the local geometries of spacetime are generated by the theory of relativity based on [[spacetime interval]]s, we can approximate ''3-space'' by the familiar [[Euclidean geometry]].

The [[Friedmann–Lemaître–Robertson–Walker metric|Friedmann–Lemaître–Robertson–Walker (FLRW) model]] using [[Friedmann equations]] is commonly used to model the universe. The FLRW model provides a curvature of the universe based on the mathematics of [[fluid dynamics]], that is, modeling the matter within the universe as a perfect fluid. Although stars and structures of mass can be introduced into an "almost FLRW" model, a strictly FLRW model is used to approximate the local geometry of the observable universe. Another way of saying this is that, if all forms of [[dark energy]] are ignored, then the curvature of the universe can be determined by measuring the average density of matter within it, assuming that all matter is evenly distributed (rather than the distortions caused by 'dense' objects such as galaxies). This assumption is justified by the observations that, while the universe is "weakly" [[homogeneity (physics)|inhomogeneous]] and [[anisotropic]] (see the [[large-scale structure of the cosmos]]), it is on average homogeneous and [[isotropic]] when analyzed at a sufficiently large spatial scale.<ref name="Dodelson2003">{{cite book |last=Dodelson |first=Scott |title=Modern Cosmology |publisher=Academic Press |date=2003 |isbn=9780122191411}}</ref>

== Global universal structure == Global structure covers the [[geometry]] and the [[topology]] of the whole universe—both the observable universe and beyond.{{sfn|Luminet|2016}} While the local geometry does not determine the global geometry completely, it does limit the possibilities, particularly a geometry of a constant curvature. The universe is often taken to be a [[geodesic manifold]], free of [[topological defect#Cosmological defects|topological defects]]; relaxing either of these complicates the analysis considerably. A global geometry is a local geometry plus a topology. It follows that a topology alone does not give a global geometry: for instance, Euclidean 3-space and [[hyperbolic geometry|hyperbolic 3-space]] have the same topology but different global geometries.

As stated in the introduction, investigations within the study of the global structure of the universe include: * whether the universe is [[infinity|infinite]] or finite in extent, * whether the geometry of the global universe is flat, positively curved, or negatively curved, and, * whether the topology is simply connected (for example, like a [[sphere]]) or else multiply connected (for example, like a [[torus]]).<ref>{{Cite book |last=Davies |first=Paul |title=Space and Time in the Modern Universe |url=https://books.google.com/books?id=SZI5AAAAIAAJ |date=1977 |publisher=[[Cambridge University Press]] |isbn=978-0-521-29151-4 |location=Cambridge}}</ref>

=== Infinite or finite === One of the unanswered questions about the universe is whether it is infinite or finite in extent.{{sfn|Luminet|2016}} For intuition, it can be understood that a finite universe has a finite volume that, for example, could be in theory filled with a finite amount of material, while an infinite universe is unbounded and no numerical volume could possibly fill it. Mathematically, the question of whether the universe is infinite or finite is referred to as [[bounded metric space|boundedness]]. An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance {{mvar|d}}, there are points that are of a distance at least {{mvar|d}} apart. A finite universe is a bounded metric space, where there is some distance {{mvar|d}} such that all points are within distance {{mvar|d}} of each other. The smallest such {{mvar|d}} is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale". ====Historical thought====

[[Ancient mythologies]] variously described the universe as finite.<ref>{{cite magazine|author=J. J. Callahan|title=The Curvature of Space in a Finite Universe|url=https://www.jstor.org/stable/24950420|magazine=[[Scientific American]]|volume=235|number= 2 (August 1976)|publisher=[[Nature America, Inc.]]: [[ITHAKA]] |jstor=24950420 }}</ref>

