How Do Patterns in Nature Relate to Mathematical Laws?
How Do Patterns in Nature Relate to Mathematical Laws?*
How many ways do you know to draw a circle? I can immediately think of the two simplest ones. We take a compass, stick the blade in the centre of the future figure, then determine the radius and rotate. The second method is to choose a suitable object of round shape and size, for example, a coffee cup, place it on a sheet of paper and outline it with a pencil. But in order to do any of the above, a few simple starting conditions are needed: someone before us must have created round objects, or someone must have created the science of geometric figures and discovered the mathematical relationships that describe the circle.
In both cases, the path goes through an initial acquaintance with the world around us, its refraction through the prism of our consciousness, and the generation of ideas that are subsequently recorded as mathematical formulas. But then a question arises: Why can natural patterns be described by mathematics at all? The most obvious answer is that nature, as a result of billions of years of evolutionary development, has developed these mathematical relationships that are most effective or consistent with the world around us. When it comes to inanimate objects, for example, river water flows along the path of least resistance, and this forms river beds. And in terms of living nature, the emphasis is on the adaptability of a given species to survive in a hostile environment or under conditions of scarce food resources, which will give it the highest chance of creating a generation. Nature solves mathematical optimisation problems through variation and selection.
But the strange thing is that the same mathematics keeps appearing across completely unrelated domains. I was very intrigued to read that a team from Northeastern University recently showed that the branching geometry of neurons, blood vessels, trees, and corals follows predictions made not by biology, but by the mathematical toolkit of string theory (Meng et al., 2026). The researchers are careful to say this does not mean the brain is quantum. It simply means that abstract mathematics, developed for one purpose, turned out to describe something else entirely — as if the mathematics had been waiting for it. In this example, the mathematical ideas have arisen purely abstractly and only later turned out to describe the nature with remarkable accuracy (Wigner, 1960).
Today, many scientists believe that natural patterns are fully covered by mathematical laws, and some researchers take the next step forward and believe that our physical reality is not merely described by mathematics but is itself a mathematical structure (Tegmark, 2014).
Theoretically, absolutely every part of the surrounding world can be mathematically, or rather statistically, described. This has not yet been fully completed for two reasons: first, we have not yet discovered all the mathematical rules that describe every natural phenomenon, and second, most natural patterns are so complex and have so many variables that it is impossible to cover them with existing computing machines.
To understand the complexity of the problem, I will point to the following well-known study, conducted at the University of Cambridge in 2016 (Martiniani et al., 2016), in which scientists considered a system of 128 soft spheres and calculated how many different stable arrangements are possible. The resulting number is about 10250. Thus, the possible configurations of tennis balls in just one basket exceed the number of all the particles in the Universe! Even if all the supercomputers on Earth worked on this task alone, they would probably need more time to solve it than has elapsed since the Big Bang. Now let’s imagine a much more complex system, such as weather, drafts, winds or the population of insects on Earth!
As a result, a deeper question arises: Why does nature follow mathematical laws? And the answer may lie in one property that connects the very large and the very small bodies. If we look more deeply and carefully at the laws that govern huge objects like galaxies and stars on the one hand and micro-objects like quantum particles on the other, we will notice one fundamental similarity: the laws of nature are invariant. This invariance (at least as far as we know) is simultaneously spatial, temporal, and (to some extent) size-independent. But invariance is not merely a passive description of nature – it actively generates its laws. Noether’s theorem shows that every symmetry corresponds to a conservation law: time symmetry gives us conservation of energy, spatial symmetry gives us conservation of momentum (Noether, 1918). In other words, the laws of nature do not simply happen to be mathematical – they are mathematical because they are symmetric.
