The narrative of Edwin Abbott's classic science fiction tale, Flatland, comes to life in contemporary physics. The story features a two-dimensional character, “A Square,” who discovers he exists within a three-dimensional world called Spaceland. Much like Mr. Square, physicists have begun to ponder whether our own universe is merely a fraction of a larger four- or ten-dimensional space. If we were to transcend into this higher-dimensional reality, we could manipulate our surroundings in extraordinary ways—such as reaching into locked containers or observing the intricate details of a human body. This perspective could potentially unveil the profound unity of the natural world.

However, recent developments suggest an unexpected twist in the dimensional narrative: space might actually have fewer dimensions than we perceive.

This notion arises from issues related to gravity, one of the fundamental forces of nature. Gravity's attractive force between two masses intensifies as they draw nearer. While this holds true for other forces like electromagnetism, gravity exhibits an unusual amplification at minuscule scales. Its strength is influenced by the masses involved and the quantum uncertainty principle, which imposes a minimum energy threshold that escalates inversely with distance. As two objects come closer, this minimum energy increases, resulting in a heightened gravitational pull. Other forces also experience quantum fluctuations, but to a lesser degree. Thus, gravity, which appears weak in everyday scenarios, parallels the strength of other forces at a minuscule scale known as the Planck scale, approximately a few trillionths of a trillionth of a meter.

At the Planck scale, gravity’s strength stabilizes, blurring the distinction between different scales. Everything appears uniform.

Conventional gravitational theory suggests that at this scale, gravity becomes so potent that matter could collapse into black holes. Given that gravity shapes spacetime, its unchecked intensification implies that the seamless continuum we experience in daily life cannot be a fundamental aspect of reality. This leads to two possibilities: either a different structure must replace it, or gravity must be constrained from escalating at tiny scales, preserving the significance of spacetime—or potentially both. Whichever the case, it indicates that our current understanding is incomplete.

The challenge of understanding the microscopic structure of spacetime has perplexed scientists since Einstein. If you gather three theorists in a room, you might hear four theories regarding its nature: a grid of discrete cells, a fluctuating foam, vibrating strings, or intertwined loops, among others.

Interestingly, theorists are discovering that many of these varied hypotheses share a commonality: at the finest scales, space appears to lose all but one dimension, effectively controlling gravity. Indeed, a one-dimensional space is the only scenario where gravity does not amplify as distance decreases.

Whether this reduction in dimensionality preserves spacetime as a fundamental aspect of nature or signals its eventual dissolution remains uncertain. Nonetheless, as theoretical physicist Frank Saueressig from Radboud University asserts, “two dimensions [one spatial and one temporal] seem to be the natural dimensions where gravity prefers to reside.”

In the context of Flatland, we might find ourselves not in Spaceland, but in Lineland.

The enigmatic nature of spatial dimensions has tangible implications, influencing how object sizes, event durations, and force strengths vary with scale. In standard three-dimensional space, doubling a sphere's radius results in an eightfold increase in volume. In four dimensions, this would increase sixteenfold; in two dimensions, only fourfold.

Consequently, by examining how volumes scale, one could infer the number of dimensions space possesses. If the scaling relationship shifts—such that doubling the sphere's radius no longer leads to an eightfold volume increase—it suggests that the dimensionality of space is not constant.

This is the hypothesis many physicists are exploring in relation to gravity.

Their speculation was spurred by a series of computer simulations of Planck-scale spacetime conducted in the mid-2000s. Renate Loll and her team at Radboud University developed an algorithm termed causal dynamical triangulations to incorporate quantum characteristics into their simulated spacetimes. Their findings indicated that space, at observable scales, appears to maintain three dimensions, as we experience. However, if gravity's strength fluctuates, space could transition into a tree-like structure with fewer dimensions or a crumpled configuration with more dimensions, akin to space freezing or boiling.

Even when space appears reassuringly three-dimensional, it may be misleading. The researchers likened a random walk through space to a gas molecule bouncing around a volume. In three-dimensional space, if you double your journey’s length, the chance of returning to your starting point diminishes by the square root of eight. Yet, their simulations revealed a slower decline near the Planck scale, suggesting there is less room to get lost—a sign of reduced dimensionality.

By volume, space is three-dimensional, but through the lens of random motion, it behaves as if it is one-dimensional or even exhibits fractional dimensions. This discrepancy signals that space becomes less accommodating, indicating a fundamental change in its nature as gravity reveals its quantum characteristics. “When normal manifolds are absent, multiple definitions of ‘dimension’ typically arise,” notes Steve Carlip from the University of California, Davis.