By way of the account of [[Diogenes Laërtius]], for [[Leucippus]] ({{Circa|5th century BC}}) the universe is spatially infinite. <ref name=DL/><ref name=SB2022>{{cite book |author=Sylvia Berryman |date=October 18, 2022 |chapter-url=https://plato.stanford.edu/entries/atomism-ancient/#LeucDemo |chapter=Ancient Atomism: 2. Ancient Greek Atomism: 2.1 Leucippus and Democritus |location=Metaphysics Research Lab, [[Stanford University centers and institutes#Center for the Study of Language and Information|CSLI]] |title=Stanford Encyclopedia of Philosophy. |url=https://plato.stanford.edu/ |editor=Edward N. Zalta |editor-link=Edward N. Zalta |quote=Leucippus held that there are an infinite number of atoms moving for all time in an infinite void, and that these can form into cosmic systems or ''kosmoi''}}</ref><ref name=DL>{{cite book |author=Diogenes Laërtius |url=https://www.gutenberg.org/files/57342/57342-h/57342-h.htm |title=The Lives and Opinions of Eminent Philosophers |chapter=BOOK IX.: III |chapter-url=https://www.gutenberg.org/files/57342/57342-h/57342-h.htm#Page_388 |page=388|translator=C. D. Yonge Laërtius |translator-link=Charles Duke Yonge |location=[[Queen's University Belfast#History|Queen’s College]], Belfast |publisher=[[George Bell & Sons|G. Bell & Sons Ltd]]: [[gutenberg.org]] |quote=These are his doctrines in general; in particular detail, they are as follow: he says that the universe is infinite, as I have already mentioned; that of it, one part is a plenum, and the other a vacuum.}}</ref> [[Eudoxus of Cnidus|Eudoxus]] ({{Circa|380 BC}}) in consideration of motion concluded that the stars were encompassed within a [[Celestial spheres|sphere]]. {{efn|ἐν τρισὶν ἐτίθετ᾽ εἶναι σφαίραις, ὧν τὴν μὲν πρώτην τὴν τῶν ἀπλανῶν ἄστρων εἶναι <ref name=MR>{{cite web |author=Molly Read |url=https://cmb.physics.wisc.edu/pub/tutorial/briefhist.html |title=A Brief History |publisher=[[University of Wisconsin, Madison]]}}</ref><ref>{{cite book |author=Ἀριστοτέλους|chapter=Λ.1073β |chapter-url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0051%3Abook%3D12%3Asection%3D1073b |title=Μετὰ τὰ Φυσικά |publisher=[[Clarendon Press]]. 1924: [[perseus.tufts.edu]] |quote=Second paragraph, 1st and 2nd lines; verified translation via Google traductor. Greek original: via iask.ai/q/Aristotle-Metaphysics-Book-I-Greek-original-39nedig: Perseus Catalog (not available): www.physics.[[NTUA|ntua.gr]]/mourmouras/greats/aristoteles/meta_ta_physica.pdf}}</ref>}}<ref> :{{cite book |title=Aristotle in 23 Volumes, Vols.17, 18|chapter=Metaphysics 12.1073b |chapter-url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0052:book=12:section=1073b&highlight=eudoxus |translator=Hugh Tredennick |location=[[Cambridge, MA]] |publisher=[[Harvard University Press]] 1933: perseus.tufts.edu}} :{{cite web |author=Todd Timberlake |date=12 May 2011 |url=https://www.compadre.org/osp/items/detail.cfm?ID=11198 |title=Computer Program Detail Page: Spheres of Eudoxus |publisher=[[American Association of Physics Teachers]] & [[National Science Foundation]]-[[National Science Digital Library]] ([[ISKME]])}} :{{cite web |author=Matthias Tomczak |location=[[Flinders University]] |url=http://gyre.umeoce.maine.edu/physicalocean/Tomczak/science+society/lectures/illustrations/lecture8/eudoxus.html |title=Lecture 8 |publisher=[[University of Maine]] Ocean Observing System}}</ref> In the concept of [[Aristotle]] (384–322 BC), there existed [[concentric]] spheres located outwardly from Earth, the furthest of which contained the stars and was sometimes known as the ''kosmos''. Outside the boundary of the furthest sphere there was no place of physically existing anything and no time, in addition to this outside the last spheres edge there was no void. <ref name=CHP>{{cite web|author=Center for History of Physics |url=https://history.aip.org/exhibits/cosmology/ideas/greekworldview.htm|title=The Greek Worldview Continuation of the Greek tradition|publisher=[[The American Institute of Physics]]}}</ref><ref name=JEG2012>{{cite web|author=Jan Edward Garrett|date=November 7, 2012|url=https://people.wku.edu/jan.garrett/341/aristotle_big_picture.htm|title=Introduction to Aristotle's Celestial and Terrestrial Physics|publisher=[[Western Kentucky University]]|quote=Sometimes this sphere is simply called the kosmos, i.e., universe or world. There is no "place" and nothing material beyond this sphere.}}</ref><ref name=DJF78>{{cite journal |author=David J. Furley |author-link=David J. Furley |title=The Greek Theory of the Infinite Universe|url=https://www.jstor.org/stable/2709119 |journal=[[Journal of the History of Ideas]] |volume=42 |date=August 1978 |number=4 (Oct. - Dec., 1981) |pages=571–585 |location=[[Cambridge]] |publisher=[[University of Pennsylvania Press]]: [[ITHAKA]] |doi=10.2307/2709119 |jstor=2709119 |url-access=subscription}}</ref><ref name=GE81> :{{cite book |author=Grant E |chapter-url=https://www.cambridge.org/core/books/abs/much-ado-about-nothing/historical-roots-of-the-medieval-concept-of-an-infinite-extracosmic-void-space/92E60C6D078DC971DDD2F5514A2614A8 |chapter=5 - The historical roots of the medieval concept of an infinite, extracosmic void space |title=Much Ado about Nothing Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution |publisher=[[Cambridge University Press]] |date=1981 |pages=105–115 |doi=10.1017/CBO9780511895326.008 |isbn=978-0-521-22983-8 }} :{{cite book |last1=Grant |first1=Edward |url=https://books.google.com/books?id=SidBQyFmgpsC |title=Much Ado about Nothing: Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution|date=29 May 1981 |isbn= 0521229839}}</ref><ref>{{cite book |author=Aristotle |title=DE CAELO |chapter=BOOK I. 9|url=https://classicalliberalarts.com/wp-content/uploads/ARISTOTLE_DE_CAELO.pdf |page=279, lines 13–16, footnote|translator1=J. L. Stocks |translator1-link=John Leofric Stocks |translator2=H. B. Wallis |location=[[St John's College, Oxford University]] |publisher=[[Humphrey Milford]] 1922 |quote=It is therefore evident that there is also no place or void or time outside the heaven. For in every place body can be present; and void is said to be that in which the presence of body, though not actual, is possible; and time is the number of movement.}}</ref>

