What If the Universe Has No Edge and No Centre?

The strange topology at the frontier of cosmology, and what a video game from 1980 has to do with the shape of everything

There is a question so large that most people never think to ask it, perhaps because the asking feels slightly ridiculous. Not “what is in the universe?”, scientists have been working on that for centuries, but something stranger and more fundamental: what shape is the universe?

Not the shape of galaxies. Not the shape of space around a black hole, though that is curved and strange in its own right. The shape of everything. The global topology of the cosmos, the way it is, or is not, connected to itself.

You might assume this is settled. You might assume that cosmologists have measured it, confirmed it, filed the answer away. They have not. The shape of the universe is an open question. And one of the leading candidates is something so familiar it lives in every arcade game ever made.

A torus.

The Pac-Man Universe

Photo by Sei on Unsplash

In Pac-Man, when you walk off the right edge of the screen, you reappear on the left. When you walk off the top, you reappear at the bottom. The screen has no boundary you can slam into. It wraps.

Mathematically, what the game designers had done, almost certainly without thinking of it this way, is give Pac-Man a torus to live in. Connect the left and right edges of a rectangle, and you get a cylinder.

Now connect the top and bottom edges of that cylinder, and you get a donut shape, a torus.

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The crucial thing is that the inhabitants of this universe, the Pac-Men, if you like, would not necessarily feel any seam. There is no wall, no border, no moment of crossing. The geometry is locally flat in every direction. The wrapping is a global property of the space, invisible from any single vantage point.

This is what cosmologists mean when they ask whether the universe has toroidal topology. Not that it is shaped like a donut from the outside, we have no “outside” to look from. Rather, that if you travelled far enough in any direction, you might return to your starting point from the opposite side. That space is finite but unbounded. That there is no edge, and no need for one.

Finite but unbounded. No edge, no centre. The idea is conceptually straightforward and almost impossible to actually picture — which is perhaps why it has taken so long to take seriously.

The Geometry of the Problem

To understand why any of this is possible, you need a brief detour through the difference between geometry and topology.

Geometry is local. It tells you about lengths, angles, curvatures. A triangle in flat space has angles summing to 180 degrees. On a sphere, they sum to more. On a saddle, they sum to less. This is what Einstein’s general relativity deals in, the curvature of spacetime, warped by mass and energy.

Topology is global. It tells you about the overall shape of a space, how it is connected. A sphere and a cube are topologically identical (you can deform one into the other without tearing). A torus is topologically distinct from a sphere, there is a hole through it, and no amount of smooth bending will get rid of that hole.

Here is the remarkable thing: general relativity is a theory of geometry, not topology. Einstein’s equations tell you the local curvature of spacetime, but they say nothing about the global topology. A universe can obey all of relativity’s laws and still be a torus. Or a sphere. Or something stranger.

The shape of the universe is, in this sense, not determined by the physics. It is an additional fact about the world that the physics leaves open.

The Evidence, Such As It Is

The best map of the early universe we have is the cosmic microwave background — the faint afterglow of light from roughly 380,000 years after the Big Bang, when the universe cooled enough for atoms to form and photons to travel freely. The CMB is our deepest photograph. It shows us, essentially, the entire observable universe at once, plastered across the sky like wallpaper.

If the universe is toroidal, if it wraps around itself, then the CMB should show a particular kind of pattern. Specifically, there should be pairs of matching circles: circles on opposite parts of the sky where the temperature fluctuations are identical, because you are looking at the same region of space from two different directions, the light having wrapped around the universal torus.

In 2003, a team led by Jean-Pierre Luminet at the Paris Observatory published a paper in Nature arguing that the CMB data from the WMAP satellite suggested exactly this kind of structure. Their model was a slightly different topology, a Poincaré dodecahedral space, but the principle was the same: a universe that wraps around itself, creating ghost images in the microwave background.

The paper caused a sensation. It also caused a great deal of careful re-examination by other teams, who found the evidence less compelling upon scrutiny. The matching circles, they argued, could be explained by chance. The alignment patterns Luminet’s team identified might be statistical artifacts.

In a finite toroidal universe, the CMB would contain matching circle pairs — the same patch of early-universe plasma visible in two directions simultaneously.

The debate has never fully resolved. Subsequent data from the Planck satellite, released in 2013 and updated in 2018, tightened the constraints. If the universe is a torus, it is a very large one, the “fundamental domain” (the size of the repeating cell) must be at least as large as the observable universe, or the matching circles would be unmistakably obvious. But large is not the same as infinite. The topology might be there, just hidden beyond what we can see.

The Missing Power Problem

There is one anomaly in the CMB that keeps cosmologists interested in the finite-universe hypothesis. It is called the missing large-scale power problem, and it is exactly what it sounds like.

The CMB has fluctuations at every scale, ripples in the temperature of the early universe, some large and sweeping, some small and tight. Standard cosmology predicts a specific distribution of these fluctuations. On almost every scale, the prediction matches observation beautifully.

Almost. On the very largest scales, fluctuations spanning more than 60 degrees of sky, the observed power is noticeably lower than the standard model predicts. There is less variation on large scales than there should be.

