Flat universe leaves 18 geometric possibilities unexplored

flat universe – Cosmologists are fairly certain the universe is flat, but that still doesn’t settle the question that keeps coming back in different forms: what is its exact shape? New work through the COMPACT collaboration argues that earlier searches in the cosmic microwave

For years, the question sounded almost settled: the universe is flat. The cosmic microwave background—the oldest light we can observe—fits a picture in which space has no overall curvature on large scales. Yet “flat” is only the starting point. Flat space can stretch without end. it can fold back on itself like a torus. or it can wrap in stranger ways that don’t show up in any simple map of reality.

And now, new research is pushing cosmologists to treat that second, harder question with fresh urgency: not whether our universe is flat, but which one of its many possible flat “shapes” it actually is.

The geometry problem begins with triangles.

Carl Friedrich Gauss. the German astronomer and mathematician who lived in the late 1700s and early 1800s. studied geometry in curved spaces and understood that the rules change when curvature changes. On a plane, the angles of a triangle add up to 180 degrees. On a sphere, the sum can be greater. An equilateral triangle on the surface of Earth can have three right angles. On other curved examples—one often compared to the “angle deficit” of a Pringles chip—the sum can even be less than 180 degrees.

Gauss’s name enters the story partly as legend: he is said to have measured distances between three German mountain peaks—Hohenhagen. Brocken. and Inselberg—and used the geometry of that triangle to infer that the angles added up close enough to 180 degrees to suggest a flat plane between the peaks. It’s a vivid method for thinking about curvature.

But it can’t simply be scaled up to the cosmos. The universe is too gigantic. Even with a large telescope. triangulating the distances between stars wouldn’t work in the way Gauss’s triangle did for mountains. Stars in our own or neighboring galaxies are too close compared to the vast scale of the universe. And there’s a second complication: the objects are moving. and gravity bends the paths that light follows on its way to us.

Instead, cosmologists use a different trick—looking back in time to the oldest radiation we can measure. Around 13.8 billion years ago. long before atoms formed. the universe was a hot. dense mix of quarks and gluons like a primordial soup. Photons couldn’t move freely through it. As the universe expanded and cooled, atomic nuclei formed first and then atoms. The cosmos became transparent. The light that escaped at that stage began its journey about 370. 000 years after the Big Bang. and it is this cosmic microwave background that fills our sky with a fossil record.

That fossil record is strikingly uniform. The signal is distributed across the sky almost the same way no matter where detectors point. From that. cosmologists build the cosmological principle: the universe must be homogeneous and isotropic—matter spread out in a uniform way. in all directions. From Einstein’s general relativity, that kind of uniformity implies that the curvature of space is constant on large scales.

With constant large-scale curvature, there are three broad cases. No curvature gives Euclidean geometry like a flat surface. Positive curvature leads to spherical geometry, like a sphere. Negative curvature leads to hyperbolic geometry, sometimes likened to the Pringles-chip shape.

To decide which case fits, cosmologists return to tiny imperfections. The cosmic microwave background is nearly uniform, but not perfectly. Those small fluctuations trace back to tiny density differences in the hot early universe. Researchers can calculate how strong the fluctuations should be depending on geometry because the curvature changes how those density waves appear in the sky.

Here’s the key comparison: if the universe is positively curved. density fluctuations should appear larger than the theoretical expectation; if negatively curved. smaller. If the universe has no curvature. the fluctuations should line up with the theoretical value—much like a flat-space triangle’s angles sum to 180 degrees.

Measurements indicate the last scenario: our universe is flat.

But flat doesn’t answer the next question, the one that keeps widening the search. In everyday life, “flat” suggests one clear form. In mathematics, it doesn’t.

To understand why, the story turns to lower dimensions. In two dimensions, flat geometry can look more than one way. A torus—a donut-like surface—is curved in the ordinary sense. but its geometry can be flat in the right way. You can imagine building one from a flat sheet: roll it until the ends meet to make a cylinder. then twist the tube until it forms a ring. The bagel appears curved, but the key geometric property is different.

In that two-dimensional flat world, there are also three other variations besides a torus: a cylinder, a Möbius strip, and a Klein bottle.

