89 billion light-years: fly straight out and you might come back from behind

Imagine flying a spaceship in a perfectly straight line, forever.

Most people picture one of two endings. Either space goes on and on with no end, or you eventually hit some kind of edge, a wall at the end of everything.

There's a third possibility, and it's stranger than both. You keep flying straight, never turning, and one day you find yourself back where you started, arriving from the opposite direction. No wall, no edge, no centre. Just a universe that closes up on itself.

If our universe is built like that, then light does the same trip. Light from our own galaxy could go all the way around and come back, and somewhere out there we'd see copies of ourselves.

Astronomers have been looking for them. This is what they found. 🕹️

Chapter 1

Asteroids in 3D

Remember the arcade game Asteroids, from 1979? Fly your ship off the right side of the screen and it pops back in on the left. Fly off the top and it reappears at the bottom. The screen has no edges. You can fly forever, and yet the screen is finite.

Mathematicians call this shape a torus: a flat sheet with its opposite edges glued together. Now do the same with a box in three dimensions. Fly out the front wall and you come back in through the back. Fly out the ceiling and you come up through the floor. That's a 3-torus, and it's a perfectly good shape for a universe. The walls aren't real places: nothing marks them, and you could draw the box around any spot you like. Only the way things connect is real.

Here's the trick to seeing what it would look like from inside. Light travels in straight lines, so as far as light is concerned, the box is repeated forever in every direction, like wallpaper. Every copy holds the same stuff at the same moment. Look far enough in the right direction and you'd see yourself, from behind. Try it 👇

Fun fact: the computer simulations cosmologists use to model the universe already work exactly like this. Their simulated universes are boxes whose walls wrap around, so most of them are little 3-tori.

Chapter 2

Flat isn't the same as endless

You may have read that the universe is "flat", and that this means it goes on forever. The first part is true. The second part doesn't follow.

"Flat" is about geometry: whether triangles add up to 180°, whether parallel lines stay parallel. (On a ball they add up to more: Nº 16 walks a triangle with three right angles.) Measurements of the oldest light in the universe, combined with maps of millions of galaxies, show that space is flat to within a fraction of a percent.

But whether space is endless is a different question, about topology: how space connects up. The Asteroids screen is perfectly flat, and it still isn't endless. Gluing edges together doesn't bend anything; it only changes what connects to what.

Einstein's equations describe how space curves, point by point. They don't say how it connects up. So the only way to find out is to look.

Chapter 3

Ten ways to close a box

How many ways can you make a flat universe? A crystallographer and two mathematicians worked out the full list in the 1930s: 18. Eight of them go on forever in at least one direction. The other 10 are finite in every direction, and they're all made by gluing the faces of a box (or a hexagonal prism) in different ways:

Straight across: the 3-torus.
With a twist: glue one pair of faces after turning it by a half, a quarter, a third or a sixth of a turn, or use three half-turns at once, at right angles. Going across a twisted pair is like following a corkscrew.
With a flip: glue faces mirror-wise. Fly the right way round one of these universes and you come back as your own mirror image. Your left glove would come home as a right glove. 🧤

You can try the half-turn twist in the Asteroids figure above (press "Half-turn"), and a flat version of the flip (press "Flat screen", then "Klein bottle").

Crystals follow the same rules: the 10 finite flat universes are exactly the 10 crystal patterns in which every step moves every point, and two of them are among the commonest ways protein molecules stack up in crystals. 💎

Hold on to the corkscrew ones. They're the loophole in this whole story.

Chapter 4

A century of ghost hunting

The idea is old. In 1900, before Einstein's general relativity, the astronomer Karl Schwarzschild wondered whether space might close up on itself, and pointed out that light could then travel all the way round. We might even see a "counter-sun" in the sky, opposite the real one. There isn't one, so he supposed the light got absorbed on the way, and used measurements of stars to argue that space, if finite, must be very big.

Since then, astronomers have searched for repeats among galaxies, galaxy clusters and quasars (the blazing centres of distant galaxies, lit up by giant black holes). One 2003 study was called, wonderfully, "Ghosts of the Milky Way": it searched for images of our own galaxy from billions of years ago, when it may have been a quasar.

It's harder than it sounds. A copy of a galaxy far away is seen as it was long ago, when it looked different, and from a different side. You can't just look for two identical pictures. And nobody has found a convincing repeat.

The bigger problem is distance. Even the most distant galaxy ever confirmed is about 34 billion light-years away, as distances stand today. To test whether the universe repeats on bigger scales, you need something farther. Only one thing we can see lies farther: the oldest light in the universe.

Chapter 5

Where's my ghost?

That oldest light is the cosmic microwave background, or CMB: the afterglow of the Big Bang, released when the universe was about 380,000 years old. It comes to us from a giant sphere around us, about 45 billion light-years away.

Wait, 45 billion? The universe is only 13.8 billion years old. The answer is that space has been expanding the whole time (Nº 1 tells how we know, with a twist). The light has travelled for 13.8 billion years, but the places it came from have since been carried much farther away. Just past the CMB, at about 46 billion light-years, is the edge of what any light could have reached us from since the Big Bang: the edge of the observable universe.

So: if space wraps around, how far away is our nearest copy, and what would we see? Slide the size of the universe 👇

A small universe would show us our own galaxy as it was long ago, perhaps even before the Sun was born. A big one would hide our copy behind the glow of the CMB, or put it so far away that its light hasn't reached us yet.

And as long as the expansion of the universe keeps speeding up, there's a distance beyond which light can never reach us, no matter how long we wait: about 63 billion light-years. Keep that number in mind. (The same speeding-up limits how far anything we send today could ever get; Nº 0 shows why that matters at the end of the world.)

