# Shape of the universe

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Local and global geometry of the universe

"Edge of the universe" redirects here. For the Bee Gees song, see [Edge of the Universe (song)](/source/Edge_of_the_Universe_(song)). For the documentary, see [Journey to the Edge of the Universe](/source/Journey_to_the_Edge_of_the_Universe).

Part of a series on Physical cosmology Big Bang Universe Age of the universe Chronology of the universe Early universe Inflation Baryogenesis Nucleosynthesis Backgrounds Gravitational wave (GWB) Microwave (CMB) Neutrino (CNB) Expansion & Future Hubble's law Redshift Expansion of the universe FLRW metric Friedmann equations Lambda-CDM model Future of an expanding universe Ultimate fate of the universe Components & Structure Components Dark energy Dark matter Photons Baryons Structure Shape of the universe Galaxy filament Galaxy formation Large quasar group Large-scale structure Reionization Structure formation Experiments Black Hole Initiative (BHI) BOOMERanG Cosmic Background Explorer (COBE) Dark Energy Survey Planck space observatory Sloan Digital Sky Survey (SDSS) 2dF Galaxy Redshift Survey ("2dF") Wilkinson Microwave Anisotropy Probe (WMAP) Scientists Aaronson Alfvén Alpher Copernicus de Sitter Dicke Ehlers Einstein Ellis Friedmann Galileo Gamow Guth Hawking Hubble Huygens Kepler Lemaître Mather Newton Penrose Penzias Rubin Schmidt Smoot Suntzeff Sunyaev Tolman Wilson Zeldovich List of cosmologists Subject history Discovery of cosmic microwave background radiation History of the Big Bang theory Timeline of cosmological theories Category Astronomy portal v t e

In physical [cosmology](/source/Cosmology), the shape of the [universe](/source/Universe) refers to both its local and global geometry. Local geometry is defined primarily by its [curvature](/source/Curvature), while the global geometry is characterised by its [topology](/source/Topology) (which itself is constrained by curvature). [General relativity](/source/General_relativity) explains how spatial curvature (local geometry) is constrained by [gravity](/source/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.[1] For example; a [multiply connected](/source/Simply_connected_space) space like a [3 torus](/source/3-torus) has everywhere zero curvature but is finite in extent, whereas a flat [simply connected](/source/Simply_connected_space) space is infinite in extent (such as [Euclidean space](/source/Euclidean_space)).[1]

Observational evidence ([WMAP](/source/WMAP), [BOOMERanG](/source/BOOMERanG_experiment), and [Planck](/source/Planck_(spacecraft)), for example) indicates that the observable universe is spatially flat to within a 0.4% [margin of error](/source/Margin_of_error) of the [curvature density parameter](/source/Friedmann_equations#Density_parameter) with an unknown global topology.[2][3] It is unknown whether the universe is simply connected like euclidean space or multiply connected like a torus.[4]

## Shape of the observable universe

Main article: [Observable universe](/source/Observable_universe)

See also: [Distance measures (cosmology)](/source/Distance_measures_(cosmology))

The universe's structure can be examined from two angles:[1]

1. **Local** geometry: This relates to the curvature of the universe, primarily concerning what we can observe.

1. **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).[5] It appears older and more [redshifted](/source/Redshift) the deeper we look into space.[6] In theory, we could look all the way back to the [Big Bang](/source/Big_Bang), but in practice, we can only see up to the [cosmic microwave background](/source/Cosmic_microwave_background) (CMB) (roughly 370000 years after the Big Bang) as anything [beyond that is opaque](/source/Recombination_(cosmology)). Studies show that the observable universe is [isotropic](/source/Isotropic) and [homogeneous](/source/Homogeneous) on the largest scales.[7]

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.[1] Different mathematical models of the universe's global geometry can be constructed, all consistent with observations and general relativity.[8] 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.[1] 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.[1] 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 information: [Curvature § Space](/source/Curvature#Space), and [Flatness problem](/source/Flatness_problem)

The [curvature](/source/Curvature_of_Riemannian_manifolds) is a quantity describing how the geometry of a space differs locally from flat space. The curvature of any locally [isotropic space](/source/Isotropic_space) (and hence of a locally isotropic universe) falls into one of the three following cases:[1]

1. Zero curvature (flat) – a drawn triangle's angles add up to 180° and the [Pythagorean theorem](/source/Pythagorean_theorem) holds; such 3-dimensional space is locally modeled by Euclidean space **E***3*.

1. Positive curvature – a drawn triangle's angles add up to more than 180°; such 3-dimensional space is locally modeled by a region of a [3-sphere](/source/N-sphere) **S***3*.

1. Negative curvature – 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](/source/Hyperbolic_space) **H***3*.

Curved geometries are in the domain of [non-Euclidean geometry](/source/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](/source/Saddle) or mountain pass. A triangle drawn on a saddle surface will have the sum of the angles adding up to less than 180°.[9]

The local geometry of the universe is determined by whether the [density parameter Ω](/source/Density_parameter#Density_parameter) is greater than, less than, or equal to 1. From top to bottom: a [spherical universe](/source/Spherical_geometry) with Ω > 1, a [hyperbolic universe](/source/Hyperbolic_geometry) with Ω < 1, and a [flat universe](/source/Euclidean_geometry) with Ω = 1. These depictions of two-dimensional surfaces are merely easily visualizable analogs to the 3-dimensional structure of (local) space.

[Proper distance](/source/Comoving_and_proper_distances) spacetime diagram of our flat [ΛCDM](/source/Lambda-CDM_model) universe. [Particle horizon](/source/Particle_horizon): green, [Hubble radius](/source/Hubble_radius): blue, [Event horizon](/source/Event_horizon): purple, [Light cone](/source/Light_cone): orange.

Hyperbolic universe with the same radiation and matter density parameters as ours, but with negative curvature instead of dark energy (ΩΛ→Ωk).

Closed universe without dark energy and with overcritical matter density, which leads to a [Big Crunch](/source/Big_Crunch). Neither the hyperbolic nor the closed examples have an Event horizon (here the purple curve is the cosmic Antipode).

[General relativity](/source/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](/source/Density_parameter), represented with Omega (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,[1]

- If Ω = 1, the universe is flat.

- If Ω > 1, there is positive curvature.

- If Ω < 1, there is negative curvature.

Scientists could experimentally calculate Ω to determine the curvature two ways. One is to count all the [mass–energy](/source/Mass%E2%80%93energy_equivalence) in the universe and take its average density, then divide that average by the critical energy density. Data from the [Wilkinson Microwave Anisotropy Probe](/source/Wilkinson_Microwave_Anisotropy_Probe) (WMAP) as well as the [Planck spacecraft](/source/Planck_(spacecraft)) give values for the three constituents of all the mass–energy in the universe – normal mass ([baryonic matter](/source/Baryonic_matter) and [dark matter](/source/Dark_matter)), relativistic particles (predominantly [photons](/source/Photon) and [neutrinos](/source/Neutrino)), and [dark energy](/source/Dark_energy) or the [cosmological constant](/source/Cosmological_constant):[10][11]

- Ωmass ≈ 0.315±0.018

- Ωrelativistic ≈ 9.24×10−5

- ΩΛ ≈ 0.6817±0.0018

- Ωtotal = Ωmass + Ωrelativistic + ΩΛ = 1.00±0.02

The actual value for critical density value is measured as *ρ*critical = 9.47×10−27 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](/source/CMB) and measuring the power spectrum and temperature [anisotropy](/source/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](/source/BOOMERanG_experiment) has determined that the sum of the angles to 180° within experimental error, corresponding to Ωtotal ≈ 1.00±0.12.[12]

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 intervals](/source/Spacetime_interval), we can approximate *3-space* by the familiar [Euclidean geometry](/source/Euclidean_geometry).

