The universe could have 18 possible shapes

What is the shape of the universe? This question is far more intriguing and truly unresolved than any debate over the shape of our planet, despite the claims of flat Earthers.

We occupy only a tiny space within a gigantic cosmos. Our perspective is limited. However, cosmologists are now almost certain that our universe is flat.

But this does not explain the exact shape of space. It could extend infinitely along all three spatial dimensions, or resemble a three-dimensional generalization of the surface of a donut, or take even wilder shapes. Flat space mathematics is surprisingly versatile and new research shakes up traditional thinking about the configuration of our cosmos.

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Triangles in the sky

Carl Friedrich Gaussa German astronomer who lived in the late 1700s and early 1800s, was one of the first mathematicians to study the geometry of curved spaces. He knew, for example, that the sum of the angles of a triangle in a plane is 180 degrees and that it is greater on a sphere. On spherical surfaces, such as that of the Earth, an equilateral triangle can, for example, consist of three right angles. Other geometries, such as the shape of a Pringles chip, may have angle sums less than 180 degrees.

The same principle applies not only to triangles on 2D surfaces but also in 3D space. Depending on the curvature of space, the sum of the angles can vary. Gauss may have considered the triangle a good starting point for studying the shape of the universe, although this is debated. He would have measured the distances between three German peaks. (Hohenhagen, Brocken and Inselberg) and determined their angles. His result: The sum was close enough to 180 degrees to suggest that there was a flat plane between the mountain peaks.

Unfortunately, although the triangle method is useful for thinking about the curvature of space, it will not answer the question of whether our universe is curved or flat. The cosmos is gigantic. Even if Gauss or another astronomer used a large telescope, triangulating the distances between stars would not work. The stars in ours or in neighboring galaxies are too close to us, relative to the vast scale of the universe. In addition, we must take into account that the objects observed are in motion and that, under the effect of gravity, the light reaching us follows partially curved trajectories.

But experts can use other tricks to infer the shape of our universe. For example, they look deep into the past, down to the oldest radiations, dating back about 13.8 billion years.

A brief history of the universe

We still don’t know exactly how our universe came to be. Fortunately precise details are not necessary to deduce its shape. Much can already be learned from the oldest light reaching us: the cosmic microwave background.

When our universe was very young, it was made of very hot and dense matter. The building blocks of atomic nuclei, quarks and gluons, floated freely in a sort of primordial soup. The medium was so dense that photons could not move freely through it.

As the universe expanded, it cooled; gradually, the first atomic nuclei and finally atoms were formed. As a result, the universe became transparent: photons could move freely. And it is this light, born around 370,000 years after the Big Bang, that we can observe.

The signal reaching us from this time is surprisingly evenly distributed across the sky, no matter where the detectors are pointed. This means that the material must have been distributed very evenly at this early stage. This observation leads to the cosmological principle: the universe must be homogeneous and isotropic. In other words, matter in the cosmos is distributed uniformly, the same in all directions. From Einstein’s equations of general relativity, it follows that the curvature of space is constant on a large scale.

This significantly restricts the possible geometry of the cosmos. If the curvature is constant, then three different cases can be distinguished:

1. No curvature: in this case you have Euclidean geometry, like on a flat surface.

2. Positive curvature: This corresponds to a spherical geometry, similar to that of a sphere.

3. Negative curvature: The geometry is hyperbolic, like a Pringles chip.

To determine which of the three cases occurs in the universe, we can still use cosmic microwave radiation. It’s almost seamless, but not quite: it contains tiny fluctuations that provide a clue to the geometry of the universe.

Small fluctuations in microwave radiation result from tiny density differences in the hot, bubbling primordial soup. And we can calculate the magnitude of these fluctuations at the beginning of the universe: the largest correspond to the greatest distance that density waves could travel.

These density fluctuations are also visible in our sky, particularly in the cosmic background. Their size depends on the geometry of the universe: if the universe is positively curved, density fluctuations should appear larger than they really are. With negative curvature, they should appear smaller. And without curvature, they must match the theoretical value exactly (in the same way that the angles of a triangle in flat space add up to 180 degrees). According to cosmologists’ measurements, the latter scenario applies to our universe.

The Universe is therefore flat, but to what extent?

Density fluctuation measurements, along with other cosmological data, suggest that our universe is flat. But that doesn’t mean we know the true shape of our universe.

Since curved 3D spaces are difficult to visualize, we can start with 2D examples. If our universe were 2D and flat, most people would imagine a flat surface. But it’s not the only 2D shape with flat geometry. Another example is the surface of a torus, which resembles a bagel or donut.