Chapter 6

Circles in the sky

In 1996 three physicists, Neil Cornish, David Spergel and Glenn Starkman, found a clever way to test the idea even when our copies are out of sight.

We see the CMB as a glowing sphere around us. In a wrapped universe, our copy sees its own glowing sphere around it. If our copy is close enough, the two spheres overlap. And two overlapping spheres meet in a circle.

Every spot on that circle is part of both spheres, so we see it twice, in two different directions. On our sky that shows up as a pair of circles with exactly the same pattern of hot and cold spots. It's like spotting the same stain on two different tiles of the wallpaper. 👇

The smaller the universe, the bigger the circles. A universe just a bit smaller than the CMB sphere is wide gives tiny circles; one much smaller gives giant ones, many pairs of them all over the sky. Watch the first pairs appear as the size slips below 1. The buttons under the map jump to real circle sizes: a famous prediction from 2003 and the searches that tested it, which the next two chapters tell.

Chapter 7

The soccer-ball universe

In 2003 the WMAP satellite made the first detailed map of the whole CMB sky, and it had a puzzle in it. On the very largest scales, the sky was strangely quiet: less pattern than expected across the widest angles. A small universe could do just that, because ripples bigger than the universe can't exist in it.

A team led by Jean-Pierre Luminet proposed a specific shape in the journal Nature: a universe built from a dodecahedron, a solid with 12 five-sided faces (the 12 black patches on a classic soccer ball sit in the same pattern), with opposite faces glued after a small twist. (It needs space to be very slightly curved, like the surface of a giant ball.) It made headlines around the world. It also made a prediction: six pairs of matched circles, each about 35° in radius.

Cornish, Spergel, Starkman and Eiichiro Komatsu had already looked. They found no matching circles bigger than 25°. Another group then claimed small matching circles; a careful re-analysis in 2007 found they weren't really there, and ruled out the "soccer ball universe", as it had become known. Later measurements showed space is also too flat for that shape at a size we could see. ⚽❌ The quiet sky itself is still there in Planck's sharper maps. Whether it's a clue or a fluke, nobody knows yet.

Chapter 8

89 billion light-years

In 2012 the same team searched WMAP's cleaner seven-year maps for matched circles of any kind: any size down to 10° in radius, anywhere on the sky, with the pattern running the same way round or mirror-reversed. They found nothing. Later searches by the Planck satellite found nothing either.

Circles smaller than 10° are hard to spot, because the glowing gas that sent out the CMB was moving, which makes the two copies of a circle look slightly different. So the searches can only say: if a loop around the universe passes through us, it's longer than about 98.5% of the CMB sphere's width, the length at which the circles would shrink to 10°. That's:

89 billion light-years

That's the number in the title. The shortest trip around the universe that passes through us, if one exists, is at least 89 billion light-years long.

Remember the number from Chapter 5? Light can never cover more than about 63 billion light-years, counted in today's distances. So no light from any copy of us has ever reached us, and none ever will. Since the Big Bang, light has made it only about halfway around even the shortest loop through us that's allowed. Our ghost is out there, if it exists, but permanently out of sight, as long as dark energy stays the same. 👻

📦 How big would the whole thing be?

If the universe were a cube just big enough to have escaped the searches, it would hold about 1.8 times the volume inside the CMB sphere. Squeeze it into the most efficient shape (a slanted box, packed like oranges on a market stall) and it could be as little as 1.3 times. There's no upper limit: space might still be infinite.

Chapter 9

The corkscrew loophole

Notice the careful wording: the 89-billion-light-year limit is about loops through us. In a 3-torus every place is the same as every other, so that settles it. In the corkscrew universes from Chapter 3, it doesn't.

In 2022 a team called COMPACT, which includes Starkman, one of the inventors of the circle method, worked out what this really allows. Going around a corkscrew universe twists you around an axis. If you happen to live on the axis, your copy is right there, a short loop away. If you live far from every axis, the twist swings your copy far off, and you see no circles at all, even though short loops exist elsewhere.

How short? From half to a sixth of the CMB sphere's width, or even less, depending on the shape: loops of maybe 45 billion light-years down to 15 or shorter, running along corkscrew axes tens of billions of light-years from us. Try it: drag yourself around, or tap where you want to stand 👇

So the search isn't over. COMPACT is now looking for subtler fingerprints across the whole sky: patterns that a finite universe would leave even when there are no circles to see. In principle, a perfect map of the CMB holds enough information to tell an endless universe from one up to about 12 to 30% wider than the CMB sphere, depending on its shape. That is a best case, not a promise of a detection. A perfect 3D map of all the matter in the observable universe could in principle reach about a quarter farther than a perfect CMB map. Galaxy surveys can only map the nearer part of it; going deeper would take radio maps of the hydrogen gas that filled space before the first galaxies. Nothing has turned up yet.

Chapter 10

The trip you'll never finish

One last twist. The CMB sphere isn't fixed: as time passes, light from slightly farther away reaches us, and the sphere grows by about one light-year every year. So could we just wait until it's big enough to test any universe?

No, not as long as the expansion keeps speeding up. Even if we wait forever, the CMB sphere never grows past about 62 billion light-years (a touch under the 63 from Chapter 5, because the CMB's light set off 380,000 years after the Big Bang), which means circles can never test a universe bigger than about 124 billion light-years around. 👇

There's another reason to suspect that any loop is far out of reach. The leading idea about the universe's first instant, called inflation, says space was stretched enormously. For the universe's shape to show up in our sky, inflation would have had to stop at just the right moment.