The [Friedmann–Lemaître–Robertson–Walker (FLRW) model](/source/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric) using [Friedmann equations](/source/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](/source/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](/source/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" [inhomogeneous](/source/Homogeneity_(physics)) and [anisotropic](/source/Anisotropic) (see the [large-scale structure of the cosmos](/source/Large-scale_structure_of_the_cosmos)), it is on average homogeneous and [isotropic](/source/Isotropic) when analyzed at a sufficiently large spatial scale.[13]

## Global universal structure

Global structure covers the [geometry](/source/Geometry) and the [topology](/source/Topology) of the whole universe—both the observable universe and beyond.[1] 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](/source/Geodesic_manifold), free of [topological defects](/source/Topological_defect#Cosmological_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 3-space](/source/Hyperbolic_geometry) 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 [infinite](/source/Infinity) 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](/source/Sphere)) or else multiply connected (for example, like a [torus](/source/Torus)).[14]

### Infinite or finite

One of the unanswered questions about the universe is whether it is infinite or finite in extent.[1] 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 [boundedness](/source/Bounded_metric_space). An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance d, there are points that are of a distance at least d apart. A finite universe is a bounded metric space, where there is some distance d such that all points are within distance d of each other. The smallest such d is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale".

#### Historical thought

[Ancient mythologies](/source/Ancient_mythologies) variously described the universe as finite.[15]

By way of the account of [Diogenes Laërtius](/source/Diogenes_La%C3%ABrtius), for [Leucippus](/source/Leucippus) (c. 5th century BC) the universe is spatially infinite. [16][17][16] [Eudoxus](/source/Eudoxus_of_Cnidus) (c. 380 BC) in consideration of motion concluded that the stars were encompassed within a [sphere](/source/Celestial_spheres). [a][20] In the concept of [Aristotle](/source/Aristotle) (384–322 BC), there existed [concentric](/source/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. [21][22][23][24][25]

From the conceptual foundation of Aristotle [b] which became the model for [Ptolemy](/source/Ptolemy) (2nd century AD) postdate the time of the completion of his work, entitled in ancient Greek; [Ὑποθέσεις τῶν πλανωμένων](/source/Ptolemy#Planetary_Hypotheses), the preferred general cosmology into the [Middle Ages](/source/Middle_Ages) was that the cosmos was finite, which was known subsequently as [Aristotelian cosmology](https://en.wikipedia.org/w/index.php?title=Aristotelian_cosmology&action=edit&redlink=1). [27][18][28][29] In [Paradiso](/source/Paradiso), [Dante Alighieri](/source/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](/source/God), an idea of reality that was the shared perception of all of the prominent thinkers of the [medieval](/source/Medieval) era. [30] [31] [Bradwardine](/source/Thomas_Bradwardine) (1344) and [Oresme](/source/Nicholas_Oresme) during the 14th century contested the Aristotelian view on the basis of infinite God.[32]

With the advent of the [heliocentric model](/source/Heliocentric_model) so scientific thought realized the possibility of an infinite universe.[33] By the use of the novel model of [Copernicus](/source/Copernicus), [Thomas Digges](/source/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](/source/Paradise).[34]

[Einstein](/source/Einstein) in consideration of his 1916 theory of [general relativity](/source/General_relativity) demonstrated in 1917 a finite universe. [35][32] The [de Sitter](/source/De_Sitter) infinite universe of 1917 was caused by the incompatibility of [Relativity](/source/Theory_of_relativity) with [Euclidean space](/source/Euclidean_space).[32] [Hilbert](/source/David_Hilbert) (1925) thought the universe was determined finite by [elliptical geometry](/source/Elliptical_geometry) or infinite by [Euclidean geometry](/source/Euclidean_geometry)[36] (i.e. flat).[37]

#### 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 [disc](/source/Disk_(mathematics)), 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](/source/3-sphere) and [3-torus](/source/3-torus), that have no edges. Mathematically, these spaces are referred to as being [compact](/source/Compact_space) without boundary. The term compact means that it is finite in extent ("bounded") and [complete](/source/Complete_metric_space). 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](/source/Differentiable_manifold). A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a [closed manifold](/source/Closed_manifold). The 3-sphere and 3-torus are both closed manifolds.[38]

#### 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[39] and applied to cosmological observations.[40][41]

In the 2000s and 2010s, it was shown that, since the universe is inhomogeneous as shown in the [cosmic web of large-scale structure](/source/Observable_universe#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.[42][43][44]

### Curvature

The curvature of the universe places constraints on the topology. If the spatial geometry is [spherical](/source/Spherical_3-manifold), 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.[39] 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](/source/Simply_connected) implies an infinite universe.[39] For example, Euclidean space is flat, simply connected, and infinite, but there are [tori](/source/Torus#Flat_torus) that are flat, multiply connected, finite, and compact (see [flat torus](/source/Flat_torus)).

In general, [local to global theorems](/source/Riemannian_geometry#Local_to_global_theorems) in [Riemannian geometry](/source/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 [Thurston geometries](/source/Geometrization_conjecture).

The latest research shows that even the most powerful future experiments (like the [SKA](/source/Square_Kilometre_Array)) 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.[45]

Final results of the *Planck* mission, released in 2018, show the cosmological curvature parameter, 1 − Ω = Ω*K* = −*Kc*2/*a*2*H*2, to be 0.0007±0.0019, consistent with a flat universe.[46] (i.e. positive curvature: *K* = +1, Ω*K* < 0, Ω > 1, negative curvature: *K* = −1, Ω*K* > 0, Ω < 1, zero curvature: *K* = 0, Ω*K* = 0, Ω = 1).

#### Universe with zero curvature

In a universe with zero curvature, the local geometry is [flat](/source/Geometrization_conjecture#Euclidean_geometry_E3). 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](/source/Torus) and [Klein bottle](/source/Klein_bottle). Moreover, in three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable.[47] These are the [Bieberbach manifolds](/source/Flat_manifold). The most familiar is the aforementioned [3-torus universe](/source/Three-torus_model_of_the_universe).

In the absence of dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion [asymptotically](/source/Asymptote) 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](/source/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 total energy](/source/Zero-energy_universe).[48]

#### Universe with positive curvature

A positively curved universe is described by [elliptic geometry](/source/Elliptic_geometry), and can be thought of as a three-dimensional [hypersphere](/source/Hypersphere), or some other [spherical 3-manifold](/source/Spherical_3-manifold) (such as the [Poincaré dodecahedral space](/source/Poincar%C3%A9_dodecahedral_space)), all of which are [quotients](/source/Quotient_space_(topology)) of the 3-sphere.