From the conceptual foundation of Aristotle {{efn|Aristotle knew of the thoughts of Leucippus (and [[Democritus]]) and considered the possibility of an infinite universe. <ref> :{{cite book |author=Ἀριστοτέλους |chapter=A.985b |chapter-url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0051:book=1:section=985b |title=Μετὰ τὰ Φυσικά |publisher=[[Clarendon Press]]. 1924: [[perseus.tufts.edu]] |quote=Greek original: via iask.ai/q/Aristotle-Metaphysics-Book-I-Greek-original-39nedig: Perseus Catalog (not available): www.physics.[[NTUA|ntua.gr]]/mourmouras/greats/aristoteles/meta_ta_physica.pdf}} :{{cite book |title=Aristotle in 23 Volumes, Vols.17, 18 |chapter=Metaphysics 1.985b |chapter-url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0052%3Abook%3D1%3Asection%3D985b|translator=Hugh Tredennick |location=[[Cambridge, MA]] |publisher=[[Harvard University Press]] 1933: perseus.tufts.edu}} :{{cite journal |author=Helge Kragh |author-link=Helge Kragh |url=https://thejournalofcosmology.com/AncientAstronomy108.html|title=Ancient Greek-Roman Cosmology: Infinite, Eternal, Finite, Cyclic, and Multiple Universes |location=[[University of Aarhus]] |journal=[[Journal of Cosmology]] |date=2010 |volume=9 |quote=A spatially infinite world was another impossibility, for by its very nature the world – meaning the heavens – revolved in a circle, and Aristotle pointed out that such motion was impossible as it would lead to an infinite velocity. What was enclosed by the outermost sphere comprised everything.}} :{{cite web |author=Jacques A. Bailly |author-link=Jacques Bailly |url=https://www.uvm.edu/~jbailly/courses/Aristotle/infinite%20in%20AristotleFromACTA.html |title=Aristotle on the Infinite, Space, and Time: Mathematical: 1) Multitude |publisher=[[uvm.edu]]}}</ref>}} which became the model for [[Ptolemy]] (2nd century AD) postdate the time of the completion of his work, entitled in ancient Greek; [[Ptolemy#Planetary Hypotheses|Ὑποθέσεις τῶν πλανωμένων]], the preferred general cosmology into the [[Middle Ages]] was that the cosmos was finite, which was known subsequently as [[Aristotelian cosmology]]. <ref> :{{cite web |author=CHRISTOPHER GRANEY |date=October 31, 2022|url=https://catholicscientists.org/articles/augustine-aquinas-and-calvin-on-the-size-of-the-moon-scripture-and-following-the-science/ |title=Augustine, Aquinas, and Calvin on the Size of the Moon, Scripture, and "Following the Science" |location=[[Specola Vaticana]] |publisher=[[The Society of Catholic Scientists]]}} :{{cite web |author=David Juste |title='Ptolemy, Planetary Hypotheses (Greek)' |url=http://ptolemaeus.badw.de/work/141. |date=10 May 2025 |website=Ptolemaeus Arabus et Latinus. Works |publisher=[[Bayerische Akademie der Wissenschaften]] |via=Center for History of Physics: history.aip.org/exhibits/cosmology}} :{{cite journal |author1=Mohan Matthen |author2=R. J. Hankinson |date=1993 |title=Aristotle's Universe: Its Form and Matter |url=https://www.jstor.org/stable/20117821 |journal=[[Synthese]] |volume= 96 |number=3 |pages=417–435 |publisher =[[Kluwer Academic Publishing]]: [[Springer Nature]]: [[ITHAKA]] |doi=10.1007/BF01064010 |jstor=20117821 }}</ref><ref name=MR/><ref name="GIN63">{{cite journal |author=G. I. Naan |date=1963 |title=On the Infinity of the Universe |url=https://www.jstor.org/stable/40400937 |journal=[[Science & Society]] |publisher=[[Sage Publications, Inc]]: [[ITHAKA]] |volume=27 |pages=176–202 |doi=10.1177/003682376302700203 |jstor=40400937 |number=2 (Spring, 1963)}}</ref><ref name=AJ6141-8_154>{{cite book |author=Alexander Jones |date=2015 |chapter=Greek Cosmology and Cosmogony|title=Handbook of Archaeoastronomy and Ethnoastronomy |editor=Ruggles, C. |pages=1549–1553 |publisher=[[Springer Nature|Springer]], New York, NY |doi=10.1007/978-1-4614-6141-8_154|isbn=978-1-4614-6141-8}}</ref> In [[Paradiso]], [[Dante Alighieri]] (1308–1320) made a universe according to the Ptolemaic mode, in which the Earth was placed at the centre of a number of spheres, the outer of which was were humanity could find the location of the realm of [[God]], an idea of reality that was the shared perception of all of the prominent thinkers of the [[medieval]] era. <ref name=LEEDS>{{cite web |author=Leeds Centre for Dante Studies & the Devers Program in Dante Studies at the [[University of Notre Dame]]| url=https://ahc.leeds.ac.uk/discover-dante/doc/paradiso |title=Paradiso|publisher=[[University of Leeds]] |quote=Ptolemaic understanding of the universe (after Ptolemy, an Alexandrian polymath of the second century A.D.). This was broadly shared by all mediaeval thinkers}}</ref> <ref>{{cite web |author=[[Beinecke Rare Book & Manuscript Library]], [[Yale University Library]] |url=https://beinecke.library.yale.edu/collections/highlights/divina-commedia-ms-428-between-1385-and-1400 |title=Divina Commedia, MS 428 |date=23 March 2021 |publisher=[[yale.edu]]}}</ref> [[Thomas Bradwardine|Bradwardine]] (1344) and [[Nicholas Oresme|Oresme]] during the 14th century contested the Aristotelian view on the basis of infinite God.<ref name=DFJOER>{{cite web |author1=DJF |author2=John J O'Connor |author3=Edmund F Robertson |title=MacTutor: The Infinite Universe |url=https://mathshistory.st-andrews.ac.uk/Astronomy/universe/ |publisher=[[st-andrews.ac.uk]] |quote=a |access-date=16 December 2025 |archive-date=4 October 2025 |archive-url=https://web.archive.org/web/20251004184845/https://mathshistory.st-andrews.ac.uk/Astronomy/universe/ |url-status=dead}}</ref>