Let's write up the ticket. 🧾

🚀 Round-trip ticket

Distanceshortest loop through us≥ 89 billion ly

Light's progress so farsince the Big Bang52%

Light's progress, everwith the expansion speeding up~70% max

Copies of you in viewfrom any era0

Arrivalnever

no edge ✦ no centre ✦ maybe no end

Somewhere far beyond everything we can see, space might quietly close up on itself.

If it does, there is another Milky Way out there, and another you: not a twin, not a double, but you, seen the long way round.

The light that would show it to us set out long ago, and it is still on its way. It always will be.

The universe keeps stretching the road faster than light can travel it.

So we'll never meet ourselves coming back. But we can still work out how far away we are.

At least 89 billion light-years, through here. Maybe less somewhere else. Maybe forever.

Part IShapes of space

Chapter 1Asteroids in three dimensions

In the 1979 arcade game Asteroids, a ship that flies off the right edge of the screen reappears on the left, and one that leaves the top comes back at the bottom. The screen has no edge and no centre. Mathematically, it is a flat torus: a rectangle with opposite sides glued together. Everything on it obeys ordinary flat geometry. Triangles have angles adding up to 180°, and parallel lines stay parallel. Yet its area is finite.

Now do the same in three dimensions. Take a box and glue each face to the face opposite: fly out through the front wall and you come back in through the back, out through the ceiling and back in through the floor. This is the 3-torus. You can't build a model of it inside our three-dimensional space, any more than a flat Asteroids screen can be rolled into a smooth doughnut without stretching, but you don't need to. From the inside it looks like ordinary space, except that going far enough in some directions brings you home.

There are three equivalent ways to picture it, and cosmologists use all of them:¹

Glued box. One box whose opposite faces are identified.
Periodic box. One box with "re-entering" walls, like the Asteroids screen. Computer simulations of the cosmos already work this way: their boxes have periodic boundaries, which makes most simulated universes 3-tori.
Tiled space. Repeat the box forever in every direction, each copy holding the same contents at the same moment. This repeated picture is the covering space, and it is how light sees the universe. Crossing a wall never bends or turns a light ray; it just keeps going straight from tile to tile.

In the tiled picture you can see the consequence. Look along any of the box's repeat directions and you see copies of yourself: not doubles, but you, seen by light that has gone around the universe. Each copy looks younger, because its light set off long ago, and it is seen from behind or from the side. Cosmologists call these copies clones or ghosts.

Figure 1🕹️ Try it

An Asteroids universe

Drag to turn

Three dimensions: a cube whose opposite faces are identified, either straight across (the 3-torus, E1) or with the top and bottom faces glued after a half-turn (the half-turn space, E2). Faces of one colour are the same face. "Unrolled" shows the covering space: the cube repeated without end, each copy holding the same contents at the same moment (in E2 every other layer is turned half a turn). "Inside" follows the ship through the covering space. Two dimensions: the flat torus and the Klein bottle, whose left and right edges are glued with a flip, the analogue of the non-orientable spaces E7–E10. A flash expands at constant speed in the covering space; where it meets its own copies, the same light arrives from two directions, which in three dimensions happens on circles.This little universe has no edges: fly out through one wall and you come back in through the opposite one. Press "Unrolled" to see it the way light sees it, as endless identical copies, and "Inside" to ride along. Then send out a flash and watch it meet itself.

The box's shape has no physical meaning: many different tile shapes describe the same universe. What matters is the set of moves that carry one tile onto the next.² And the 3-torus need not be a cube. Its three repeat lengths can differ and its edges can meet at angles other than 90°.

Chapter 2Flat doesn't mean infinite

Two different properties of space are easy to confuse.

Geometry is local: how space curves, whether triangles add up to 180°, whether parallel lines stay parallel. (On a sphere a triangle's angles add up to more: Nº 16 walks one with three right angles.) General relativity ties it to the matter and energy in the universe, through equations that hold point by point.
Topology is global: how space is connected. Are there loops you cannot shrink to a point? Does going straight bring you back?

Einstein's equations are local, so they don't fix the topology.² Measuring the geometry doesn't settle it either. The cosmic microwave background (CMB) and the pattern of galaxies show that space is flat to within a fraction of a percent: the curvature parameter is Ω_K = 0.0007 ± 0.0019 from Planck with earlier galaxy surveys,³ and 0.0023 ± 0.0011 when DESI's 2025 galaxy maps are combined with the CMB, which DESI does not count as a significant departure from flatness.⁴ It is often said that this makes the universe infinite. The COMPACT collaboration's 2026 review calls that rule simply "false".¹

The correct rules are:

Positive curvature (like the surface of a sphere, but in three dimensions): always finite.
Zero curvature (flat): finite or infinite. Ten finite flat topologies exist, each in a continuous range of sizes and proportions.
Negative curvature (saddle-shaped): finite or infinite. There are infinitely many finite ones.

A flat torus has no edge, no centre and no curvature anywhere, and still has a finite volume. Gluing faces adds no bending; it only changes how the parts connect. So the question "is space finite?" is open, and it has to be answered by looking.

Chapter 3Eighteen flat universes

How many ways are there to build a flat space? In the 1930s, a crystallographer and two mathematicians found the full list: there are exactly eighteen, labelled E1 to E18 by cosmologists.^5,6 Ten are finite in every direction. Five are finite in two directions and infinite in the third (the "chimney" spaces); two are finite in one direction only (the "slab" spaces); and E18 is ordinary infinite space.

The ten finite ones are built from a box, or from a prism with a hexagonal base, by gluing its faces in different ways:

E1, the 3-torus: glue opposite faces straight across.
E2 to E5, the turn spaces: glue one pair of faces with a twist of a half-turn, quarter-turn, third-turn or sixth-turn. Travelling across that pair is a corkscrew motion: a shift plus a rotation.
E6, the Hantzsche–Wendt space: three perpendicular half-turn corkscrews.
E7 to E10, the Klein spaces: glue some faces with a mirror flip.