[Poincaré dodecahedral space](/source/Homology_sphere#Cosmology) is a positively curved space, colloquially described as "soccerball-shaped", as it is the [quotient](/source/Quotient_space_(topology)) of the 3-sphere by the [binary icosahedral group](/source/Binary_icosahedral_group), which is very close to [icosahedral symmetry](/source/Icosahedral_symmetry), the symmetry of a soccer ball. This was proposed by [Jean-Pierre Luminet](/source/Jean-Pierre_Luminet) and colleagues in 2003[40][49] and an optimal orientation on the sky for the model was estimated in 2008.[41]

#### 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-manifolds](/source/Hyperbolic_3-manifold), and their classification is not completely understood.[50] Those of finite volume can be understood via the [Mostow rigidity theorem](/source/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](/source/Pseudosphere), a canonical model of hyperbolic geometry. An example is the [Picard horn](/source/Picard_horn), a negatively curved space, colloquially described as "funnel-shaped".[51]

#### 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.[13] 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](/source/Closed_manifold) (i.e., compact without boundary) and [open manifold](/source/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](/source/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric) (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

- [de Sitter space](/source/De_Sitter_space) – Maximally symmetric Lorentzian manifold with a positive cosmological constant

- [Ekpyrotic universe](/source/Ekpyrotic_universe) – Cosmological model—A string-theory-related model depicting a [five-dimensional](/source/Five-dimensional), [membrane](/source/Brane)-shaped universe; an alternative to the [Hot Big Bang Model](/source/Big_Bang), whereby the universe is described to have originated when two membranes collided at the fifth dimension

- [Extra dimensions in string theory](/source/String_Theory#Extra_dimensions) – Theory of subatomic structurePages displaying short descriptions of redirect targets for 6 or 7 extra space-like dimensions all with a *compact* topology

- [History of the center of the Universe](/source/History_of_the_center_of_the_Universe) – Historical concept in cosmologyPages displaying short descriptions of redirect targets

- [Holographic principle](/source/Holographic_principle) – Principle in theoretical physics

- [List of cosmology paradoxes](/source/List_of_paradoxes#Cosmology) – List of statements that appear to contradict themselves

- [Spacetime topology](/source/Spacetime_topology) – Topological structure of 4D spacetime

- [Theorema Egregium](/source/Theorema_Egregium) – Result of differential geometry proved by Gauss—The "remarkable theorem" discovered by [Gauss](/source/Carl_Friedrich_Gauss), which showed there is an intrinsic notion of curvature for surfaces. This is used by [Riemann](/source/Bernhard_Riemann) to generalize the (intrinsic) notion of curvature to higher-dimensional spaces

- [Three-torus model of the universe](/source/Three-torus_model_of_the_universe) – Cartesian product of 3 circlesPages displaying short descriptions of redirect targets

- [Zero-energy universe](/source/Zero-energy_universe) – Hypothesis that the total amount of energy in the universe is exactly zero

## Notes

1. **[^](#cite_ref-20)** ἐν τρισὶν ἐτίθετ᾽ εἶναι σφαίραις, ὧν τὴν μὲν πρώτην τὴν τῶν ἀπλανῶν ἄστρων εἶναι [18][19]

1. **[^](#cite_ref-28)** Aristotle knew of the thoughts of Leucippus (and [Democritus](/source/Democritus)) and considered the possibility of an infinite universe. [26]

## References

1. ^ [***a***](#cite_ref-FOOTNOTELuminet2016_1-0) [***b***](#cite_ref-FOOTNOTELuminet2016_1-1) [***c***](#cite_ref-FOOTNOTELuminet2016_1-2) [***d***](#cite_ref-FOOTNOTELuminet2016_1-3) [***e***](#cite_ref-FOOTNOTELuminet2016_1-4) [***f***](#cite_ref-FOOTNOTELuminet2016_1-5) [***g***](#cite_ref-FOOTNOTELuminet2016_1-6) [***h***](#cite_ref-FOOTNOTELuminet2016_1-7) [***i***](#cite_ref-FOOTNOTELuminet2016_1-8) [***j***](#cite_ref-FOOTNOTELuminet2016_1-9) [Luminet 2016](#CITEREFLuminet2016).

1. **[^](#cite_ref-NASA_Shape_2-0)** ["Will the Universe expand forever?"](https://map.gsfc.nasa.gov/universe/uni_shape.html). [NASA](/source/NASA). 24 January 2014. Retrieved 16 March 2015.

1. **[^](#cite_ref-Fermi_Flat_3-0)** Biron, Lauren (7 April 2015). ["Our universe is Flat"](http://www.symmetrymagazine.org/article/april-2015/our-flat-universe?email_issue=725). *Symmetry Magazine*. [FermiLab](/source/FermiLab)/[SLAC](/source/SLAC_National_Accelerator_Laboratory).

1. **[^](#cite_ref-Akrami2024_Promise_future_4-0)** Yashar Akrami; Stefano Anselmi; Craig J. Copi; Johannes R. Eskilt; Andrew H. Jaffe (April 2024). ["Promise of Future Searches for Cosmic Topology"](https://doi.org/10.1103%2FPHYSREVLETT.132.171501). *[Physical Review Letters](/source/Physical_Review_Letters)*. **132** 171501. [arXiv](/source/ArXiv_(identifier)):[2210.11426](https://arxiv.org/abs/2210.11426). [doi](/source/Doi_(identifier)):[10.1103/PHYSREVLETT.132.171501](https://doi.org/10.1103%2FPHYSREVLETT.132.171501). [ISSN](/source/ISSN_(identifier)) [0031-9007](https://search.worldcat.org/issn/0031-9007). [PMID](/source/PMID_(identifier)) [38728711](https://pubmed.ncbi.nlm.nih.gov/38728711). [Wikidata](/source/WDQ_(identifier)) [Q136902920](https://www.wikidata.org/wiki/Q136902920). While unambiguous indicators of topology have yet to be detected, ... Much more can be done to discover, or constrain, the topology of space.

1. **[^](#cite_ref-5)** Crane, Leah (29 June 2024). de Lange, Catherine (ed.). ["How big is the universe, really?"](https://www.newscientist.com/article/mg26234970-500-how-big-is-the-universe-the-shape-of-space-time-could-tell-us/). *New Scientist*. p. 31.

1. **[^](#cite_ref-NASA_WMAP_Overview_6-0)** ["WMAP Overview"](https://science.nasa.gov/mission/wmap/wmap-overview/). *NASA Science*. NASA. Retrieved 25 May 2026.

1. **[^](#cite_ref-PrimackStructure_7-0)** Primack, Joel R. ["Formation of Structure in the Universe"](https://ned.ipac.caltech.edu/level5/Primack/Primack1_2.html). *NASA/IPAC Extragalactic Database*. Caltech. Retrieved 25 May 2026.

1. **[^](#cite_ref-Akrami2024_8-0)** Akrami, Yashar; Anselmi, Stefano; Copi, Craig J.; et al. (2024). ["Promise of Future Searches for Cosmic Topology"](https://doi.org/10.1103%2FPhysRevLett.132.171501). *Physical Review Letters*. **132** (17) 171501. [arXiv](/source/ArXiv_(identifier)):[2210.11426](https://arxiv.org/abs/2210.11426). [doi](/source/Doi_(identifier)):[10.1103/PhysRevLett.132.171501](https://doi.org/10.1103%2FPhysRevLett.132.171501).