With the advent of the [[heliocentric model]] so scientific thought realized the possibility of an infinite universe.<ref>{{cite book |author=Sun Kwok |date=22 October 2021 |chapter-url=https://link.springer.com/chapter/10.1007/978-3-030-80260-8_16 |chapter=Is the Universe Finite?: Abstract |title=Our Place in the Universe - II The Scientific Approach to Discovery |edition=1 |location=[[University of British Columbia]]| doi=10.1007/978-3-030-80260-8 |isbn=978-3-030-80260-8 |publisher=[[Springer Nature|Springer Nature Switzerland AG]] |bibcode=2021opus.book.....K |quote=After the development of the heliocentric theory, the hypothesis of the daily rotation of the celestial sphere was replaced by the hypothesis of the rotation of the Earth. This removes the need for the stars to lie at the same distance and rotate together, which in turn opens the possibility that the stars may have different distances from Earth and that the Universe could be infinite in size}}</ref> By the use of the novel model of [[Copernicus]], [[Thomas Digges]] in: ''A perfit description of the caelestiall orbs'', published 1576, made explanation of such a model universe, at that time presenting an idealogical break in concept from the then tradition of the reality of a celestial outer realm known as [[Paradise]].<ref name=JDB2005>{{cite book |author=John D. Barrow |author-link=John D. Barrow |url=https://books.google.com/books?id=KSNJJU6cRyMC |pages=116–117|title=The Infinite Book: A Short Guide to the Boundless, Timeless and Endless |chapter=chapter seven Is the Universe Infinite? |publisher=Jonathon Cape, [[Vintage Books]], [[Random House]]|date= 2005 |isbn=0099443724 |edition=reprint |via=plus.maths.org/content/do-infinities-exist-nature-0 University of Cambridge}}</ref>