Finite in	How many	Orientable	Non-orientable
All three directions	10	E1–E6	E7–E10
Two directions (chimneys)	5	E11, E12	E13–E15
One direction (slabs)	2	E16	E17
None (ordinary space)	1	E18
Total	18	10	8

Following COMPACT's numbering.⁶

Crystallographers got there first. The set of moves that carries one tile onto the next is what they call a space group, the mathematics that classifies the symmetries of crystals. A universe needs a space group in which every move except doing nothing shifts every point; a move that left any point in place would leave a pinch or a crease in space there. Of the 219 types of space group (counting mirror-image pairs as one), exactly ten pass this test, and they are E1 to E10. The crystallographer Werner Nowacki found them in 1934 by checking the space groups one by one.^5,7 Two of the ten turn up constantly in biology: the Hantzsche–Wendt space has the symmetry of P2₁2₁2₁, the commonest arrangement of molecules in protein crystals, and the half-turn space has that of P2₁, the runner-up.^8,9

"Non-orientable" means that some loops reverse handedness. Carry a left glove around one and it comes back as a right glove; the Klein-bottle mode of Figure 1 shows the two-dimensional version. Some physicists doubt that such spaces are compatible with particle physics, whose weak force distinguishes left from right, but the question isn't settled.⁶

One distinction matters a great deal later. In the 3-torus every point is equivalent: everyone sees the same pattern of clones. It is homogeneous. In the turn spaces it isn't. A corkscrew has an axis. A point on the axis is simply carried forward, so its clone is just the corkscrew's length away; a point far from the axis is also swung around, so its clone is farther. Where you stand changes what you see. Chapter 9 shows why that is the loophole.

What about curved space?

Curved spaces have their own catalogues. Positively curved finite spaces are a 3-sphere divided by a finite group of symmetries; the most famous is the Poincaré dodecahedral space, a dodecahedron with opposite faces glued after a twist of 36°. Negatively curved finite spaces come in infinite variety. The smallest orientable one is the Weeks manifold, whose volume is about 0.94 times the cube of its curvature radius.¹⁰

But space is so nearly flat that most finite curved spaces would have to be enormous: using the 95% curvature range from Planck 2018 combined with baryon acoustic oscillations (Planck's own data allow much more positive curvature), a plain 3-sphere is at least about 900 times the volume inside the CMB sphere, the dodecahedral space about 8 times and the smallest orientable hyperbolic space (the Weeks manifold) about 25 times (this page's arithmetic). DESI's 2025 data leave the hyperbolic figure unchanged and lean, at about two standard deviations, against positive curvature altogether.⁴ Only spaces whose gluing includes a very small twist, such as lens spaces made with a cyclic group of high order (roughly 19 or more for that curvature range; this page's arithmetic), could still be small enough to see.¹¹ A COMPACT preprint posted on 8 October 2026 goes further: however small the curvature, some lens spaces and observer positions put the shortest loop through us between 0.985 and 1.1 times the CMB sphere's diameter, too long for circles big enough to find, but short enough that an ideal map could still reveal the topology through the statistical signs of Chapter 10.¹² Most of this page is therefore about flat shapes.

Part IILooking for ourselves

Chapter 4A century of ghost hunting

The idea is older than modern cosmology.

1900: Karl Schwarzschild, before general relativity, considers that space might be curved. If it is closed, he prefers "elliptic" space (a 3-sphere with opposite points identified, already a non-trivial topology) to the spherical kind. Light would then circle the universe and show us a "counter-sun" opposite the real one; he explains its absence by absorption, and uses parallaxes and star counts to put lower limits on the radius of curvature of space.¹³
1917: Willem de Sitter, citing Schwarzschild, suggests that Einstein's closed universe should be elliptic rather than spherical: arguably the first non-trivial topology proposed in relativistic cosmology.¹⁴
1924: Alexander Friedmann points out that the equations don't decide whether the universe is finite, even when space is negatively curved.¹⁵
1971: George Ellis's review "Topology and cosmology" sets out the programme.¹⁶
1983: Li-Zhi Fang and Humitaka Sato propose a torus about 600 megaparsecs (2 billion light-years) across to explain a claimed periodicity in quasar redshifts. The periodicity didn't survive.¹⁷
1984: William Thurston and Jeffrey Weeks explain three-dimensional manifolds in Scientific American. That article and the 1995 review below are later credited with helping to start the modern era.¹⁸
1995–96: Marc Lachièze-Rey and Jean-Pierre Luminet's review "Cosmic topology",¹⁹ and Roland Lehoucq and colleagues' "cosmic crystallography": if space is small, histograms of the separations between galaxy clusters should show spikes at the repeat lengths.²⁰

Hunting for ghost images directly is harder than it sounds. Every copy of an object is seen at a different distance, so at a different age, and from a different direction. You can't just look for two identical pictures. In 2003 a paper titled "Ghosts of the Milky Way" searched quasar catalogues for images of our own galaxy from a possible early quasar phase. It found candidate pairs, but no more than chance would produce.²¹ Later searches of Sloan Digital Sky Survey quasars found nothing convincing either.²²

The deeper limit is reach. Large quasar catalogues thin out beyond redshifts of about 4 or 5, roughly 25 billion light-years. The most distant galaxy confirmed so far, MoM-z14, at redshift 14.44, is 33.8 billion light-years away today (this page's arithmetic)²³. To find out whether space repeats on scales of tens of billions of light-years, astronomers need something farther away. The farthest thing anyone can see is the cosmic microwave background.