1. **[^](#cite_ref-GuardianTopology2024_9-0)** Ball, Philip (8 June 2024). ["'We're trying to find the shape of space': scientists wonder if the universe is like a doughnut"](https://www.theguardian.com/science/article/2024/jun/08/shape-universe-topology-doughnut-flat-curvature-dimensions). *The Guardian*. Retrieved 25 May 2026.

1. **[^](#cite_ref-10)** ["Density Parameter, Omega"](http://hyperphysics.phy-astr.gsu.edu/hbase/astro/denpar.html). *hyperphysics.phy-astr.gsu.edu*. Retrieved 2015-06-01.

1. **[^](#cite_ref-11)** Ade, P. A. R.; [Aghanim, N.](/source/Nabila_Aghanim); Armitage-Caplan, C.; et al. ([Planck Collaboration](/source/Planck_Collaboration)) (November 2014). "Planck 2013 results. XVI. Cosmological parameters". *[Astronomy & Astrophysics](/source/Astronomy_%26_Astrophysics)*. **571**: A16. [arXiv](/source/ArXiv_(identifier)):[1303.5076](https://arxiv.org/abs/1303.5076). [Bibcode](/source/Bibcode_(identifier)):[2014A&A...571A..16P](https://ui.adsabs.harvard.edu/abs/2014A&A...571A..16P). [doi](/source/Doi_(identifier)):[10.1051/0004-6361/201321591](https://doi.org/10.1051%2F0004-6361%2F201321591). [ISSN](/source/ISSN_(identifier)) [0004-6361](https://search.worldcat.org/issn/0004-6361). [S2CID](/source/S2CID_(identifier)) [118349591](https://api.semanticscholar.org/CorpusID:118349591).

1. **[^](#cite_ref-12)** de Bernardis, P.; Ade, P. A. R.; Bock, J. J.; et al. (April 2000). "A flat Universe from high-resolution maps of the cosmic microwave background radiation". *[Nature](/source/Nature_(journal))*. **404** (6781): 955–959. [arXiv](/source/ArXiv_(identifier)):[astro-ph/0004404](https://arxiv.org/abs/astro-ph/0004404). [Bibcode](/source/Bibcode_(identifier)):[2000Natur.404..955D](https://ui.adsabs.harvard.edu/abs/2000Natur.404..955D). [doi](/source/Doi_(identifier)):[10.1038/35010035](https://doi.org/10.1038%2F35010035). [ISSN](/source/ISSN_(identifier)) [0028-0836](https://search.worldcat.org/issn/0028-0836). [PMID](/source/PMID_(identifier)) [10801117](https://pubmed.ncbi.nlm.nih.gov/10801117). [S2CID](/source/S2CID_(identifier)) [4412370](https://api.semanticscholar.org/CorpusID:4412370).

1. ^ [***a***](#cite_ref-Dodelson2003_13-0) [***b***](#cite_ref-Dodelson2003_13-1) Dodelson, Scott (2003). *Modern Cosmology*. Academic Press. [ISBN](/source/ISBN_(identifier)) [9780122191411](https://en.wikipedia.org/wiki/Special:BookSources/9780122191411).

1. **[^](#cite_ref-14)** Davies, Paul (1977). [*Space and Time in the Modern Universe*](https://books.google.com/books?id=SZI5AAAAIAAJ). Cambridge: [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [978-0-521-29151-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-29151-4).

1. **[^](#cite_ref-15)** J. J. Callahan. ["The Curvature of Space in a Finite Universe"](https://www.jstor.org/stable/24950420). *[Scientific American](/source/Scientific_American)*. Vol. 235, no. 2 (August 1976). [Nature America, Inc.](/source/Nature_America%2C_Inc.): [ITHAKA](/source/ITHAKA). [JSTOR](/source/JSTOR_(identifier)) [24950420](https://www.jstor.org/stable/24950420).

1. ^ [***a***](#cite_ref-DL_16-0) [***b***](#cite_ref-DL_16-1) Diogenes Laërtius. ["BOOK IX.: III"](https://www.gutenberg.org/files/57342/57342-h/57342-h.htm#Page_388). [*The Lives and Opinions of Eminent Philosophers*](https://www.gutenberg.org/files/57342/57342-h/57342-h.htm). Translated by [C. D. Yonge Laërtius](/source/Charles_Duke_Yonge). [Queen’s College](/source/Queen's_University_Belfast#History), Belfast: [G. Bell & Sons Ltd](/source/George_Bell_%26_Sons): [gutenberg.org](/source/Gutenberg.org). p. 388. 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.

1. **[^](#cite_ref-SB2022_17-0)** Sylvia Berryman (October 18, 2022). ["Ancient Atomism: 2. Ancient Greek Atomism: 2.1 Leucippus and Democritus"](https://plato.stanford.edu/entries/atomism-ancient/#LeucDemo). In [Edward N. Zalta](/source/Edward_N._Zalta) (ed.). [*Stanford Encyclopedia of Philosophy*](https://plato.stanford.edu/). Metaphysics Research Lab, [CSLI](/source/Stanford_University_centers_and_institutes#Center_for_the_Study_of_Language_and_Information). 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*{{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: CS1 maint: location missing publisher ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_location_missing_publisher))

1. ^ [***a***](#cite_ref-MR_18-0) [***b***](#cite_ref-MR_18-1) Molly Read. ["A Brief History"](https://cmb.physics.wisc.edu/pub/tutorial/briefhist.html). [University of Wisconsin, Madison](/source/University_of_Wisconsin%2C_Madison).

1. **[^](#cite_ref-19)** Ἀριστοτέλους. ["Λ.1073β"](https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0051%3Abook%3D12%3Asection%3D1073b). *Μετὰ τὰ Φυσικά*. [Clarendon Press](/source/Clarendon_Press). 1924: [perseus.tufts.edu](/source/Perseus.tufts.edu). 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.gr](/source/NTUA)/mourmouras/greats/aristoteles/meta_ta_physica.pdf

1. **[^](#cite_ref-21)** 1. ["Metaphysics 12.1073b"](https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0052:book=12:section=1073b&highlight=eudoxus). *Aristotle in 23 Volumes, Vols.17, 18*. Translated by Hugh Tredennick. [Cambridge, MA](/source/Cambridge%2C_MA): [Harvard University Press](/source/Harvard_University_Press) 1933: perseus.tufts.edu. 1. Todd Timberlake (12 May 2011). ["Computer Program Detail Page: Spheres of Eudoxus"](https://www.compadre.org/osp/items/detail.cfm?ID=11198). [American Association of Physics Teachers](/source/American_Association_of_Physics_Teachers) & [National Science Foundation](/source/National_Science_Foundation)-[National Science Digital Library](/source/National_Science_Digital_Library) ([ISKME](/source/ISKME)). 1. Matthias Tomczak. ["Lecture 8"](http://gyre.umeoce.maine.edu/physicalocean/Tomczak/science+society/lectures/illustrations/lecture8/eudoxus.html). [Flinders University](/source/Flinders_University): [University of Maine](/source/University_of_Maine) Ocean Observing System.

