Beyond Flatland: Exploring the Possible Shapes of the Universe

Preksha Sanjay Madhva, ROBOTICS ENGINEER

5 minute read

The universe, vast and enigmatic, continues to challenge our understanding of reality. While we can observe its expansion and the distribution of galaxies, its overall shape remains one of cosmology's most profound mysteries. Is it flat like a pancake, curved like a sphere, or something far more exotic? Perhaps the universe is twisted like a pretzel or a Mobius strip? The answer, if we can find it, holds the key to the ultimate fate of everything.

Cosmologists have developed various theories about the universe's geometry, each with its own implications for the universe’s past, present, and future. The universe's shape isn't just a fun thought experiment; it's tied to fundamental physics, like Einstein's theory of general relativity which links the universe's geometry to its matter and energy content. The geometry of the cosmos, often described as its "shape," dictates the ultimate fate and experience of a hypothetical intergalactic traveler. So let's take a look at the main contenders by taking an adventure across the universe.

The Flat Universe or “The Universe with No Edge” Idea:

Imagine a flat surface, except you keep the three dimensions so it is not a single 2D plane. Despite the slight misnomer of a name, that's essentially what a flat universe would be like. In this scenario, familiar rules of geometry apply where straight lines stay straight and triangles add up to 180 degrees. In a flat universe, our voyager party would embark on an unending journey, akin to traversing an infinite plane. Their path, however straight, would never curve back upon itself, and parallel paths would remain eternally so, stretching into the vastness without convergence or divergence and never returning to the origin of their journey. This implies that the universe likely stretches on endlessly, possessing infinite extent.

Visualization of the geometry of a flat universe. Source: NASA / WMAP Science Team

Visualization of a right-handed three-dimensional Cartesian coordinate system. Source: Wikipedia

Observations of the cosmic microwave background (CMB), the afterglow of the Big Bang, strongly suggest that the universe is remarkably close to flat. The Wilkinson Microwave Anisotropy Probe (WMAP) and Planck satellite data have provided detailed maps of the CMB, showing that its fluctuations are consistent with a flat geometry. Flatness necessitates a precise balance of matter and energy, known as critical density. Current observations suggest that the expansion of the universe is accelerating, likely leading to an eternal expansion and a "Big Freeze." Since a flat universe only requires Euclidean geometry, it also simplifies some cosmological calculations and models, such as the standard model of cosmology - the Lambda Cold Dark Matter model.

The Closed Universe or “The Giant Beach Ball” Theory:

A closed universe is like the surface of a sphere – positively curved. In this case, straight lines eventually meet, i.e. parallel lines eventually converge, and the angles of a triangle add up to more than 180 degrees. Thus, a closed universe with the topology of a sphere would lead our traveler on a grand, looping expedition. No matter the initial direction, their path would inevitably curve, bringing them full circle back to their point of origin, a testament to the finite, yet unbounded nature of this cosmic sphere.

Visualization of the geometry of a closed universe. Source: NASA / WMAP Science Team

A closed universe would be finite in size, possessing a definite volume. Its ultimate fate would likely be a "Big Crunch," where the universe collapses. A closed universe requires a higher-than-critical density of matter and energy, and parallel light rays would eventually converge. While the closed universe concept is less favored by current data, which supports a flat universe, it's not entirely off the table as a slightly closed universe as such a model would explain anomalies such as the CMB light being bent more than expected (enhanced lensing) if we used the flat universe model.

The Open Universe or “The Saddle-Shaped” Concept:

On the flip side, an open universe is negatively curved, like a saddle or a Pringles chip. In this geometry, parallel lines all diverge, and the angles of a triangle add up to less than 180 degrees. Thus, an open universe, with its hyperbolic, negatively curved geometry, would present a contrasting scenario. Here, the traveler would also venture into infinity, but their path would diverge, with parallel lines spreading further and further apart, creating an ever-expanding gulf between them.

Visualization of the geometry of a saddle universe. Source: NASA / WMAP Science Team

Current observations do not strongly support an open universe, however, some models like “bubble nucleation” suggest an end result of an open universe. Like a flat universe, an open universe is predicted to possess an infinite extent. It would expand forever, requiring a lower-than-critical density of matter and energy. Parallel light rays would diverge. In some versions, depending on the behavior of dark matter, the universe could end in a "Big Rip," where everything is torn apart by the accelerating expansion.

The Torus-Shaped Universe or “The Cosmic Donut”:

Now, for something a bit more unusual: the torus-shaped universe. Picture a donut. This model suggests the universe has a finite volume but no edges, and if you traveled far enough in a straight line, you'd end up back where you started. Thus this model proposes that the universe has the topology of a three-dimensional torus, or a "donut" shape. Our travelers would end up in a unique situation. By venturing far enough in any direction, they would find themselves reappearing from the opposite side, a consequence of the universe's donut-like topology. The traveler could travel around the edge of the donut and arrive at the origin, or take a shortcut by journeying in the direction of the hole till they loop back to the starting point. This journey, while seemingly paradoxical, highlights the intricate and often counterintuitive nature of cosmic geometry, where the very fabric of spacetime produces a kind of cosmic “hall-of-mirrors-effect” that dictates the traveler's ultimate destination.

Such a universe cannot really be visualized like the image above, as donuts have two-dimensional surfaces, and the proposed toroidal universe would be curved not just through space, but through spacetime. However, it is simpler, and also rather fun, to visualize our universe as a sort of cosmic donut.

This universe would be finite and boundless, possessing a limited volume but no edges. A key characteristic is the "wraparound" effect, creating repeating patterns. Observers could see multiple images of the same objects like multiple galaxies, as light would have traveled through different paths around the torus. The torus shape would imprint specific patterns on the CMB, and if proven, would place constraints on the minimum size of the universe.

General Implications and The Ongoing Quest for Discovery:

While basic Euclidean geometries are frequently discussed, the universe's true form may encompass more complex topologies. The Multiconnected Universe theory or the “Multiverse Theory”, for instance, postulates a finite universe that appears infinite due to "wraparound" effects and other theories explore non-trivial topologies involving spacetime curvature. The universe's geometry is intrinsically linked to the nature and distribution of dark matter and dark energy, providing valuable clues regarding the fundamental laws of physics, particularly the interplay between gravity and spacetime. Establishing the universe's shape is a primary objective of observational cosmology, driving the development of increasingly sophisticated observational techniques. This geometry influences the evolutionary trajectory of the universe, including galaxy formation and its ultimate fate, and prompts profound philosophical inquiries concerning the nature of reality and our place within the cosmos. Cosmologists are continually refining measurements of the Cosmic Microwave Background and galaxy distribution, anticipating crucial insights from future advanced telescopes and gravitational wave detectors.

Unraveling the universe's geometry transcends mere abstract consideration; it is essential for understanding the fundamental forces that govern the cosmos. Through the refinement of observational techniques, the utilization of advanced telescopes, and the anticipation of insights from gravitational wave detection, cosmologists are progressively approaching a comprehensive understanding of the universe's grand design. This pursuit ultimately seeks to shine a light upon the fundamental nature of reality and our position within it. As our tools gradually improve, we're slowly but surely getting closer to unraveling this cosmic mystery. It’s definitely worth solving!

Recommended column to read next: Space Questions for the Everyday - Volume I