And the same principle operates at every scale – simultaneously in the microcosm and macrocosm. To begin our exploration, let’s take outer space as an example. Measurements show that it has an almost ideal flat geometry with a density parameter Ω≈1.0007 ± 0.0019, which means that the Universe, at least in its visible part, is extremely close to being flat, and suggests large-scale geometric simplicity. Let’s also look at the microworld. The wave function of almost all quantum particles is extremely accurately described by the equation of Erwin Schrödinger, which suggests that at the microlevel we have symmetry too (Schrödinger, 1926). The wave function describes the particle not by a position in space, but by a mathematical formula in an abstract Hilbert space that has nothing to do with the geometry of the physical world we live in. Physicists have debated for more than a century whether it is a model of reality or reality itself, and the fact that the question remains open is itself telling. Moreover, the entire Standard Model of particle physics is built not by enumerating the observed forces, but by requiring that the equations remain symmetrical under certain transformations. The forces themselves arise from the requirement of symmetry. And here is the twist in the scientific approach: instead of mathematics describing nature, as it has been for centuries, we now know that the requirements of symmetry effectively write the laws of nature from within.
This is the invariance I referred to earlier, and it makes the mathematical description of nature possible, because mathematics seeks to discover what remains unchanged behind the apparent variety of forms.
To make the complexity even greater, we must also consider the role that information plays in physics. Quantum mechanics, since the time of Stephen Hawking, tells us that information cannot be destroyed (Hawking, 1976). This is so because the equations of quantum mechanics are reversible. That is, just as the current quantum state can influence the future, so too can the future quantum state uniquely determine the past. However, this would not be possible if the information contained in that state were destroyed. According to the holographic principle, the maximum information in a volume of space is determined not by the volume itself, but by its boundary (Susskind, 1995). This means that the structure of space itself can arise from quantum information. At some point, the boundary between pure mathematics, quantum mechanics, and natural models seems to completely disappear.
If precise mathematical relationships between information, energy, matter, and space-time genuinely exist in the world around us, then nature and reality itself may be a quantum information process. This is increasingly starting to look like a matrix, and for us, who are inside our space-time, it should be difficult or rather impossible to determine what exactly reality is.
Such a prospect is both inspiring and frightening, because it brings up the question: how do we humans actually differ from the mathematical or real matrix that surrounds us? Since it is known that our logic and thinking are highly structured statistically, this would mean that to some extent it is possible to describe them mathematically. Then what would be the difference between our thinking as a natural phenomenon and the artificial intelligence as a product of abstract mathematics? I personally hope that there is something that distinguishes us.
If we return to the initial question – how to draw a circle – we can imagine how a machine would approach it. It would probably apply the formula we know from school, calculate the coordinates of the points and generate an accurate circle – geometrically perfect and completely devoid of meaning.
At the same time, Leonardo da Vinci has a different approach that no artificial intelligence would. By inscribing the human body within the shapes of the circle and the square, he illustrates the insight that the dimensions of the individual limbs of a human being follow mathematical laws, while at the same time all human beings are composed of the same proportions and differ from each other only in their variation therein.
Perhaps this is the most intriguing pattern of all: a universe governed by equations has produced minds capable of asking why those equations exist at all.
References
Hawking, S. (1976) ‘Breakdown of Predictability in Gravitational Collapse’, Physical Review D, 14(10), pp. 2460–2473.
Martiniani, S., Schrenk, K.J., Stevenson, J.D., Wales, D.J. and Frenkel, D. (2016) ‘Turning intractable counting into sampling: computing the configurational entropy of three-dimensional jammed packings’, Physical Review E, 93(1), p. 012906.
Meng, X., Piazza, B., Both, C. et al. (2026) ‘Surface optimization governs the local design of physical networks’, Nature, 649, pp. 315–322.
Noether, E. (1918) ‘Invariante Variationsprobleme’, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, pp. 235–257.
Schrödinger, E. (1926) ‘Quantisierung als Eigenwertproblem’, Annalen der Physik, 79, pp. 361–376.
Susskind, L. (1995) ‘The World as a Hologram’, Journal of Mathematical Physics, 36(11), pp. 6377–6396.
Tegmark, M. (2014) Our Mathematical Universe. New York: Knopf.
Wigner, E. (1960) ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, Communications on Pure and Applied Mathematics, 13(1), pp. 1–14.
*The essay was written for The Math Minds Underground Essay Competition 2026