1. **[^](#cite_ref-CHP_22-0)** Center for History of Physics. ["The Greek Worldview Continuation of the Greek tradition"](https://history.aip.org/exhibits/cosmology/ideas/greekworldview.htm). [The American Institute of Physics](/source/The_American_Institute_of_Physics).

1. **[^](#cite_ref-JEG2012_23-0)** Jan Edward Garrett (November 7, 2012). ["Introduction to Aristotle's Celestial and Terrestrial Physics"](https://people.wku.edu/jan.garrett/341/aristotle_big_picture.htm). [Western Kentucky University](/source/Western_Kentucky_University). Sometimes this sphere is simply called the kosmos, i.e., universe or world. There is no "place" and nothing material beyond this sphere.

1. **[^](#cite_ref-DJF78_24-0)** [David J. Furley](/source/David_J._Furley) (August 1978). ["The Greek Theory of the Infinite Universe"](https://www.jstor.org/stable/2709119). *[Journal of the History of Ideas](/source/Journal_of_the_History_of_Ideas)*. **42** (4 (Oct. - Dec., 1981)). [Cambridge](/source/Cambridge): [University of Pennsylvania Press](/source/University_of_Pennsylvania_Press): [ITHAKA](/source/ITHAKA): 571–585. [doi](/source/Doi_(identifier)):[10.2307/2709119](https://doi.org/10.2307%2F2709119). [JSTOR](/source/JSTOR_(identifier)) [2709119](https://www.jstor.org/stable/2709119).

1. **[^](#cite_ref-GE81_25-0)** 1. Grant E (1981). ["5 - The historical roots of the medieval concept of an infinite, extracosmic void space"](https://www.cambridge.org/core/books/abs/much-ado-about-nothing/historical-roots-of-the-medieval-concept-of-an-infinite-extracosmic-void-space/92E60C6D078DC971DDD2F5514A2614A8). *Much Ado about Nothing Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution*. [Cambridge University Press](/source/Cambridge_University_Press). pp. 105–115. [doi](/source/Doi_(identifier)):[10.1017/CBO9780511895326.008](https://doi.org/10.1017%2FCBO9780511895326.008). [ISBN](/source/ISBN_(identifier)) [978-0-521-22983-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-22983-8). 1. Grant, Edward (29 May 1981). [*Much Ado about Nothing: Theories of Space and Vacuum from the Middle Ages to the Scientific Revolution*](https://books.google.com/books?id=SidBQyFmgpsC). [ISBN](/source/ISBN_(identifier)) [0521229839](https://en.wikipedia.org/wiki/Special:BookSources/0521229839).

1. **[^](#cite_ref-26)** Aristotle. "BOOK I. 9". [*DE CAELO*](https://classicalliberalarts.com/wp-content/uploads/ARISTOTLE_DE_CAELO.pdf) (PDF). Translated by [J. L. Stocks](/source/John_Leofric_Stocks); H. B. Wallis. [St John's College, Oxford University](/source/St_John's_College%2C_Oxford_University): [Humphrey Milford](/source/Humphrey_Milford) 1922. p. 279, lines 13–16, footnote. 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.

1. **[^](#cite_ref-27)** 1. Ἀριστοτέλους. ["A.985b"](https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0051:book=1:section=985b). *Μετὰ τὰ Φυσικά*. [Clarendon Press](/source/Clarendon_Press). 1924: [perseus.tufts.edu](/source/Perseus.tufts.edu). Greek original: via iask.ai/q/Aristotle-Metaphysics-Book-I-Greek-original-39nedig: Perseus Catalog (not available): www.physics.[ntua.gr](/source/NTUA)/mourmouras/greats/aristoteles/meta_ta_physica.pdf 1. ["Metaphysics 1.985b"](https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0052%3Abook%3D1%3Asection%3D985b). *Aristotle in 23 Volumes, Vols.17, 18*. Translated by Hugh Tredennick. [Cambridge, MA](/source/Cambridge%2C_MA): [Harvard University Press](/source/Harvard_University_Press) 1933: perseus.tufts.edu. 1. [Helge Kragh](/source/Helge_Kragh) (2010). ["Ancient Greek-Roman Cosmology: Infinite, Eternal, Finite, Cyclic, and Multiple Universes"](https://thejournalofcosmology.com/AncientAstronomy108.html). *[Journal of Cosmology](/source/Journal_of_Cosmology)*. **9**. [University of Aarhus](/source/University_of_Aarhus). 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. 1. [Jacques A. Bailly](/source/Jacques_Bailly). ["Aristotle on the Infinite, Space, and Time: Mathematical: 1) Multitude"](https://www.uvm.edu/~jbailly/courses/Aristotle/infinite%20in%20AristotleFromACTA.html). [uvm.edu](/source/Uvm.edu).

1. **[^](#cite_ref-29)** 1. CHRISTOPHER GRANEY (October 31, 2022). ["Augustine, Aquinas, and Calvin on the Size of the Moon, Scripture, and "Following the Science""](https://catholicscientists.org/articles/augustine-aquinas-and-calvin-on-the-size-of-the-moon-scripture-and-following-the-science/). [Specola Vaticana](/source/Specola_Vaticana): [The Society of Catholic Scientists](/source/The_Society_of_Catholic_Scientists). 1. David Juste (10 May 2025). ["'Ptolemy, Planetary Hypotheses (Greek)'"](http://ptolemaeus.badw.de/work/141.). *Ptolemaeus Arabus et Latinus. Works*. [Bayerische Akademie der Wissenschaften](/source/Bayerische_Akademie_der_Wissenschaften) – via Center for History of Physics: history.aip.org/exhibits/cosmology. 1. Mohan Matthen; R. J. Hankinson (1993). ["Aristotle's Universe: Its Form and Matter"](https://www.jstor.org/stable/20117821). *[Synthese](/source/Synthese)*. **96** (3). [Kluwer Academic Publishing](/source/Kluwer_Academic_Publishing): [Springer Nature](/source/Springer_Nature): [ITHAKA](/source/ITHAKA): 417–435. [doi](/source/Doi_(identifier)):[10.1007/BF01064010](https://doi.org/10.1007%2FBF01064010). [JSTOR](/source/JSTOR_(identifier)) [20117821](https://www.jstor.org/stable/20117821).

1. **[^](#cite_ref-GIN63_30-0)** G. I. Naan (1963). ["On the Infinity of the Universe"](https://www.jstor.org/stable/40400937). *[Science & Society](/source/Science_%26_Society)*. **27** (2 (Spring, 1963)). [Sage Publications, Inc](/source/Sage_Publications%2C_Inc): [ITHAKA](/source/ITHAKA): 176–202. [doi](/source/Doi_(identifier)):[10.1177/003682376302700203](https://doi.org/10.1177%2F003682376302700203). [JSTOR](/source/JSTOR_(identifier)) [40400937](https://www.jstor.org/stable/40400937).

1. **[^](#cite_ref-AJ6141-8_154_31-0)** Alexander Jones (2015). "Greek Cosmology and Cosmogony". In Ruggles, C. (ed.). *Handbook of Archaeoastronomy and Ethnoastronomy*. [Springer](/source/Springer_Nature), New York, NY. pp. 1549–1553. [doi](/source/Doi_(identifier)):[10.1007/978-1-4614-6141-8_154](https://doi.org/10.1007%2F978-1-4614-6141-8_154). [ISBN](/source/ISBN_(identifier)) [978-1-4614-6141-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4614-6141-8).

