The universe could have 18 possible shapes

The question of the universe’s shape is much more captivating and unresolved than the controversy surrounding the shape of our planet, despite flat-Earth claims.

We inhabit a minuscule part of an immense cosmos, limiting our perspective. Yet, cosmologists are now quite confident that the universe is flat.

However, this doesn’t clarify the precise shape of space. It might stretch infinitely across three spatial dimensions, resemble a three-dimensional donut surface, or assume even more exotic forms. The mathematics behind flat space is incredibly flexible, and recent studies are challenging conventional views about the universe’s structure.

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Triangles in the Sky

Carl Friedrich Gauss, a German astronomer from the late 1700s and early 1800s, was among the first to explore geometry in curved spaces. He understood that in a plane, the angles of a triangle total 180 degrees, but on a sphere, they exceed this. For example, an equilateral triangle on Earth’s spherical surface can have three right angles. Other geometries, like a Pringles chip, can have angle sums less than 180 degrees.

This concept applies not only to 2D triangles but also in 3D space. The sum of angles varies with the curvature of space. Gauss might have considered triangles a useful starting point for exploring the universe’s shape, though this is debated. He reportedly measured the distances between three German mountain peaks (Hohenhagen, Brocken, and Inselberg) to determine their angles. His findings suggested a flat plane between the peaks, as the angle sum was nearly 180 degrees.

Although the triangle method helps conceptualize space curvature, it doesn’t resolve whether our universe is curved or flat. The cosmos is vast, and measuring distances between stars is impractical due to their relative proximity and the universe’s expanse. Additionally, objects’ motion and gravity-induced light curvature complicate measurements.

However, scientists use other methods to infer the universe’s shape, such as examining ancient radiation dating back approximately 13.8 billion years.

A Brief History of the Universe

The exact origin of our universe remains unknown, but understanding its shape doesn’t require precise details. The cosmic microwave background, the oldest light reaching us, offers valuable insights.

In its infancy, the universe was composed of extremely hot, dense matter. Quarks and gluons, the building blocks of atomic nuclei, floated freely in a primordial soup so dense that photons couldn’t move freely.

As the universe expanded and cooled, atomic nuclei and atoms formed, rendering the universe transparent and allowing photons to travel unimpeded. This light, emitted about 370,000 years after the big bang, is observable today.

The cosmic microwave background is remarkably uniform across the sky, indicating that matter was evenly distributed in the early universe. This supports the cosmological principle: the universe is homogeneous and isotropic, meaning matter is evenly distributed in all directions. Einstein’s general relativity equations suggest that space curvature is constant on large scales.

This limits potential cosmic geometries to three possibilities:

1. No curvature: a Euclidean, flat geometry.

2. Positive curvature: a spherical geometry, like a sphere.

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

To identify the universe’s geometry, scientists use cosmic microwave radiation, which is nearly homogeneous but contains tiny fluctuations that hint at the universe’s shape.

These small fluctuations arise from density differences in the primordial soup. Calculating their early strength reveals the maximum distance density waves could travel.

Density fluctuations are visible in the cosmic background. Their apparent size depends on the universe’s geometry: positive curvature makes them appear larger, negative curvature smaller, and no curvature matches theoretical values (like a triangle’s angles summing to 180 degrees in flat space). Cosmologists’ measurements support the latter scenario for our universe.

So the Universe Is Flat—But How Flat?

Density fluctuation data and other cosmological evidence indicate a flat universe, but the true shape remains uncertain.

Visualizing curved 3D spaces is challenging, so we begin with 2D examples. A 2D flat universe might be imagined as a flat surface, but it could also resemble a torus, like a bagel or donut.

A bagel appears curved, but it’s not in an essential sense. Theoretically, a torus can be formed by taking a flat, stretchy sheet of paper, gluing opposite sides to create a cylinder, and then twisting it into a ring.

Three other variations of flat 2D space exist: a cylinder, a Möbius strip, and a Klein bottle.

In three dimensions, the possibilities multiply. In 1934, mathematician Werner Nowacki proved there are 18 different flat 3D shapes. If our universe is flat, it must have one of these shapes.

Some candidates can be eliminated because eight of the 18 are “nonorientable.” In a nonorientable universe, a rocket would eventually return to its starting point in a mirrored form: right becomes left, and vice versa. Experts say such universes violate physics laws.

This leaves 10 potential shapes for the universe:

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

2. A 3D torus: imagine gluing together the opposite faces of a cube.

3. A half-twist torus: similar to #2, but one pair of surfaces is twisted 180 degrees, like a Möbius strip.

4. A quarter-twist torus: like #2, but surfaces are joined with a 90-degree twist.

5. A third-twist prism: using a six-sided prism instead of a cube, with opposite faces glued and one face rotated 120 degrees.

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

7. A shape called a Hantzsche-Wendt manifold, consisting of two cubes stacked with complex face connections.

8. A space with infinitely many flat planes twisted relative to each other.

9. An infinitely tall “chimney”: four surfaces form a parallelogram, with opposite surfaces glued together.

10. Like #9, but one pair of surfaces is rotated 180 degrees.

Though these shapes share flat geometry, each has unique traits. Experts can search for clues to identify the universe’s precise shape using detailed cosmological data.

Infinitely Many Copies of Ourselves

Many possible universe shapes are compact, meaning they don’t extend infinitely. They share a characteristic of repetition. In a torus-shaped universe, light from Earth would eventually loop back to Earth, creating a reflection.

However, given the universe’s vastness and light’s finite speed, any returning light from our solar system or galaxy would likely be unrecognizable due to changes over time. The cosmos might be so vast that light hasn’t had time to complete the journey.

Clues to a compact universe could exist. The cosmos’s shape affects how matter and light interacted in the early universe, reflected in the cosmic microwave background radiation. Researchers have searched for repeating structures within it, like identical circular patterns, indicating a compact universe. They had to consider geometry: the radiation we receive on spherical Earth forms a spherical surface, but the universe could have a more complex shape, traceable in the data.

During the 2000s and 2010s, experts found no identical circular structures in cosmic microwave background data. As a result, most cosmologists concluded the universe likely has a simple structure: flat and infinitely extending in three dimensions. Research on the universe’s shape stalled due to a lack of new evidence until the Collaboration for Observations, Models, and Predictions of Anomalies and Cosmic Topology (COMPACT) launched in 2022.

The collaboration compares the latest cosmic microwave background data with various potential universe shapes. They’ve found the absence of identical circular structures in the background isn’t as restrictive as once thought. It might not be possible to identify these structures in a compact universe. Additionally, experts are exploring other cosmological data features that suggest complex universe shapes. The COMPACT team continues to analyze the data and develop models, with promising results expected in the coming months and years.

This suggests the universe could be more intricate than previously believed. Understanding the cosmos’s shape is not just academic; the topology of spacetime was likely set by quantum processes shortly after the big bang. Knowing the universe’s shape could reveal more about these early processes—or so scientists hope.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.