[[Einstein]] in consideration of his 1916 theory of [[general relativity]] demonstrated in 1917 a finite universe. <ref name=COR2017>{{cite journal |author=Cormac O'Raifeartaigh |author-link=Cormac O'Raifeartaigh|date=February 3, 2017 |url=https://physicstoday.aip.org/news/albert-einstein-and-the-origins-of-modern-cosmology |title=Albert Einstein and the origins of modern cosmology |journal=[[Physics Today]] |issue=2 |article-number=12150 |publisher=[[American Center for Physics|AIP]] |doi=10.1063/PT.5.9085 |bibcode=2017PhT..2017b2150O |quote= {{quote|“[[Cosmological|Kosmologische]] Betrachtungen zur allgemeinen Relativitätstheorie”}} {{quote|The outcome of those deliberations was Einstein’s “[[Cosmological considerations]]” paper of 1917. His ingenious breakthrough was to postulate that we inhabit a universe of closed spatial geometry. Relativity could deliver a satisfactory model of the known universe if it was assumed that the cosmos had the geometry of a three-dimensional sphere—unbounded spatially, yet finite in content.}} |url-access=subscription}}</ref><ref name=DFJOER/> The [[de Sitter]] infinite universe of 1917 was caused by the incompatibility of [[Theory of relativity|Relativity]] with [[Euclidean space]].<ref name=DFJOER/> [[David Hilbert|Hilbert]] (1925) thought the universe was determined finite by [[elliptical geometry]] or infinite by [[Euclidean geometry]]<ref> :{{cite web |author=David Hilbert |author-link=David Hilbert |url=https://lawrencecpaulson.github.io/papers/on-the-infinite.pdf |title=On the infinite |page=186 |publisher=[[Cambridge University Press]]: [[Lawrence Paulson|lawrencecpaulson]] |date=5 June 2012 |quote=Einstein has shown that euclidean geometry must be abandoned...all the results of astronomy are perfectly compatible with the postulate that the universe is elliptical.}} :{{cite journal |title=Über das Unendliche - On the infinite DAVID HILBERT (in: Philosophy of mathematics) |journal=[[Mathematische Annalen]]|volume= 95|date=1926|url=https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Philosophy/Philosophy.html |edition=2nd |editor1=Paul Benacerraf |editor2=Hilary Putnam |translator1=Erna Putnam |translator2=Gerald J. Massey |location=[[Göttingen]]: Berlin: (Cambridge London New York New Rochelle Melbourne Sydney) |publisher=[[Springer Verlag]]: ([[Cambridge University Press]]: math.[[dartmouth.edu]] |quote=German language title: jamesrmeyer.com/infinite/hilbert-uber-das-unendliche)}}</ref> (i.e. flat).<ref name=JS252001>{{cite web |author=Joseph Silk |author-link=Joseph Silk |url=https://www.esa.int/Science_Exploration/Space_Science/Is_the_Universe_finite_or_infinite_An_interview_with_Joseph_Silk |title=Is the Universe finite or infinite? An interview with Joseph Silk |date=2 May 2001 |location=[[University of Oxford]] |publisher=[[European Space Agency]]}}</ref>

====Current thought==== ===== With or without boundary ===== Assuming a finite universe, the universe can either have an edge or no edge. Many finite mathematical spaces, e.g., a [[disk (mathematics)|disc]], have an edge or boundary. Spaces that have an edge are difficult to treat, both conceptually and mathematically. Namely, it is difficult to state what would happen at the edge of such a universe. For this reason, spaces that have an edge are typically excluded from consideration.

However, there exist many finite spaces, such as the [[3-sphere]] and [[3-torus]], that have no edges. Mathematically, these spaces are referred to as being [[compact space|compact]] without boundary. The term compact means that it is finite in extent ("bounded") and [[complete metric space|complete]]. The term "without boundary" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a [[differentiable manifold]]. A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a [[closed manifold]]. The 3-sphere and 3-torus are both closed manifolds.<ref name="Lee2011">{{cite book |last=Lee |first=John M. |title=Introduction to Topological Manifolds |edition=2nd |publisher=Springer |date=2011 |isbn=9781441979391}}</ref>

===== Observational methods ===== In the 1990s and early 2000s, empirical methods for determining the global topology using measurements on scales that would show multiple imaging were proposed<ref name="Luminet1995" /> and applied to cosmological observations.<ref name="Nat03" /><ref name="RBSG08" />

In the 2000s and 2010s, it was shown that, since the universe is inhomogeneous as shown in the [[observable universe#Large-scale structure|cosmic web of large-scale structure]], acceleration effects measured on local scales in the patterns of the movements of galaxies should, in principle, reveal the global topology of the universe.<ref name="RBBSJ2007">{{cite Q|Q68598777}}</ref><ref name="RR09">{{cite Q|Q68676519}}</ref><ref name="ORB12">{{cite Q|Q96692451}}</ref>

=== Curvature === The curvature of the universe places constraints on the topology. If the spatial geometry is [[spherical 3-manifold|spherical]], i.e., possess positive curvature, the topology is compact. For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite.<ref name="Luminet1995">{{harvnb|Lachièze-Rey|Luminet|1995}}</ref> Many textbooks erroneously state that a flat or hyperbolic universe implies an infinite universe; however, the correct statement is that a flat universe that is also [[simply connected]] implies an infinite universe.<ref name="Luminet1995" /> For example, Euclidean space is flat, simply connected, and infinite, but there are [[torus#Flat torus|tori]] that are flat, multiply connected, finite, and compact (see [[flat torus]]).