1. **[^](#cite_ref-LEEDS_32-0)** Leeds Centre for Dante Studies & the Devers Program in Dante Studies at the [University of Notre Dame](/source/University_of_Notre_Dame). ["Paradiso"](https://ahc.leeds.ac.uk/discover-dante/doc/paradiso). [University of Leeds](/source/University_of_Leeds). Ptolemaic understanding of the universe (after Ptolemy, an Alexandrian polymath of the second century A.D.). This was broadly shared by all mediaeval thinkers

1. **[^](#cite_ref-33)** [Beinecke Rare Book & Manuscript Library](/source/Beinecke_Rare_Book_%26_Manuscript_Library), [Yale University Library](/source/Yale_University_Library) (23 March 2021). ["Divina Commedia, MS 428"](https://beinecke.library.yale.edu/collections/highlights/divina-commedia-ms-428-between-1385-and-1400). [yale.edu](/source/Yale.edu).

1. ^ [***a***](#cite_ref-DFJOER_34-0) [***b***](#cite_ref-DFJOER_34-1) [***c***](#cite_ref-DFJOER_34-2) DJF; John J O'Connor; Edmund F Robertson. ["MacTutor: The Infinite Universe"](https://web.archive.org/web/20251004184845/https://mathshistory.st-andrews.ac.uk/Astronomy/universe/). [st-andrews.ac.uk](/source/St-andrews.ac.uk). Archived from [the original](https://mathshistory.st-andrews.ac.uk/Astronomy/universe/) on 4 October 2025. Retrieved 16 December 2025. a

1. **[^](#cite_ref-35)** Sun Kwok (22 October 2021). ["Is the Universe Finite?: Abstract"](https://link.springer.com/chapter/10.1007/978-3-030-80260-8_16). *Our Place in the Universe - II The Scientific Approach to Discovery* (1 ed.). [University of British Columbia](/source/University_of_British_Columbia): [Springer Nature Switzerland AG](/source/Springer_Nature). [Bibcode](/source/Bibcode_(identifier)):[2021opus.book.....K](https://ui.adsabs.harvard.edu/abs/2021opus.book.....K). [doi](/source/Doi_(identifier)):[10.1007/978-3-030-80260-8](https://doi.org/10.1007%2F978-3-030-80260-8). [ISBN](/source/ISBN_(identifier)) [978-3-030-80260-8](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-80260-8). 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

1. **[^](#cite_ref-JDB2005_36-0)** [John D. Barrow](/source/John_D._Barrow) (2005). "chapter seven Is the Universe Infinite?". [*The Infinite Book: A Short Guide to the Boundless, Timeless and Endless*](https://books.google.com/books?id=KSNJJU6cRyMC) (reprint ed.). Jonathon Cape, [Vintage Books](/source/Vintage_Books), [Random House](/source/Random_House). pp. 116–117. [ISBN](/source/ISBN_(identifier)) [0099443724](https://en.wikipedia.org/wiki/Special:BookSources/0099443724) – via plus.maths.org/content/do-infinities-exist-nature-0 University of Cambridge.

1. **[^](#cite_ref-COR2017_37-0)** [Cormac O'Raifeartaigh](/source/Cormac_O'Raifeartaigh) (February 3, 2017). ["Albert Einstein and the origins of modern cosmology"](https://physicstoday.aip.org/news/albert-einstein-and-the-origins-of-modern-cosmology). *[Physics Today](/source/Physics_Today)* (2) 12150. [AIP](/source/American_Center_for_Physics). [Bibcode](/source/Bibcode_(identifier)):[2017PhT..2017b2150O](https://ui.adsabs.harvard.edu/abs/2017PhT..2017b2150O). [doi](/source/Doi_(identifier)):[10.1063/PT.5.9085](https://doi.org/10.1063%2FPT.5.9085). "[Kosmologische](/source/Cosmological) Betrachtungen zur allgemeinen Relativitätstheorie" The outcome of those deliberations was Einstein's "[Cosmological considerations](/source/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.

1. **[^](#cite_ref-38)** 1. [David Hilbert](/source/David_Hilbert) (5 June 2012). ["On the infinite"](https://lawrencecpaulson.github.io/papers/on-the-infinite.pdf) (PDF). [Cambridge University Press](/source/Cambridge_University_Press): [lawrencecpaulson](/source/Lawrence_Paulson). p. 186. 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. 1. Paul Benacerraf; Hilary Putnam, eds. (1926). ["Über das Unendliche - On the infinite DAVID HILBERT (in: Philosophy of mathematics)"](https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Philosophy/Philosophy.html). *[Mathematische Annalen](/source/Mathematische_Annalen)*. **95**. Translated by Erna Putnam; Gerald J. Massey (2nd ed.). [Göttingen](/source/G%C3%B6ttingen): Berlin: (Cambridge London New York New Rochelle Melbourne Sydney): [Springer Verlag](/source/Springer_Verlag): ([Cambridge University Press](/source/Cambridge_University_Press): math.[dartmouth.edu](/source/Dartmouth.edu). German language title: jamesrmeyer.com/infinite/hilbert-uber-das-unendliche)

1. **[^](#cite_ref-JS252001_39-0)** [Joseph Silk](/source/Joseph_Silk) (2 May 2001). ["Is the Universe finite or infinite? An interview with Joseph Silk"](https://www.esa.int/Science_Exploration/Space_Science/Is_the_Universe_finite_or_infinite_An_interview_with_Joseph_Silk). [University of Oxford](/source/University_of_Oxford): [European Space Agency](/source/European_Space_Agency).

1. **[^](#cite_ref-Lee2011_40-0)** Lee, John M. (2011). *Introduction to Topological Manifolds* (2nd ed.). Springer. [ISBN](/source/ISBN_(identifier)) [9781441979391](https://en.wikipedia.org/wiki/Special:BookSources/9781441979391).

1. ^ [***a***](#cite_ref-Luminet1995_41-0) [***b***](#cite_ref-Luminet1995_41-1) [***c***](#cite_ref-Luminet1995_41-2) [Lachièze-Rey & Luminet 1995](#CITEREFLachièze-ReyLuminet1995)

1. ^ [***a***](#cite_ref-Nat03_42-0) [***b***](#cite_ref-Nat03_42-1) [Luminet, Jean-Pierre](/source/Jean-Pierre_Luminet); Weeks, Jeffrey R.; Riazuelo, Alain; Lehoucq, Roland; Uzan, Jean-Philippe (October 2003). "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background". *[Nature](/source/Nature_(journal))*. **425** (6958): 593–595. [arXiv](/source/ArXiv_(identifier)):[astro-ph/0310253](https://arxiv.org/abs/astro-ph/0310253). [Bibcode](/source/Bibcode_(identifier)):[2003Natur.425..593L](https://ui.adsabs.harvard.edu/abs/2003Natur.425..593L). [doi](/source/Doi_(identifier)):[10.1038/nature01944](https://doi.org/10.1038%2Fnature01944). [ISSN](/source/ISSN_(identifier)) [0028-0836](https://search.worldcat.org/issn/0028-0836). [PMID](/source/PMID_(identifier)) [14534579](https://pubmed.ncbi.nlm.nih.gov/14534579). [S2CID](/source/S2CID_(identifier)) [4380713](https://api.semanticscholar.org/CorpusID:4380713).

