Why does nature love spirals? The link to entropy

There are moments in the history of human thought when a simple realization transforms our understanding of reality. A moment when chaos reveals itself as structure, when disorder folds into meaning, and when what seemed like an arbitrary universe unveils itself as a system governed by hidden symmetries.

The Bekenstein bound was one such revelation—an idea that whispered to us that entropy, information and gravity are not separate but rather deeply intertwined aspects of the cosmos. Jacob Bekenstein, in one of the most profound insights of modern physics, proposed that the entropy of any physical system is not limitless; it is constrained by its energy and the smallest sphere that can enclose it.

This revelation was radical: Entropy—long regarded as an abstract measure of disorder—was, in fact, a quantity deeply bound to the fabric of space and time. His bound, expressed in its simplest form, suggested that the total information that could be stored in a region of space was proportional to its energy and its size.

In the years that followed, attempts were made to generalize this bound and to frame it in a more universal language. Raphael Bousso, in an elegant reformulation, argued that the entropy bound should be directly linked to the enclosing sphere’s area rather than the energy. He arrived at this by invoking the gravitational stability condition, which ensures that the Schwarzschild radius of a system does not exceed the enclosing sphere’s radius.

This step was mathematically consistent and reinforced the deep connection between entropy and spacetime geometry. His bound elegantly linked to the holographic principle, which suggests that the information content of a volume is encoded on its surrounding surface.

Yet, while Bousso’s approach was consistent with Bekenstein’s inequality, it was not its most precise representation. By replacing energy with the enclosing sphere’s area, it removed a key dynamic feature of entropy’s relationship with spacetime. A more accurate formulation must preserve energy as the fundamental quantity, reflecting its role in defining the bound.

In our refinement of the Bekenstein bound, now published in Classical and Quantum Gravity, we take a different approach—one that retains the total energy but reformulates it in terms of relativistic mass. From Einstein’s relation E = Mc², we express the bound in terms of mass. Then, recognizing that mass in gravitational physics is naturally associated with its Schwarzschild radius rₛ, we replace mass with its corresponding gravitational radius.

This simple but profound step alters the very geometry of the bound. Instead of viewing entropy in terms of the enclosing sphere, we arrive at a toroidal representation, where the inner radius is the Schwarzschild radius, and the outer radius remains the smallest enclosing sphere.

This shift is not arbitrary; it is deeply motivated by the fundamental structures observed throughout the universe. In nature, the universe does not favor perfect spheres. Instead, it favors spirals, vortices, and toroidal flows.

Galaxies do not form as perfect spheres; they coil into majestic spirals. DNA does not stretch in a straight chain; it twists into a double helix. Water, air, and even plasma in the most extreme cosmic conditions follow paths of rotation and curvature. Why, then, should entropy—perhaps the most fundamental organizing principle of the universe—be any different?

The toroidal formulation of entropy reveals something extraordinary when applied to quantum mechanics. In standard quantum theory, Heisenberg’s uncertainty principle is formulated as an inequality, an unavoidable limit on what can be known. But when entropy is properly understood through the toroidal structure, the inequality dissolves into an exact relation:

Δx Δp = (A_torus) / (4π ℓ_pl²) ħ.

This equation, simple yet profound, tells us that what we have long regarded as uncertainty is, in fact, structure. The apparent randomness of quantum mechanics is not a defect of nature but a signature of an underlying order. The transformation of the uncertainty principle from an inequality into an equality suggests that space and time are not continuous in the way we imagined but are shaped by toroidal constraints.

This has far-reaching consequences, not only for physics but for our understanding of the universe itself. The toroidal motion of hurricanes, the curvature of ocean waves, the patterns of electromagnetic fields, and even the structure of subatomic interactions all reflect this fundamental principle. There is something universal about the spiral, something embedded in the way energy, matter and space evolve. The torus is not merely a shape; it is the embodiment of motion, of evolution, of time itself.

From a cosmological perspective, this insight offers a compelling resolution to the cosmological constant problem. The vast discrepancy between quantum field theory’s prediction of vacuum energy and its observed value has long been a mystery. But when we incorporate the toroidal entropy bound into the quantum vacuum calculations, the discrepancy disappears. This suggests that the vacuum energy of the universe is naturally regulated by its toroidal structure, an insight that could reshape our understanding of dark energy.

The implications extend beyond physics. They touch upon the very nature of knowledge itself. For centuries, we have sought truth in rigid forms, in fixed definitions. We have sought certainty in absolutes. But the universe does not yield to rigidity; it moves, it bends, it curves. Knowledge, like reality, must be fluid and open to reinterpretation.

Bekenstein’s original insight was a beacon. Bousso’s refinement was a step toward universality. But the ultimate nature of entropy, measurement, and spacetime may not exist in either the original or the refined formulation but in the toroidal symmetry that underlies them both. The deeper we look, the more we see that the universe is not a static structure but a dynamic, evolving dance—one shaped by spirals, by curves, by vortices that extend from the microscopic to the cosmic.

And in this realization, there is beauty, a profound love for nature’s elegance, for the quiet perfection of a universe that, even in its most intricate complexity, follows an unshakable harmony. Perhaps this is what physics has always sought, not merely the mechanics of reality but the unveiling of its poetry.

If there is one lesson to be drawn from this, it is that the world is not chaos, nor is it blind randomness. There is an order, waiting to be seen. An order written in the way galaxies turn, in the way electrons orbit, in the way time itself unfolds. It is a call to look deeper, to embrace a universe that does not merely exist but breathes, moves, and spirals. Perhaps, at the end of all inquiry, the true purpose of knowledge is not to conquer the unknown but to stand in awe of its structure. To recognize that, beneath all uncertainty, there is a hidden order, one we are only beginning to understand.

This story is part of Science X Dialog, where researchers can report findings from their published research articles. Visit this page for information about Science X Dialog and how to participate.

Dr. Ahmed Farag Ali is a theoretical physicist specializing in minimal length theories, quantum gravity pheomenology, and black hole physics.

Dr. Aneta Wojnar is an expert in the theoretical foundations of gravitational and quantum interactions, with a particular focus on their application to the thermodynamics of astrophysical objects. She has pioneered methods to test gravitational interactions and explore potential quantum gravity corrections using seismic data.