Chapter 5Your nearest ghost

First, some distances. Cosmologists measure them comoving: how far apart two places are today, after the expansion of the universe has carried them apart (Nº 1 explains the expansion itself, and why it could equally be told as all our rulers shrinking). Light from the most distant objects set off when they were much closer, so the "light-travel distance" (13.8 billion light-years for the oldest light) is not the distance between two places at any single moment. In comoving terms, with the Planck 2018 parameters:³

What	Distance today
The most distant confirmed galaxy, MoM-z14 (light left 13.5 billion years ago)	33.8 Gly
The last-scattering surface, where the CMB comes from (light left about 380,000 years after the Big Bang)	45.25 Gly
The particle horizon: the farthest light could have come from since the Big Bang	46.1 Gly
The farthest light emitted today will ever reach	16.7 Gly
The farthest any light can ever travel, from the Big Bang to the infinite future	62.8 Gly

This page's arithmetic, from the Friedmann equation with radiation, matter and a constant cosmological constant. The last-scattering radius is Planck's r_*/θ_* = 13.87 Gpc. The last two rows assume dark energy never changes. The 16.7 is the same limit that decides how far a bubble of true vacuum born today could ever spread, the subject of Nº 0, which quotes 16.6, 46.2 and 62.8 for these distances from Planck 2018 combined with baryon acoustic oscillations (H₀ = 67.66 km/s/Mpc, Ω_m = 0.311); the difference is about 1%.

The last two rows exist because the expansion of the universe is speeding up. Light set off today keeps moving outwards, but the expansion keeps stretching the road ahead of it, so its progress, counted in today's distances, slows towards a limit: it can never cover more than 16.7 billion light-years of them. Even light that set off at the Big Bang will never get farther than 62.8.

Now suppose space is a 3-torus with a loop length L. The nearest copy of the Milky Way is then a distance L away, and what we see of it depends on L:

If L were 5 billion light-years, we'd see the Milky Way as it was 4.3 billion years ago, with a young Sun in it, from a different side (this page's arithmetic).
At 30 billion light-years, as it was when the universe was 600 million years old: hardly a galaxy at all.
Beyond 45.25, the copy would be hidden behind the glowing plasma of the early universe.
Beyond 46.1, no light from it has reached us yet.
Beyond 62.8, none ever will.

Figure 2👻 Try it

Where's my ghost?

Drag a copy · tap to inspect

How far around the universe is?

Light from the nearest copy left it

The universe was then

Copies of the Milky Way in our sky

Light's trip round the loop

In the nearest copy you'd see

A slice through a 3-torus with loop length L, centred on us, in comoving distances today. Copies of the Milky Way sit on a square grid of spacing L; each is drawn (schematically) as our galaxy looked when the light now arriving left it, from the age of the universe at that comoving distance in flat ΛCDM with Planck 2018 parameters (this page's arithmetic; the Sun formed at an age of about 9.2 billion years). The count of copies in our sky is for the full three-dimensional lattice. Beyond the last-scattering surface (45.25 billion light-years) the view is blocked by the opaque early plasma; beyond the particle horizon (46.1) no light has arrived yet; and with a constant cosmological constant, no light emitted at any time ever crosses more than 62.8. The circle searches put the shortest loop through us at 89.1 or more.If space wraps around, light from our own galaxy could go all the way round and come back. Drag a copy (or the slider) to choose how big the universe is, and tap any copy to see what it would show us. If space does wrap around, the real universe is at least 89 billion light-years around through us, far beyond the dashed ring at about 63 billion light-years, past which no light can ever come back while the expansion keeps speeding up.

The measurements of Part III put our nearest copy at 89.1 billion light-years or more. That is not just "behind the CMB". It is nearly twice the particle horizon. No light from any copy of us, from any era, has reached us since the Big Bang. If dark energy stays constant, none ever will: our ghost is permanently out of sight. Light has made it only about halfway (52%) around even the shortest loop through us that is allowed, and it never will make more than about 70% (this page's arithmetic).

That's why galaxy-by-galaxy ghost hunting stopped being the main tool. The CMB offers a cleverer test, which works even when the copies themselves are out of sight.

Chapter 6Circles in the sky

In 1996 Neil Cornish, David Spergel and Glenn Starkman noticed something simple.²⁴ We see the CMB coming from a thin sphere around us, the last-scattering surface, of radius χ = 45.25 billion light-years. In a wrapped universe, each of our clones has its own such sphere. If a clone is close enough, less than one CMB-sphere diameter away, the two spheres overlap. Two spheres intersect in a circle.

Every point on that circle is at the right distance to be on both spheres, so we see it twice: once looking one way, and again, via the loop, looking another way. The result is a pair of circles on the sky carrying the same pattern of hot and cold spots. The signature is, in the authors' phrase, "writ large on the microwave background sky".

For a clone at distance L, simple geometry gives the circles' angular radius α:

cos α = L / (2χ) = L / D D = 2χ = 90.5 billion light-years is the diameter of the last-scattering sphere. A small universe gives big circles; as L approaches D the circles shrink to points and vanish.