This is exactly what you would expect in a finite universe. If space wraps around itself at some scale, fluctuations larger than that scale simply cannot exist, there is no room for them. The universe cannot contain a wave longer than the universe. A toroidal topology would naturally suppress large-scale power, because the largest fluctuations would be cut off by the size of the space itself.

The missing power problem might be a statistical fluke. The universe is a single realisation of a random process, and single realisations are allowed to be unusual. But it might also be a signal. Cosmologists are not sure.

The universe cannot contain a wave longer than the universe. That sentence, once you sit with it, is one of the strangest you will encounter in physics.

What It Would Feel Like

Suppose the universe is a torus. Suppose, further, that it is small enough that light has had time to traverse it — that the wrapping scale is within the observable universe. What would we see?

We would see ourselves. Or rather, we would see ghost images of the Milky Way, ancient light from our own galaxy that has wrapped all the way around the cosmos and returned from the other direction. We would see the same galaxy cluster appearing in multiple places in the sky. We would see a universe that, on the largest scales, has a tiling structure — repetitions of the same deep-field patch, arranged with the precise symmetry of a mathematical wallpaper pattern.

Astronomers have looked for this. They have not found it. Which either means the universe is not a torus, or means the torus is large enough that the repetitions fall outside our observational horizon, the sphere beyond which light has not had time to reach us since the Big Bang.

This second possibility is not falsifiable, at least not with current technology. It is the epistemological frustration at the heart of cosmology: the universe might be finite, might be periodic, might wrap around itself in a structure of mathematical elegance, and we simply cannot see far enough to know.

The Mathematics Behind the Shape

A torus, mathematically, is a quotient space. You take the Euclidean plane and identify points that differ by integer multiples of some basis vectors. Every point at position (x, y) is the same as every point at (x + L, y) and (x, y + L), where L is the period. The plane tiles; the torus is the tile.

This is exactly the geometry of the arcade screen. And it means the torus is, locally, perfectly flat. There is no curvature. A triangle drawn anywhere inside it has angles summing exactly to 180 degrees. The topology is nontrivial, the space is connected to itself in a way a flat infinite plane is not, but the geometry is Euclidean.

This matters for cosmology, because measurements suggest the universe is, to extremely high precision, spatially flat. The total density of matter and energy is very close to the critical density that produces a flat geometry. A toroidal universe would be consistent with flatness. An infinite flat universe would also be consistent with flatness. The geometry cannot distinguish them.

The higher-dimensional generalisations are straightforward. A three-torus, the shape cosmologists actually have in mind when they discuss toroidal topology, is what you get by taking a three-dimensional box and identifying opposite faces. Walk out through the right wall and you come back in through the left. Walk out through the ceiling and you fall in through the floor. The geometry remains flat. The topology wraps in all three spatial directions.

At this point, a reasonable person might ask: why does it matter? If the torus is large enough that we cannot see the repetitions, if the topology leaves no observable fingerprint we can currently detect, what is the practical difference between a finite toroidal universe and an infinite flat one?

The question is fair, and the answer is partly philosophical. An infinite universe has infinite matter, infinite energy, infinitely many planets, and — by the logic of combinatorics, infinitely many exact copies of every arrangement of atoms that has ever existed, including you, reading this, on a planet almost identical to Earth but orbiting a star in a galaxy you would not recognise. The infinite universe forces a strange kind of multiplication of existence that many physicists find uncomfortable.

A finite universe avoids this. It has a specific amount of matter, a specific number of galaxies, a specific size. It is bounded without having a boundary. It is, in some sense, more aesthetically satisfying, a cosmos with a definite quantity to it, rather than an unimaginable and unbounded profusion.

There is also the deeper mathematical question. The universe is not just matter moving through space, it is a manifold, a topological object with definite mathematical properties. Asking what those properties are is not frivolous. It is asking, in the most basic sense, what kind of thing the universe is.

We live inside a shape we cannot step outside of to examine. All we have are the shadows it casts on our instruments, anomalies in ancient light, missing fluctuations, circles that may or may not match.

The Shape We Cannot See

Photo by Andy Holmes on Unsplash

The universe, as best we can measure, is 93 billion light-years across, in its observable portion. What lies beyond that horizon we do not know, because no signal from there has had time to reach us. The topology of the universe could announce itself only in that unreachable beyond, writing its signature in a pattern of ghost images we cannot receive.

Or it might have left its mark in the data we already have. The missing power on large angular scales. The specific alignments of the CMB’s quadrupole and octopole moments. These are not settled anomalies, they might be statistical noise. But they are also exactly what a small, finite, toroidal universe would produce.

We are in the position of a person trying to determine the shape of a room by listening to the echoes. We have the echoes. We are not sure what shape they imply.

What we know is this: the universe obeys mathematics. Its geometry is described by equations. Its history is written in light. And somewhere in that light, in the ancient, barely-warm afterglow of the Big Bang still washing over our satellites, there may be a clue about whether space wraps, whether the cosmos is finite, whether the universe is, in its deepest structure, a shape that any child who ever played Pac-Man has already, unknowingly, inhabited.