1. ^ [***a***](#cite_ref-RBSG08_43-0) [***b***](#cite_ref-RBSG08_43-1) Lew, B.; Roukema, B.; Szaniewska, Agnieszka; Gaudin, Nicolas E. (May 2008). "A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data". *[Astronomy & Astrophysics](/source/Astronomy_%26_Astrophysics)*. **482** (3): 747–753. [arXiv](/source/ArXiv_(identifier)):[0801.0006](https://arxiv.org/abs/0801.0006). [Bibcode](/source/Bibcode_(identifier)):[2008A&A...482..747L](https://ui.adsabs.harvard.edu/abs/2008A&A...482..747L). [doi](/source/Doi_(identifier)):[10.1051/0004-6361:20078777](https://doi.org/10.1051%2F0004-6361%3A20078777). [ISSN](/source/ISSN_(identifier)) [0004-6361](https://search.worldcat.org/issn/0004-6361). [S2CID](/source/S2CID_(identifier)) [1616362](https://api.semanticscholar.org/CorpusID:1616362).

1. **[^](#cite_ref-RBBSJ2007_44-0)** Boudewijn François Roukema; Bajtlik S.; Biesiada M.; Szaniewska A.; Jurkiewicz H. (March 2007). "A weak acceleration effect due to residual gravity in a multiply connected universe". *[Astronomy & Astrophysics](/source/Astronomy_%26_Astrophysics)*. **463** (3): 861–871. [arXiv](/source/ArXiv_(identifier)):[astro-ph/0602159](https://arxiv.org/abs/astro-ph/0602159). [Bibcode](/source/Bibcode_(identifier)):[2007A&A...463..861R](https://ui.adsabs.harvard.edu/abs/2007A&A...463..861R). [doi](/source/Doi_(identifier)):[10.1051/0004-6361:20064979](https://doi.org/10.1051%2F0004-6361%3A20064979). [ISSN](/source/ISSN_(identifier)) [0004-6361](https://search.worldcat.org/issn/0004-6361). [Zbl](/source/Zbl_(identifier)) [1118.85330](https://zbmath.org/?format=complete&q=an:1118.85330). [Wikidata](/source/WDQ_(identifier)) [Q68598777](https://www.wikidata.org/wiki/Q68598777).

1. **[^](#cite_ref-RR09_45-0)** Boudewijn François Roukema; Rozanski P. T. (2009). "The residual gravity acceleration effect in the Poincare dodecahedral space". *[Astronomy & Astrophysics](/source/Astronomy_%26_Astrophysics)*. **502**: 27–35. [arXiv](/source/ArXiv_(identifier)):[0902.3402](https://arxiv.org/abs/0902.3402). [Bibcode](/source/Bibcode_(identifier)):[2009A&A...502...27R](https://ui.adsabs.harvard.edu/abs/2009A&A...502...27R). [doi](/source/Doi_(identifier)):[10.1051/0004-6361/200911881](https://doi.org/10.1051%2F0004-6361%2F200911881). [ISSN](/source/ISSN_(identifier)) [0004-6361](https://search.worldcat.org/issn/0004-6361). [Zbl](/source/Zbl_(identifier)) [1177.85087](https://zbmath.org/?format=complete&q=an:1177.85087). [Wikidata](/source/WDQ_(identifier)) [Q68676519](https://www.wikidata.org/wiki/Q68676519).

1. **[^](#cite_ref-ORB12_46-0)** Jan J Ostrowski; Boudewijn F Roukema; Zbigniew P Buliński (30 July 2012). "A relativistic model of the topological acceleration effect". *[Classical and Quantum Gravity](/source/Classical_and_Quantum_Gravity)*. **29** (16) 165006. [arXiv](/source/ArXiv_(identifier)):[1109.1596](https://arxiv.org/abs/1109.1596). [doi](/source/Doi_(identifier)):[10.1088/0264-9381/29/16/165006](https://doi.org/10.1088%2F0264-9381%2F29%2F16%2F165006). [ISSN](/source/ISSN_(identifier)) [0264-9381](https://search.worldcat.org/issn/0264-9381). [Zbl](/source/Zbl_(identifier)) [1253.83052](https://zbmath.org/?format=complete&q=an:1253.83052). [Wikidata](/source/WDQ_(identifier)) [Q96692451](https://www.wikidata.org/wiki/Q96692451).

1. **[^](#cite_ref-47)** Vardanyan, Mihran; Trotta, Roberto; Silk, Joseph (21 July 2009). ["How flat can you get? A model comparison perspective on the curvature of the Universe"](https://doi.org/10.1111%2Fj.1365-2966.2009.14938.x). *[Monthly Notices of the Royal Astronomical Society](/source/Monthly_Notices_of_the_Royal_Astronomical_Society)*. **397** (1): 431–444. [arXiv](/source/ArXiv_(identifier)):[0901.3354](https://arxiv.org/abs/0901.3354). [Bibcode](/source/Bibcode_(identifier)):[2009MNRAS.397..431V](https://ui.adsabs.harvard.edu/abs/2009MNRAS.397..431V). [doi](/source/Doi_(identifier)):[10.1111/j.1365-2966.2009.14938.x](https://doi.org/10.1111%2Fj.1365-2966.2009.14938.x). [S2CID](/source/S2CID_(identifier)) [15995519](https://api.semanticscholar.org/CorpusID:15995519).

1. **[^](#cite_ref-48)** [Aghanim, N.](/source/Nabila_Aghanim); Akrami, Y.; Ashdown, M.; et al. ([Planck Collaboration](/source/Planck_Collaboration)) (September 2020). "Planck 2018 results: VI. Cosmological parameters". *[Astronomy & Astrophysics](/source/Astronomy_%26_Astrophysics)*. **641**: A6. [arXiv](/source/ArXiv_(identifier)):[1807.06209](https://arxiv.org/abs/1807.06209). [Bibcode](/source/Bibcode_(identifier)):[2020A&A...641A...6P](https://ui.adsabs.harvard.edu/abs/2020A&A...641A...6P). [doi](/source/Doi_(identifier)):[10.1051/0004-6361/201833910](https://doi.org/10.1051%2F0004-6361%2F201833910). [ISSN](/source/ISSN_(identifier)) [0004-6361](https://search.worldcat.org/issn/0004-6361). [S2CID](/source/S2CID_(identifier)) [119335614](https://api.semanticscholar.org/CorpusID:119335614).

1. **[^](#cite_ref-ConwayRossetti2003_49-0)** Conway, John Horton; Rossetti, Juan Pablo (2003). "Describing the platycosms". [arXiv](/source/ArXiv_(identifier)):[math/0311476](https://arxiv.org/abs/math/0311476).

1. **[^](#cite_ref-50)** [*A Universe From Nothing lecture by Lawrence Krauss at AAI*](https://www.youtube.com/watch?v=7ImvlS8PLIo). 2009. [Archived](https://ghostarchive.org/varchive/youtube/20211215/7ImvlS8PLIo) from the original on 2021-12-15. Retrieved 17 October 2011 – via [YouTube](/source/YouTube).