Figure 3⭕ Try it

Circles in the sky

A cube-shaped universe

Size of the universe, compared with the CMB sphere's diameter

Once around

Biggest circle

Matched pairs

Verdict

Left: our last-scattering sphere (radius χ = 45.25 billion light-years) inside one cell of a cubic 3-torus of side L. Where the sphere crosses a face it meets the sphere of our copy beyond that face, in a circle of angular radius α with cos α = L/(2χ); opposite faces are the same face, so each circle appears twice. Right: the whole sky (Mollweide projection) with a toy temperature pattern made of waves that fit the box. Each colour marks one matched pair; face diagonals add pairs below L = 0.707 × 2χ and body diagonals below 0.577 × 2χ. The strips unroll the selected pair: the two match exactly, a circle chosen at random does not. The score beside the lower two strips is the circle searches' original (1998) match statistic (1 for a perfect match, 0 on average for unrelated circles). Real circles match less well (Doppler and late-time effects). Search limits: WMAP first-year 25° (2004), Planck temperature 20° (2013), Planck polarisation 15° (2015), WMAP seven-year 10° for any arrangement (2012).We see the cosmic microwave background as a glowing sphere all around us. If the universe is a box narrower than that sphere, the sphere pokes out through the walls, and because opposite walls are the same wall, each place where it pokes through shows up twice on our sky: a pair of circles with the same pattern. Shrink the universe and watch the circles appear, then tap a circle to compare the pair. The score shows how well each strip matches circle A: 1 is a perfect match, and close to 0 means no match at all.

The pattern of circles tells you the shape. In a 3-torus the two circles of each pair sit on opposite sides of the sky. In a turn space the pattern on one circle is rotated relative to its partner. In an orientable space the patterns run clockwise around one circle and anticlockwise around the other; in a non-orientable space they run the same way.

The match is never perfect. The plasma at last scattering was moving, which adds a Doppler signal that depends on the viewing direction, and the light from the two circles crosses different structures on its way to us. Small circles suffer most, which is why searches stop at about 10° radius.²⁵

Part IIITwenty years of nothing

Chapter 7The soccer-ball universe

In 2003 the first WMAP maps of the CMB showed something odd: there was less structure than expected on the very largest angles, and almost no correlation between points more than about 60° apart. A small universe could produce that kind of quietness, because waves longer than the universe can't exist in it.

Jean-Pierre Luminet, Jeffrey Weeks and colleagues proposed a specific shape in Nature: the Poincaré dodecahedral space, a slightly positively curved universe built from a dodecahedron with opposite faces glued after a 36° twist.²⁶ It fitted the weak large-angle correlations and predicted a total density 1.3% above the flat value. It also predicted six pairs of matched circles, about 35° in radius.

Cornish, Spergel, Starkman and Eiichiro Komatsu had already searched the WMAP maps. They found no matched circles larger than 25° in radius, which excluded the dodecahedral model's 35° circles, and any 3-torus smaller than about 24 Gpc (78 billion light-years, by that paper's reckoning).²⁷ In 2004 another group reported six matched pairs at about 11°, with the predicted 36° twist.²⁸ In 2007 Key, Cornish, Spergel and Starkman repeated the analysis with an optimised statistic and showed that the signal disappeared once it was properly weighted. They ruled out what they called the "infamous 'soccer ball universe'".²⁵

Curvature measurements finished the job. The model needed Ω_K ≈ −0.013, and its authors wrote that finding a density below 1.01 times the critical value (Ω_K above −0.01) would refute it. Planck with earlier galaxy surveys gives 0.0007 ± 0.0019, more than five standard deviations beyond that line, and DESI's 2025 maps combined with the CMB give 0.0023 ± 0.0011, more than ten (this page's arithmetic).^26,3,4 A dodecahedral universe is still possible in principle, but only with far less curvature, which makes it much too large to see.

Chapter 8Eighty-nine billion light-years

The strongest general limit came in 2012. Pascal Vaudrevange, Starkman, Cornish and Spergel searched the WMAP seven-year maps for matched circle pairs of any arrangement: radii from 10° to 90°, centres from 11° to 180° apart, matching in either sense. After discarding spurious pairs through two contaminated patches of sky, they found none. No pair rose above a detection threshold the authors set, by their own rough estimate, at the 3σ (99.7%) level.²⁹ Since the smallest circles they could see had a 10° radius, and cos 10° = 0.985:

L_{through us} > 0.985 × D = 0.985 × 90.5 ≈ 89.1 billion light-years The fraction 0.985 is the measurement; the conversion to light-years uses Planck 2018 and is uncertain by about 0.2 billion light-years (this page's arithmetic). The 2012 paper itself quoted "approximately 26 Gpc", about 85 billion light-years, using earlier parameters. COMPACT's review notes that the limit holds only if the clone doesn't lie too close to the plane of our Galaxy.¹

That's the title of this page: with that one proviso, the shortest closed loop through our position, in any shape of flat universe, is at least eighty-nine billion light-years long. A cube-shaped universe could still be a little smaller than the CMB sphere is wide, 89.1 against 90.5, with circles under 10° that no one could find.

Planck's own circle searches were narrower, looking only for back-to-back pairs. Its 2013 search of the temperature maps ruled out loops shorter than 0.94 D at 99% confidence;³⁰ its 2015 search of the polarisation maps raised that limit to 0.97 D, for orientations not hidden by the polarisation mask.³¹ Planck also compared whole simulated skies with the real one for a handful of fixed shapes, which pushes some of them, such as the dodecahedral space, to at least 1.03 D (by a likelihood-ratio criterion rather than a standard confidence level).³⁰ COMPACT's 2026 review adds one more null result: an unpublished search over all possible circle pairs by the authors of the 2012 search (cited as a 2014 private communication) found no significant match either.¹

Search	Circles	Loop through us at least
WMAP first year (2004)	back-to-back, radius above 25°	0.91 D = 82 Gly
WMAP seven-year (2012)	any arrangement, radius above 10°	0.985 D = 89.1 Gly
Planck temperature (2013)	back-to-back, above about 20°	0.94 D = 85 Gly
Planck polarisation (2015)	back-to-back, above about 15°	0.97 D = 88 Gly

Lengths in billions of light-years use D = 90.5 (this page's arithmetic).