In general, [[Riemannian geometry#Local to global theorems|local to global theorems]] in [[Riemannian geometry]] relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in [[geometrization conjecture|Thurston geometries]].

The latest research shows that even the most powerful future experiments (like the [[Square Kilometre Array|SKA]]) will not be able to distinguish between a flat, open and closed universe if the true value of cosmological curvature parameter is smaller than 10−4. If the true value of the cosmological curvature parameter is larger than 10−3 we will be able to distinguish between these three models even now.<ref>{{Cite journal |last1=Vardanyan |first1=Mihran |last2=Trotta |first2=Roberto |last3=Silk |first3=Joseph |date=21 July 2009 |title=How flat can you get? A model comparison perspective on the curvature of the Universe |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=397 |issue=1 |pages=431–444 |arxiv=0901.3354 |bibcode=2009MNRAS.397..431V |doi=10.1111/j.1365-2966.2009.14938.x |s2cid=15995519 |doi-access=free}}</ref>

Final results of the ''Planck'' mission, released in 2018, show the cosmological curvature parameter, {{nowrap|1=1 − Ω = Ω''K'' = −''Kc''2/''a''2''H''2}}, to be {{val|0.0007|0.0019}}, consistent with a flat universe.<ref>{{Cite journal |last1=Aghanim |first1=N. |author-link1=Nabila Aghanim |last2=Akrami |first2=Y. |last3=Ashdown |first3=M. |last4=Aumont |first4=J. |last5=Baccigalupi |first5=C. |last6=Ballardini |first6=M. |last7=Banday |first7=A. J. |last8=Barreiro |first8=R. B. |last9=Bartolo |first9=N. |last10=Basak |first10=S. |last11=Battye |first11=R. |last12=Benabed |first12=K. |last13=Bernard |first13=J.-P. |last14=Bersanelli |first14=M. |last15=Benoit-Levy |first15=A. |display-authors=3 |date=September 2020 |title=Planck 2018 results: VI. Cosmological parameters |journal=[[Astronomy & Astrophysics]] |volume=641 |pages=A6 |arxiv=1807.06209 |bibcode=2020A&A...641A...6P |doi=10.1051/0004-6361/201833910 |issn=0004-6361 |s2cid=119335614 |collaboration=[[Planck Collaboration]] |last16=Bernard |first16=J. P. |last17=Bersanelli |first17=M. |last18=Bielewicz |first18=P. |last19=Bonaldi |first19=A. |last20=Bonavera |first20=L. |last21=Bond |first21=J. R. |last22=Borrill |first22=J. |last23=Bouchet |first23=F. R. |last24=Boulanger |first24=F. |last25=Bucher |first25=M. |last26=Burigana |first26=C. |last27=Butler |first27=R. C. |last28=Calabrese |first28=E. |last29=Cardoso |first29=J. F.}}</ref> (i.e. positive curvature: {{nowrap|1=''K'' = +1}}, {{nowrap|Ω''K'' < 0}}, {{nowrap|Ω > 1}}, negative curvature: {{nowrap|1=''K'' = −1}}, {{nowrap|Ω''K'' > 0}}, {{nowrap|Ω < 1}}, zero curvature: {{nowrap|1=''K'' = 0}}, {{nowrap|1=Ω''K'' = 0}}, {{nowrap|1=Ω = 1}}).

==== Universe with zero curvature ==== In a universe with zero curvature, the local geometry is [[geometrization conjecture#Euclidean geometry E3|flat]]. The most familiar such global structure is that of Euclidean space, which is infinite in extent. Flat universes that are finite in extent include the [[torus]] and [[Klein bottle]]. Moreover, in three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable.<ref name="ConwayRossetti2003">{{cite arXiv |last1=Conway |first1=John Horton |last2=Rossetti |first2=Juan Pablo |date=2003 |title=Describing the platycosms |eprint=math/0311476 }}</ref> These are the [[flat manifold|Bieberbach manifolds]]. The most familiar is the aforementioned [[three-torus model of the universe|3-torus universe]].

In the absence of dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion [[asymptote|asymptotically]] approaching zero. With dark energy, the expansion rate of the universe initially slows down, due to the effect of gravity, but eventually increases. The [[ultimate fate of the universe]] is the same as that of an open universe in the sense that space will continue expanding forever.