Then come the three-dimensional possibilities, and with them the dramatic widening of the list.

In 1934, mathematician Werner Nowacki proved that there are 18 different flat three-dimensional shapes. If our universe is truly flat, then it must match one of those 18 candidates.

Some can be removed immediately. Eight of the 18 are “nonorientable.” In a nonorientable universe. if you fly through it you would eventually return to your starting point but in a mirrored form: your right becomes left and vice versa. According to experts, such universes contradict the laws of physics.

That leaves 10 different flat forms the universe could still have. They include: an infinitely extended three-dimensional space with x. y. and z axes; a three-dimensional generalization of a torus created by gluing opposite faces of a cube; a half-twist torus and a quarter-twist torus that differ by twisting one pair of surfaces by 180 degrees or 90 degrees; a third-twist prism and a sixth-twist prism. built from a six-sided prism where opposite faces are glued with rotations of 120 degrees or 60 degrees; a Hantzsche-Wendt manifold described as two cubes stacked on top of each other with faces joined in a complex way; and spaces with infinitely many flat planes or an infinitely tall “chimney” formed by four surfaces arranged as the sides of a parallelogram. with opposite surfaces glued. The list also includes a version of that chimney where one of the pairs is rotated by 180 degrees.

All of these share the same underlying flat geometry, but each comes with its own distinctive global features. The problem is that “global features” are exactly what the universe doesn’t advertise easily.

To see why, imagine light repeating.

Many of the candidate shapes are compact, meaning they don’t extend outward infinitely. If space is compact and torus-shaped, light from Earth would eventually reach Earth again. You would see your reflection—perhaps more than once—because of the repetition built into the geometry.

Yet the universe isn’t waiting for us with instant rewinds. Light travels at a finite speed, and our cosmos is gigantic. Even if light from our solar system or galaxy returned someday. we likely wouldn’t recognize the image: by then. the shape at that time probably would not resemble the surroundings we know today. There’s also the possibility that the universe is so vast that light simply hasn’t had time to traverse it.

Still, researchers look for subtler traces.

In compact universes, the shape of the cosmos affects how matter and light interacted in the early universe. That influence should show up in the cosmic microwave background radiation. One approach has been to search for repeating structures within it—identical circular arrangements that could indicate a compact universe.

Those searches required geometric care. Because the radiation is received on the spherical Earth, the signal appears on a spherical surface. But the universe might have a more complex shape, so any repeated traces should appear within the spherical data in a particular way.

When experts searched for identical circular structures in cosmic microwave background radiation data during the 2000s and 2010s. they found nothing. Because the searches didn’t reveal repetition. most cosmologists assumed a fairly simple structure: a flat universe extending infinitely in all three spatial dimensions.

For a while, the effort stalled. The reason wasn’t a lack of effort—it was a lack of new evidence.

That changed in 2022, when the Collaboration for Observations, Models and Predictions of Anomalies and Cosmic Topology—COMPACT—was launched.

COMPACT researchers compare the latest cosmic microwave background data with the different possible shapes of the universe. Their central finding is unsettling for anyone who thought the earlier searches had settled the issue. The lack of evidence for identical circular structures in the cosmic microwave background is far less restrictive than previously thought. In other words, in a compact universe, it’s quite plausible that researchers might not identify these structures at all.

The team is still analyzing the data and building models. Their work now focuses on other features in cosmological data that could point toward complex shapes of the universe, with exciting new results expected in the coming months and years.

Put together, the new approach suggests something more than a technical adjustment. It points toward a reality that may be more complex than earlier searches led people to assume. The topology of spacetime is believed to have been determined by quantum processes shortly after the Big Bang. If cosmologists could identify the universe’s shape more precisely, it could help reveal more about those early processes.

That’s the hope driving the next round of scrutiny: not just confirming that space is flat, but figuring out which of the 18 flat three-dimensional options actually matches the cosmos we’re inside.

The universe may not be curved. But for now, it’s still not known which flat form it wears.

universe shape flat universe cosmic microwave background COMPACT collaboration topology of spacetime Werner Nowacki nonorientable manifolds cosmological principle cosmic geometry