The spacetime continuum may indeed be fragmented, indicating that the smooth expanse we perceive in our daily lives cannot represent a fundamental reality.

Causal dynamical triangulations, like computer simulations in any scientific field, operate as a black box: they receive equations as input and produce numerical output, but do not elucidate their internal workings. “What it actually means remains a mystery,” Loll admits. The theory that closely aligns with these dimensional transformations is known as asymptotic safety, a lesser-known candidate for quantum gravity originating in the late 1970s. This theory posits that gravity possesses an intrinsic regulator, making it “safe.” Unique characteristics of gravity that remain undetected in everyday life emerge near the Planck scale, ensuring that the force reaches a finite maximum strength instead of escalating indefinitely. Many particle physicists favor asymptotic safety because it suggests gravity behaves similarly to forces governing atomic nuclei, which are known to diminish at short distances.

According to asymptotic safety, space reduces its dimensions not due to flattening like a crushed can, but rather because quantum fluctuations alter the standard rules governing particle movement. For instance, a particle’s random trajectory might retain a memory of its path, evading previously traversed areas—similar to the '70s video game Snake, where players navigate a dot while avoiding its own trail. Saueressig suggests that this constraint could increase the probability of returning to the starting point, creating the impression of traversing fewer dimensions.

As if eliminating dimensions weren’t radical enough, asymptotic safety implies an even more profound transformation of space. If gravity stabilizes at small scales, preventing the emergence of Planck-scale black holes, then the spacetime continuum remains intact, contrary to standard theory's predictions. It becomes genuinely continuous, allowing limitless zooming. However, upon zooming in, one becomes disoriented. With gravity—and possibly other forces—no longer intensifying, scales appear indistinguishable. Each segment resembles the entirety, embodying the essence of a fractal structure.

Martin Reuter, a leading figure in asymptotic safety at the University of Mainz, likens this universe to a turbulent fluid, with eddies nested within each other. Distance loses its fixed significance, as the length between points is contingent on the zoom level, similar to how the coastline of Britain grows longer with more detailed measurements.

While one-dimensional space appears to rescue gravity from its own escalation, this alone is insufficient to persuade physicists that space genuinely sheds dimensions at smaller scales. The primary criterion for any theory is its internal consistency. Advocates of asymptotic safety and other dimensional reduction theories openly acknowledge their current inability to demonstrate this.

Some critics assert that asymptotic safety may not function, particularly when addressing black hole physics. The complexity of these cosmic phenomena scales with their external surface area, whereas complexity in any continuum-based theory, including asymptotic safety, should scale with internal volume. Proponents counter that these scaling relationships could shift if spatial dimensions change, potentially resolving these discrepancies.

Even if asymptotic safety does not emerge as the ultimate theory, it may still serve as a stepping stone toward a hypothetical theory where space arises from a fundamentally spaceless level. “Between ‘no spacetime’ and ‘normal spacetime,’ there may exist ‘unusual spacetime,’” remarks Daniele Oriti from the Max Planck Institute for Gravitational Physics in Potsdam, Germany.

Laboratory experiments are unlikely to resolve this debate in the near future—or indeed much of anything regarding quantum gravity, given that the Planck scale is far smaller than current instruments can analyze. Observations of the cosmos may yield better insights. Reuter notes that the early universe appeared scale-invariant, with density variations of all sizes exhibiting similar magnitudes. This observation supports the asymptotic safety theory and a spacetime that is a fundamental aspect of nature. However, alternative theories provide different explanations for scale invariance.

Even if all theories proposing dimensional loss are ultimately proven incorrect, they might still hint at a profound characteristic of quantum gravity. “This is part of what makes it so intriguing,” Carlip notes. “It may reveal something fundamental about quantum gravity that these diverse approaches have in common, though we do not yet comprehend what that is.”

In the preface to the 1884 edition of Flatland, Abbott, through A Square, lamented how we are “all alike the Slaves of our respective Dimensional prejudices.” Today, physicists are learning to shed their own dimensional biases. After all, as Mr. Square observes, “one touch of Nature makes all worlds akin.”

George Musser is a writer specializing in physics and cosmology, author of The Complete Idiot’s Guide to String Theory and Spooky Action at a Distance (forthcoming from Farrar, Straus and Giroux). He served as a senior editor at Scientific American for 14 years and has received the American Institute of Physics Science Writing Award, among other accolades.