A flat universe can have [[zero-energy universe|zero total energy]].<ref>{{cite AV media |date=2009 |title=A Universe From Nothing lecture by Lawrence Krauss at AAI |url=https://www.youtube.com/watch?v=7ImvlS8PLIo |url-status=live |archive-url=https://ghostarchive.org/varchive/youtube/20211215/7ImvlS8PLIo |archive-date=2021-12-15 |access-date=17 October 2011 |via=[[YouTube]]}}{{cbignore}}</ref>

==== Universe with positive curvature ==== A positively curved universe is described by [[elliptic geometry]], and can be thought of as a three-dimensional [[hypersphere]], or some other [[spherical 3-manifold]] (such as the [[Poincaré dodecahedral space]]), all of which are [[Quotient space (topology)|quotient]]s of the 3-sphere.

[[Homology sphere#Cosmology|Poincaré dodecahedral space]] is a positively curved space, colloquially described as "soccerball-shaped", as it is the [[Quotient space (topology)|quotient]] of the 3-sphere by the [[binary icosahedral group]], which is very close to [[icosahedral symmetry]], the symmetry of a soccer ball. This was proposed by [[Jean-Pierre Luminet]] and colleagues in 2003<ref name="Nat03">{{Cite journal |last1=Luminet |first1=Jean-Pierre |author-link1=Jean-Pierre Luminet |last2=Weeks |first2=Jeffrey R. |last3=Riazuelo |first3=Alain |last4=Lehoucq |first4=Roland |last5=Uzan |first5=Jean-Philippe |date=October 2003 |title=Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background |journal=[[Nature (journal)|Nature]] |volume=425 |issue=6958 |pages=593–595 |arxiv=astro-ph/0310253 |bibcode=2003Natur.425..593L |doi=10.1038/nature01944 |issn=0028-0836 |pmid=14534579 |s2cid=4380713}}</ref><ref name="physwebLum03">{{Cite web |last=Dumé |first=Isabelle |date=8 October 2003 |title=Is the universe a dodecahedron? |url=https://physicsworld.com/a/is-the-universe-a-dodecahedron/ |website=[[Physics World]] |language=en-GB}}</ref> and an optimal orientation on the sky for the model was estimated in 2008.<ref name="RBSG08">{{Cite journal |last1=Lew |first1=B. |last2=Roukema |first2=B. |last3=Szaniewska |first3=Agnieszka |last4=Gaudin |first4=Nicolas E. |date=May 2008 |title=A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data |journal=[[Astronomy & Astrophysics]] |volume=482 |issue=3 |pages=747–753 |arxiv=0801.0006 |bibcode=2008A&A...482..747L |doi=10.1051/0004-6361:20078777 |issn=0004-6361 |s2cid=1616362}}</ref>

==== Universe with negative curvature ==== A hyperbolic universe, one of a negative spatial curvature, is described by hyperbolic geometry, and can be thought of locally as a three-dimensional analog of an infinitely extended saddle shape. There is a great variety of [[hyperbolic 3-manifold]]s, and their classification is not completely understood.<ref name="Thurston1997">{{cite book |last=Thurston |first=William P. |title=Three-Dimensional Geometry and Topology |volume=1 |publisher=Princeton University Press |date=1997 |isbn=9780691083049}}</ref> Those of finite volume can be understood via the [[Mostow rigidity theorem]]. For hyperbolic local geometry, many of the possible three-dimensional spaces are informally called "horn topologies", so called because of the shape of the [[pseudosphere]], a canonical model of hyperbolic geometry. An example is the [[Picard horn]], a negatively curved space, colloquially described as "funnel-shaped".<ref name="Aurich0403597">{{cite journal |last= Aurich |first= Ralf|author2=Lustig, S. |author3=Steiner, F. |author4=Then, H. |title= Hyperbolic Universes with a Horned Topology and the CMB Anisotropy |journal= [[Classical and Quantum Gravity]] |volume= 21 |issue= 21 |pages= 4901–4926 |date= 2004 |doi= 10.1088/0264-9381/21/21/010 |bibcode= 2004CQGra..21.4901A |arxiv= astro-ph/0403597|s2cid= 17619026}}</ref>

==== Curvature: open or closed ==== When cosmologists speak of the universe as being "open" or "closed", they most commonly are referring to whether the curvature is negative or positive, respectively.<ref name="Dodelson2003"/> These meanings of open and closed are different from the mathematical meaning of open and closed used for sets in topological spaces and for the mathematical meaning of open and closed manifolds, which gives rise to ambiguity and confusion. In mathematics, there are definitions for a [[closed manifold]] (i.e., compact without boundary) and [[open manifold]] (i.e., one that is not compact and without boundary). A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closed or open manifold. For example, in the [[Friedmann–Lemaître–Robertson–Walker metric|Friedmann–Lemaître–Robertson–Walker]] (FLRW) model, the universe is considered to be without boundaries, in which case "compact universe" could describe a universe that is a closed manifold.