How big must the whole universe be? That depends on its shape. A cube-shaped universe at the limit holds 89.1³ cubic billion light-years, 1.82 times the volume inside the CMB sphere. But a cube isn't the most economical shape. If no two copies of us can be closer than 89.1 billion light-years, then balls of radius 44.6 billion light-years (half that distance) around each copy can't overlap, and the densest possible packing of equal balls, proved by Thomas Hales in 1998 and published in 2005, sets a floor.³² The tightest flat universes (for example a skewed 3-torus with face-centred-cubic repeats) hold at least 1.29 times the CMB sphere's volume (this page's arithmetic). There is no upper limit. Space may also be infinite.

Part IVThe loophole

Chapter 9The corkscrew loophole

The 89-billion-light-year limit is about loops through us. In the 3-torus every point is equivalent, so the limit applies to every loop in the universe. In most other shapes it doesn't.

In 2022 the COMPACT collaboration, which includes Glenn Starkman, one of the inventors of the circle method, worked out what the circle searches really exclude.^33,2 The answer is: much less than had been assumed. Take the half-turn space E2. Going once around its corkscrew moves you a distance L_B along the axis and turns you half a turn about it. If you sit on an axis, your clone is just L_B away. If you sit a distance ρ from the axis, the half-turn swings your clone to the other side of it, and the distance becomes

d = √(L_B² + (2ρ)²) For a general turn by angle θ the sideways part is 2ρ sin(θ/2). Doing the corkscrew N times (N = 2, 4, 3 or 6 in E2 to E5) is a straight shift of NL_B, which every observer feels.

So a universe can have loops much shorter than the CMB sphere's diameter, as long as they run along corkscrew axes far from us. The COMPACT paper puts it this way: in all but the simplest cases, the shortest loop "can avoid us", and can be shorter than the CMB sphere's diameter "by a factor ranging from 2 to at least 6".² In the half-turn space the shortest loop can be half the CMB diameter; in the sixth-turn space about a sixth; in a slab with a rotation, there is no limit at all. In the non-orientable spaces, the shortest loop anywhere can be as little as half the shortest loop through us.¹

Figure 4🌀 Try it

The corkscrew loophole

Drag yourself around

Length of the corkscrew loop

Length of the straight loops

Shortest loop in this universe

Shortest loop through you

Your distance from the nearest corkscrew axis

Do you see circles?

Observers who'd see circles

The half-turn space E2 in its covering space. Going once along a corkscrew moves you a distance L_B up its axis and turns you half a turn about it, so a point at distance ρ from an axis has a copy √(L_B² + 4ρ²) away, on the far side of the axis one layer up (and one down); two straight loops of length L_A run across, and the axes form a grid of spacing L_A/2. Shaded tubes: observers whose shortest loop is shorter than the last-scattering diameter D = 90.5 billion light-years, who would see matched circles (those between 0.985 D and D would see only circles under 10°, too small for the searches). Lengths in units of D; the fraction of observers comes from sampling (this page's arithmetic, following COMPACT Part I).In some shapes of universe, the shortcut around space twists like a corkscrew, and how short it looks depends on where you stand. Near a corkscrew axis your copy is close by, just one twist up, and you'd see circles; far from every axis the twist swings your copy far away. Drag or tap yourself into the gaps between the tubes. (Gly is short for a billion light-years; D is the CMB sphere's width, 90.5 Gly.)

The short loops can't be near us, though. With the shortest loops at their minimum, the nearest axis carrying those shortest loops must be at least about 39 billion light-years away in the half-turn space and about 88 in the sixth-turn space (this page's arithmetic). And tilting a 3-torus doesn't help: it is still homogeneous, so its shortest loop still has to pass the circle test. Tilt matters for a different reason: Planck's full-sky likelihood tests used only cubic tori, so for tilted or unequal tori the circle limit is essentially the only constraint, and the range just beyond it is largely unexplored.³³

COMPACT's summary of the state of play is blunt. Current constraints "exclude only some topologies, parameter ranges, and observer positions".¹

Chapter 10Beyond circles

Circles are only the loudest signature. A finite universe also changes the statistics of the whole sky. In infinite space, fluctuations come in waves of every wavelength and every direction. In a 3-torus of side L, only waves that fit the box exist, with wavevectors 2πn/L: a continuous range becomes a lattice. Spots in different directions become correlated in specific, calculable ways, even when the universe is somewhat larger than the CMB sphere and no circles exist.³³

COMPACT has been computing these correlations for every flat shape, any size, any orientation and any observer position.^34,6 Some of the results:

Reach. In principle, a perfect map of the CMB's temperature carries enough information to tell a cube-shaped universe from an infinite one up to about 1.12 D (about 101 billion light-years), and an idealised three-dimensional survey of the observable universe up to about 1.37 D. These are best-case limits, set by whether the information is there at all, not guaranteed detections. For the best cases among the other shapes, the CMB could reach about 1.2–1.3 D.^1,35 How far these searches reach also depends on the assumed spectrum of primordial fluctuations on the largest scales (a 2026 preprint).³⁶
Parity. Some shapes make the sky's even and odd patterns correlate in ways usually blamed on new physics. Topology could produce a "parity violation" with no parity-violating physics at all.³⁷
Machine learning. Trained classifiers can tell small tori from infinite space with high accuracy when the orientation is known; unknown orientation is the main difficulty.³⁸
Prospects. A comprehensive analysis of existing Planck data might reach about 1.05 D, and a future mission such as LiteBIRD, by adding much sharper maps of the CMB's polarisation, in principle as far as about 1.2 D for any flat shape and for any curved one that is searched for.¹

None of this has found anything yet. One minority analysis, using topological statistics of the Planck maps, reported a "further hint" of a torus of two to three Hubble lengths across,³⁹ but a universe that small would produce large circles, which aren't there.