A bagel looks curved, but in a crucial sense, it’s not. You could, in theory, form a torus by taking a flat (and unusually stretchy) sheet of paper and gluing opposite sides together to create a cylinder. You can then twist this sheet so that the open ends of the cylinder meet, creating a hollow ring or torus.

In fact, there are three other variations of a two-dimensional flat space: a cylinder, a Möbius strip and a Klein bottle.

In three dimensions, the possibilities are even more diverse. In 1934, the mathematician Werner Nowacki demonstrated that there are 18 different flat 3D shapes. If our universe is truly flat, then it has one of these 18 shapes.

We can exclude certain candidates because eight out of 18 are “non-directable”. If you were to fly a rocket through a non-steerable universe, you would eventually return to where you started, but in mirror form: your right would now be your left, and vice versa. According to experts, such universes contradict the laws of physics.

This leaves 10 different shapes the universe can have:

1. An infinitely extended 3D space with x, y And z axes.

2. A 3D generalization of the torus: in this case, we can imagine sticking the opposite faces of a cube together.

3. A half-twisted torus: Same as #2, but a pair of surfaces are twisted 180 degrees, like a Möbius strip.

4. A quarter twist torus: Same as #2, but a pair of surfaces are joined by rotating them 90 degrees.

5. A third twist prism: instead of looking at the faces of a cube, you can also use a six-sided prism. Here, opposite faces are also glued together, but one face is rotated 120 degrees.

6. A sixth twist prism: same as #5, but one side is rotated 60 degrees.

7. A form called Hantzsche-Wendt collector which consists of two cubes stacked on top of each other, whose faces are connected together in a complex way.

8. A space made up of an infinity of flat planes that can be twisted relative to each other.

9. A space made up of an infinitely high “chimney”: four surfaces arranged like the sides of a parallelogram. Opposing surfaces are glued together.

10. Same as number 9, but one of the pairs of surfaces rotates 180 degrees.

All of these shapes share the same flat geometry but each has its own unique characteristics. Experts can therefore look for clues and evidence of these features to determine the precise shape of the universe using increasingly detailed cosmological data.

An infinite number of copies of ourselves

Many of these candidates for the shape of the universe are compact, meaning they don’t expand outward infinitely. Rather, one striking characteristic they share is repetition. In a torus-shaped universe, for example, light from our Earth would eventually reach Earth again, allowing us to see our reflection.

That said, our universe is gigantic and light travels at a finite speed. This means that even if light from our solar system or galaxy were to reach us again one day, we probably wouldn’t recognize the image. Indeed, its form at that time probably bore little resemblance to our current environment. Additionally, our cosmos might be so vast that light simply hasn’t had enough time to travel through it.

But there could be other clues if we live in a compact universe. The shape of the cosmos influences, in other things, the way matter and light interact in the early universe. And this should be reflected in the cosmic microwave background radiation. The researchers looked for repetitive structures within it, such as identical circular arrangements that would indicate a compact universe. To do this, they had to take into account some geometric considerations: because we receive microwave radiation on the spherical Earth, the signal has the shape of a spherical surface. Our universe could, however, have a more complex shape, and traces of this should be reflected in the spherical data we receive.

When experts looked for identical circular structures in cosmic microwave background radiation data during the 2000s and 2010s, they found nothing. Therefore, most cosmologists assumed that the universe had a fairly simple structure: it would be flat and extend infinitely in all three spatial dimensions. Research into the shape of the universe has stalled due to a lack of new evidence…until the launch of the Collaboration for Observations, Models and Predictions of Anomalies and Cosmic Topology (COMPACT) in 2022.

Researchers in the collaboration are comparing the latest data on cosmological microwave radiation with the different possible shapes of the universe. They discovered that the lack of evidence for identical circular structures in the cosmic microwave melts is much less restrictive than previously thought. In fact, it is entirely plausible that we would not identify any of these structures in a compact universe. Additionally, experts are working to identify other features of cosmological data that could indicate complex shapes for the universe. The COMPACT team always analyzes the data and develops suitable models. Exciting new results are expected in the months and years to come.

All this means that the universe could be much more complex than previously thought. And the question of the shape of our cosmos is not just academic. The topology of space-time was probably determined by quantum processes that occurred shortly after the Big Bang. Therefore, if we knew the shape of the universe more precisely, we could learn more about the complex processes at its beginning – or so we hope.

This article was originally published in spectrum of science and has been reproduced with permission. It was translated from the original German version with the help of artificial intelligence and reviewed by our editors..