== See also == {{div col}} * {{annotated link|de Sitter space}} * {{annotated link|Ekpyrotic universe}}—A string-theory-related model depicting a [[five-dimensional]], [[brane|membrane]]-shaped universe; an alternative to the [[Big Bang|Hot Big Bang Model]], whereby the universe is described to have originated when two membranes collided at the fifth dimension * {{annotated link|String Theory#Extra dimensions|Extra dimensions in string theory}} for 6 or 7 extra space-like dimensions all with a ''compact'' topology * {{annotated link|History of the center of the Universe}} * {{annotated link|Holographic principle}} * {{annotated link|List of paradoxes#Cosmology|List of cosmology paradoxes}} * {{annotated link|Spacetime topology}} * {{annotated link|Theorema Egregium}}—The "remarkable theorem" discovered by [[Carl Friedrich Gauss|Gauss]], which showed there is an intrinsic notion of curvature for surfaces. This is used by [[Bernhard Riemann|Riemann]] to generalize the (intrinsic) notion of curvature to higher-dimensional spaces * {{annotated link|Three-torus model of the universe}} * {{annotated link|Zero-energy universe}} {{div col end}} == Notes == {{notelist}} == References == {{reflist}}

== External links == * [http://icosmos.co.uk Geometry of the Universe] at icosmos.co.uk * {{Cite journal |last1=Levin |first1=Janna |last2=Scannapieco |first2=Evan |last3=Silk |first3=Joseph |name-list-style=amp |date=September 1998 |title=The topology of the universe: the biggest manifold of them all |journal=[[Classical and Quantum Gravity]] |volume=15 |issue=9 |pages=2689–2697 |arxiv=gr-qc/9803026 |bibcode=1998CQGra..15.2689L |doi=10.1088/0264-9381/15/9/015 |issn=0264-9381 |s2cid=119080782}} * {{Cite journal |last1=Lachièze-Rey |first1=Marc |last2=Luminet |first2=Jean-Pierre |author-link2=Jean-Pierre Luminet |date=March 1995 |title=Cosmic topology |journal=[[Physics Reports]] |language=en |volume=254 |issue=3 |pages=135–214 |arxiv=gr-qc/9605010 |bibcode=1995PhR...254..135L |doi=10.1016/0370-1573(94)00085-H |s2cid=119500217}} * {{Cite journal |last=Luminet |first=Jean-Pierre |date=15 January 2016 |title=The Status of Cosmic Topology after Planck Data |journal=[[Universe (journal)|Universe]] |language=en |volume=2 |issue=1 |page=1 |arxiv=1601.03884 |bibcode=2016Univ....2....1L |doi=10.3390/universe2010001 |issn=2218-1997 |s2cid=7331164 |doi-access=free}} * {{cite web|first=Sean|last=Markey|date=8 October 2003|url=http://news.nationalgeographic.com/news/2003/10/1008_031008_finiteuniverse.html|title=Universe is Finite, 'Soccer Ball'-Shaped, Study Hints|archive-url=https://web.archive.org/web/20031010171053/http://news.nationalgeographic.com/news/2003/10/1008_031008_finiteuniverse.html|archive-date=10 October 2003|website=[[National Geographic News]]}} Possible wrap-around dodecahedral shape of the universe * Classification of [http://star-www.st-and.ac.uk/~kdh1/cos/cos.html possible universes] in the [[Lambda-CDM]] model. * {{Cite journal |last=Fagundes |first=Helio V. |date=December 2002 |title=Exploring the global topology of the universe |journal=[[Brazilian Journal of Physics]] |volume=32 |issue=4 |pages=891–894 |arxiv=gr-qc/0112078 |bibcode=2002BrJPh..32..891F |doi=10.1590/S0103-97332002000500012 |issn=0103-9733 |s2cid=119495347}} * {{cite web|last=Grime|first=James|title={{pi}}39 (Pi and the size of the Universe)|url=http://www.numberphile.com/videos/pi_universe.html|work=Numberphile|publisher=[[Brady Haran]]|access-date=2013-04-07|archive-date=2015-04-30|archive-url=https://web.archive.org/web/20150430002504/http://www.numberphile.com/videos/pi_universe.html}} * [http://blogs.scientificamerican.com/degrees-of-freedom/2011/07/25/what-do-you-mean-the-universe-is-flat-part-i What do you mean the universe is flat?] Scientific American Blog explanation of a flat universe and the curved spacetime in the universe.

[[Category:Differential geometry]] [[Category:General relativity]] [[Category:Physical cosmological concepts]] [[Category:Unsolved problems in astronomy]] [[Category:Big Bang]]