Chapter 11Ceilings and open questions

The CMB sphere isn't fixed. As time passes, light from slightly farther away arrives, and the sphere grows by about one light-year per year. But it doesn't grow forever. Light set off from here today can only ever cover 16.7 billion light-years, so the sphere's radius tends to 45.25 + 16.69 ≈ 61.9 (a little less than the 62.8 of Chapter 5, which counts light that set off at the Big Bang itself rather than 380,000 years later), and the longest loop matched circles could ever test tends to about 124 billion light-years (this page's arithmetic). By the time the reach passes 122, about 50 billion years from now, the CMB will have cooled to about 0.15 K.

Figure 5⏳ Try it

The growing sky

Drag the copy

Time from now

How far around the universe is (through us)

Radius of the CMB sphere

Longest loop circles could test

Temperature of the CMB

Circles for this universe

Comoving distances in flat ΛCDM (Planck 2018) with a constant cosmological constant (this page's arithmetic). Our last-scattering sphere and that of our copy a loop length away grow by the distance light covers after today; they overlap, and matched circles exist, once twice the radius exceeds the loop length. The radius tends to 45.25 + 16.69 ≈ 61.9 billion light-years (dashed), a little under the 62.8 that light from the Big Bang itself can cover, so the longest loop matched circles could ever test tends to 123.9. Statistical methods reach somewhat further. If dark energy weakens enough for the expansion to stop speeding up (DESI analyses of 2025 and 2026 hint, at about three standard deviations, that it may be weakening), these ceilings disappear.The CMB sphere we see slowly grows as light from farther away arrives. Press play and watch ours and our copy's grow. As long as the universe's expansion keeps speeding up, they never grow past about 62 billion light-years, so circles can never test a universe more than about 124 billion light-years around.

Billions of light-years (today's distances)	What it is
16.7	Farthest light set off today will ever reach
33.8	Most distant confirmed galaxy (MoM-z14)
45.25	Radius of the CMB sphere
46.1	Edge of the observable universe
62.8	Farthest any light can ever travel
89.1	Shortest loop through us allowed by the circle searches
90.5	Diameter of the CMB sphere: circles vanish beyond this
~95–118	Reach of statistical searches, existing data to an ideal CMB map
~124	The most matched circles in the CMB can ever test

This page's arithmetic, flat ΛCDM with Planck 2018 parameters and a constant cosmological constant. The "never" statements on this page assume dark energy doesn't change. DESI analyses of 2025 and 2026 hint, at about three standard deviations, that dark energy may be weakening; that is short of the five-sigma discovery standard.⁴ If it weakened enough for the expansion to stop speeding up for good, these ceilings would disappear and light would eventually get round any loop; if it strengthened, they would shrink. Nobody yet knows which.

Some consequences worth stating plainly:

No light ray will ever go all the way around a 3-torus universe consistent with the data: every loop is at least 89 billion light-years, and light can never cover more than 62.8.
The guaranteed "private" region around us, with no repeat of anything in it, is a ball of radius 44.6 billion light-years: 96% of the volume inside the CMB sphere.
In the turn spaces, short loops could exist far away, 15 to 45 billion light-years long, but their axes would have to lie tens of billions of light-years from us.

And the open questions:

Is space finite? Unknown. Nothing has been detected in any channel.
How much has really been excluded? Circle limits have been translated shape by shape only for the orientable flat spaces and lens spaces. A full likelihood search of Planck data over all shapes, sizes, orientations and positions is under way.
Are the CMB's large-angle oddities real? If they are, a universe just larger than the CMB sphere is one possible explanation.
What would it say about inflation? If inflation happened, a loop short enough to see would be something of a coincidence: inflation would have had to last just the right time, since a little longer would have stretched any loop far beyond our horizon. A detection would therefore tell us how long inflation lasted, much as observable curvature would.¹
Can galaxy surveys and 21-cm maps help? In principle, an ideal three-dimensional map of the whole observable universe reaches about a quarter farther than a perfect CMB map; galaxy surveys cover only its nearer part, so going deeper would take 21-cm maps of the hydrogen before the first galaxies.

COMPACT's review allows that the universe may be so large that we never learn its shape. But the evidence could already be sitting in data we have. Either way, the answer to "how far is it around the universe?" is no longer "infinitely far, obviously". It is: at least eighty-nine billion light-years through here, and possibly much less somewhere else.

Chapter 12Glossary and sources

Topology: How a space is connected: which loops can be shrunk to a point, and whether going straight can bring you back.
Geometry, curvature: The local shape of space: how triangles and parallel lines behave. Measured to be flat to within a fraction of a percent.
3-torus: A box with opposite faces glued: finite, flat, with no edge or centre.
Covering space: The picture in which the universe's cell is repeated forever. Light travels in straight lines through it.
Clone, ghost: An image of an object seen by light that has gone around the universe.
Comoving distance: The distance between two places today, after the expansion has carried them apart.
Last-scattering surface, CMB sphere: The sphere, 45.25 billion light-years away today, from which the cosmic microwave background reaches us.
Particle horizon: The farthest any light could have come from since the Big Bang: 46.1 billion light-years, the edge of the observable universe.
Matched circles: Pairs of circles on the sky with the same pattern, where our CMB sphere intersects a copy of itself.
Homogeneous space: One in which every point is equivalent. The 3-torus is; the turn spaces are not.
Orientable: A space in which no loop turns a left glove into a right one.

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