Imagine a map of every galaxy whose light we can observe. Would it map all of space? No: the limit of our observations does not reveal what lies beyond it or determine the Universe’s total volume.
We do not yet know whether all of space has a finite volume. Observations strongly constrain some geometric and topological models, but do not prove infinity. Nearly zero spatial curvature does not prove it either. The question concerns space on a particular slice of cosmic time, rather than an eternity in time. Copi and colleagues: a modern review of cosmic topology, section I.
The observable region and all of space are different questions
Light travels at a finite speed, and the Universe has a history of expansion. Our data therefore cover a limited region of the cosmic past. Its boundary limits the information we can obtain; it is not a wall. Describing all of space requires a model that extends beyond direct observations.
Three statements are easy to confuse: “we cannot see farther,” “there is no space farther out,” and “space continues infinitely.” The first alone establishes neither of the others. Cosmology therefore asks separately about geometry — how distances and curvature work — and topology, the global way in which parts of space connect. Planck: geometry and global properties, introduction.
Flat space can have a finite volume
Start with a two-dimensional example. On an infinite plane, a grid can continue without end. Now take a square and impose a different rule: identify its right edge with its left, and its top with its bottom. Crossing one boundary of the drawn cell brings us to the corresponding position on the opposite side. These are the same points of space, rather than a jump across an external void.
This space is called a flat two-dimensional torus. It has a finite area and no physical edge. Within a small patch, distances and angles behave just as they do on the plane. The global difference appears in loops: in certain directions, a locally straight journey can close on itself.
For a three-dimensional example, identify opposite faces of a parallelepiped in the same way. This is a mathematical possibility, rather than an established shape of our Universe. Curvature describes local geometry, whereas the identifications change global connectivity. Planck 2015: toroidal models, §3.1; COMPACT review: definitions and the periodic domain, section II.
The diagram compares two flat teaching geometries. Matching colours on opposite sides indicate an identification rule, rather than a material frame around the Universe.
The same local geometry, different global rules
Within a small patch, both models have the same flat rules for distances and angles.
An infinite plane
The drawing ends at the frame. The grid continues beyond it; the area of the whole space is infinite.
A flat two-dimensional torus
Right edge ↔ left edge; top ↔ bottom. Paired sides represent the same points, not physical walls.
The area is finite: one square represents the whole space. Arrows show two paths continuing across identified sides.
An ordinary doughnut in three-dimensional space does not have the precise geometry of this flat example: its surface has varying intrinsic curvature. The square with identified sides provides the required model without that misleading impression. Similarly, a finite cosmos does not require a physical space “outside” in which it must sit.
How global connectivity could leave a trace
In some compact models, light from one object can reach us along different paths. We would see several images of it in different directions. But a galaxy changes over time: different travel times and viewing angles make identification difficult. Two similar galaxies are not sufficient evidence of a closed cosmos. COMPACT: searches for repeated images, section I.
Another tool is the cosmic microwave background, ancient light that we observe at microwave wavelengths. For certain topologies, the same region of the last-scattering surface could appear on the sky along a pair of circles. Researchers seek matching temperature or polarization patterns along those circles, accounting for noise, foreground emission from our Galaxy and the physics of the signal. They are not looking for a drawn, bright circular line. Planck: the matched-circles method, §4.2.
A closed spatial loop also does not promise that a spacecraft could complete a circuit of the Universe. Such a journey depends on size, expansion and available time; the geometric property alone cannot settle that.
A negative search places limits on models
The Planck 2015 study, published in 2016, found no statistically significant matched circle pairs in the maps tested. This constrained particular compact models and scales under the conditions of the analysis. It does not mean that every topology, size, orientation and possible observer position has been examined. Planck: results, §5.1.1.
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Reading a lower bound on size
A particular Planck polarization analysis of a cubic three-dimensional torus, using the fiducial flat ΛCDM model, obtained an approximate 99% confidence lower bound:
(L)/(2)≳0.97χ_(rec).
L is the side length of the cubic fundamental domain, and χ_(rec) is the comoving distance to the last-scattering surface. Both quantities use the same comoving distance convention. For a cube, L/2 is the radius of its largest inscribed sphere.
Multiplying both sides by two gives:
L≳1.94χ_(rec).
This is a lower bound, rather than a statement that L = ∞. The finite value L = 2χ_(rec) satisfies the inequality; that illustrates its meaning without establishing that this model describes our cosmos.
The result depends on the type of torus, the circle search, the sky mask and whether the required orientation can be detected. Its confidence level is not the probability that the Universe is infinite. This is a specific historical result, rather than the latest universal limit for every possible space. Primary Planck paper: conditions and result, §5.1.1.
The 2026 COMPACT review stresses that the absence of circles does not exhaust the search. Global topology can alter statistical correlations in the microwave background even when a simple circle signal is absent. Some parameters and observer positions remain to be tested; whether topology can be determined in general remains an open question. COMPACT: search limits and prospects, sections III–IV.
A finite space without an edge is mathematically possible, while observations have not proved infinity. We can rule out specific models and refine the allowed scales. That is useful knowledge even while the overall answer remains open. Revisit local curvature in the article about space and curvature.