1. **[^](#cite_ref-physwebLum03_51-0)** Dumé, Isabelle (8 October 2003). ["Is the universe a dodecahedron?"](https://physicsworld.com/a/is-the-universe-a-dodecahedron/). *[Physics World](/source/Physics_World)*.

1. **[^](#cite_ref-Thurston1997_52-0)** Thurston, William P. (1997). *Three-Dimensional Geometry and Topology*. Vol. 1. Princeton University Press. [ISBN](/source/ISBN_(identifier)) [9780691083049](https://en.wikipedia.org/wiki/Special:BookSources/9780691083049).

1. **[^](#cite_ref-Aurich0403597_53-0)** Aurich, Ralf; Lustig, S.; Steiner, F.; Then, H. (2004). "Hyperbolic Universes with a Horned Topology and the CMB Anisotropy". *[Classical and Quantum Gravity](/source/Classical_and_Quantum_Gravity)*. **21** (21): 4901–4926. [arXiv](/source/ArXiv_(identifier)):[astro-ph/0403597](https://arxiv.org/abs/astro-ph/0403597). [Bibcode](/source/Bibcode_(identifier)):[2004CQGra..21.4901A](https://ui.adsabs.harvard.edu/abs/2004CQGra..21.4901A). [doi](/source/Doi_(identifier)):[10.1088/0264-9381/21/21/010](https://doi.org/10.1088%2F0264-9381%2F21%2F21%2F010). [S2CID](/source/S2CID_(identifier)) [17619026](https://api.semanticscholar.org/CorpusID:17619026).

## External links

- [Geometry of the Universe](http://icosmos.co.uk) at icosmos.co.uk

- Levin, Janna; Scannapieco, Evan & Silk, Joseph (September 1998). "The topology of the universe: the biggest manifold of them all". *[Classical and Quantum Gravity](/source/Classical_and_Quantum_Gravity)*. **15** (9): 2689–2697. [arXiv](/source/ArXiv_(identifier)):[gr-qc/9803026](https://arxiv.org/abs/gr-qc/9803026). [Bibcode](/source/Bibcode_(identifier)):[1998CQGra..15.2689L](https://ui.adsabs.harvard.edu/abs/1998CQGra..15.2689L). [doi](/source/Doi_(identifier)):[10.1088/0264-9381/15/9/015](https://doi.org/10.1088%2F0264-9381%2F15%2F9%2F015). [ISSN](/source/ISSN_(identifier)) [0264-9381](https://search.worldcat.org/issn/0264-9381). [S2CID](/source/S2CID_(identifier)) [119080782](https://api.semanticscholar.org/CorpusID:119080782).

- Lachièze-Rey, Marc; [Luminet, Jean-Pierre](/source/Jean-Pierre_Luminet) (March 1995). "Cosmic topology". *[Physics Reports](/source/Physics_Reports)*. **254** (3): 135–214. [arXiv](/source/ArXiv_(identifier)):[gr-qc/9605010](https://arxiv.org/abs/gr-qc/9605010). [Bibcode](/source/Bibcode_(identifier)):[1995PhR...254..135L](https://ui.adsabs.harvard.edu/abs/1995PhR...254..135L). [doi](/source/Doi_(identifier)):[10.1016/0370-1573(94)00085-H](https://doi.org/10.1016%2F0370-1573%2894%2900085-H). [S2CID](/source/S2CID_(identifier)) [119500217](https://api.semanticscholar.org/CorpusID:119500217).

- Luminet, Jean-Pierre (15 January 2016). ["The Status of Cosmic Topology after Planck Data"](https://doi.org/10.3390%2Funiverse2010001). *[Universe](/source/Universe_(journal))*. **2** (1): 1. [arXiv](/source/ArXiv_(identifier)):[1601.03884](https://arxiv.org/abs/1601.03884). [Bibcode](/source/Bibcode_(identifier)):[2016Univ....2....1L](https://ui.adsabs.harvard.edu/abs/2016Univ....2....1L). [doi](/source/Doi_(identifier)):[10.3390/universe2010001](https://doi.org/10.3390%2Funiverse2010001). [ISSN](/source/ISSN_(identifier)) [2218-1997](https://search.worldcat.org/issn/2218-1997). [S2CID](/source/S2CID_(identifier)) [7331164](https://api.semanticscholar.org/CorpusID:7331164).

- Markey, Sean (8 October 2003). ["Universe is Finite, 'Soccer Ball'-Shaped, Study Hints"](https://web.archive.org/web/20031010171053/http://news.nationalgeographic.com/news/2003/10/1008_031008_finiteuniverse.html). *[National Geographic News](/source/National_Geographic_News)*. Archived from [the original](http://news.nationalgeographic.com/news/2003/10/1008_031008_finiteuniverse.html) on 10 October 2003. Possible wrap-around dodecahedral shape of the universe

- Classification of [possible universes](http://star-www.st-and.ac.uk/~kdh1/cos/cos.html) in the [Lambda-CDM](/source/Lambda-CDM) model.

- Fagundes, Helio V. (December 2002). "Exploring the global topology of the universe". *[Brazilian Journal of Physics](/source/Brazilian_Journal_of_Physics)*. **32** (4): 891–894. [arXiv](/source/ArXiv_(identifier)):[gr-qc/0112078](https://arxiv.org/abs/gr-qc/0112078). [Bibcode](/source/Bibcode_(identifier)):[2002BrJPh..32..891F](https://ui.adsabs.harvard.edu/abs/2002BrJPh..32..891F). [doi](/source/Doi_(identifier)):[10.1590/S0103-97332002000500012](https://doi.org/10.1590%2FS0103-97332002000500012). [ISSN](/source/ISSN_(identifier)) [0103-9733](https://search.worldcat.org/issn/0103-9733). [S2CID](/source/S2CID_(identifier)) [119495347](https://api.semanticscholar.org/CorpusID:119495347).

- Grime, James. ["π39 (Pi and the size of the Universe)"](https://web.archive.org/web/20150430002504/http://www.numberphile.com/videos/pi_universe.html). *Numberphile*. [Brady Haran](/source/Brady_Haran). Archived from [the original](http://www.numberphile.com/videos/pi_universe.html) on 2015-04-30. Retrieved 2013-04-07.

- [What do you mean the universe is flat?](http://blogs.scientificamerican.com/degrees-of-freedom/2011/07/25/what-do-you-mean-the-universe-is-flat-part-i) Scientific American Blog explanation of a flat universe and the curved spacetime in the universe.

[Portals](https://en.wikipedia.org/wiki/Wikipedia:Contents/Portals):
- [Astronomy](https://en.wikipedia.org/wiki/Portal:Astronomy)
- [Stars](https://en.wikipedia.org/wiki/Portal:Stars)
- [Outer space](https://en.wikipedia.org/wiki/Portal:Outer_space)
- [Physics](https://en.wikipedia.org/wiki/Portal:Physics)

---
Adapted from the Wikipedia article [Shape of the universe](https://en.wikipedia.org/wiki/Shape_of_the_universe) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Shape_of_the_universe?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.