The beauty of numbers. Mathematical constants in nature

3D model of the endoplasmic reticulum of a eukaryotic cell with a Terassaki ramp that connects the flat sheets of the membrane

In 2013, a group of molecular biologists from the United States investigated a very interesting form of the endoplasmic reticulum - an organoid inside a eukaryotic cell. The membrane of this organoid consists of flat sheets connected by spiral “ramps”, as if calculated in a 3D modeling program. These are the so-called Terazaki ramps. Three years later, astrophysicists noticed the work of biologists. They were amazed: after all, exactly such structures are present inside neutron stars. The so-called "nuclear paste" consists of parallel sheets connected by spiral forms.

The amazing structural similarity of living cells and neutron stars - where did it come from? It is obvious that there is no direct connection between living cells and neutron stars. Just a coincidence?


Model of spiral junctions between flat sheets of the membrane in a eukaryotic cell
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There is an assumption that the laws of nature act on all objects of the micro and macrocosm in such a way that some of the most optimal forms and configurations appear as if by themselves. In other words, the objects of the physical world obey the hidden mathematical laws underlying the entire universe.

Let's look at some more examples that confirm this theory. These are examples when different in essence material objects exhibit similar properties.

For example, acoustic black holes first observed in 2011 exhibit the same properties that real black holes should have in theory. In the first experimental acoustic black hole, the Bose-Einstein condensate of 100 thousand rubidium atoms was spun up to supersonic speed in such a way that certain parts of the condensate overcame the sound barrier, and the neighboring parts did not. The boundary of these parts of the condensate modeled the black hole event horizon, where the flow velocity is exactly equal to the speed of sound. At temperatures around absolute zero, the sound begins to behave like quantum particles — phonons (a fictional quasiparticle personifies the quantum of the vibrational motion of the atoms of a crystal). It turned out that the “sound” black hole absorbs particles in the same way as a real black hole absorbs photons. Thus, the flow of fluid acts on the sound in the same way that a real black hole acts on light. In principle, a sound black hole with phonons can be considered as a peculiar model of a real curvature in space-time.

If we look more broadly at the structural similarities in various physical phenomena, then we can see an amazing order in natural chaos. All the various natural phenomena, in fact, are described by simple basic rules. Mathematical rules.

Take the fractals. These are self-similar geometric forms that can be divided into parts so that each part is at least approximately a reduced copy of the whole. One example is the famous Barnsley fern.



Barnsley Fern is constructed using four affine transformations of the form:



This particular sheet is generated with the following factors:









In the surrounding nature, such mathematical formulas are found everywhere - in clouds, trees, mountain ranges, ice crystals, flickering flames, in the seashore. These are examples of fractals, the structure of which is described by relatively simple mathematical calculations.

Galileo Galilei said back in 1623: “All science is recorded in this great book - I mean the Universe - which is always open for us, but which cannot be understood without having learned to understand the language in which it is written. And it is written in the language of mathematics, and its letters are triangles, circles and other geometric figures, without which it is impossible for a person to make out a single word of it; without them, it is like wandering in darkness. "

In fact, mathematical rules manifest themselves not only in the geometry and visual outlines of natural objects, but also in other laws. For example, in the nonlinear dynamics of the population size, the growth rate of which is dynamically reduced when approaching the natural limit of the ecological niche. Or in quantum physics.

As for the most famous mathematical constants — for example, the numbers pi — it is quite natural that it is widely found in nature, because the corresponding geometric forms are the most rational and suitable for many natural objects. In particular, the fundamental physical constant is the number 2π. It shows what is equal to the angle of rotation in radians, contained in one full rotation when the body rotates. Accordingly, this constant is commonly found in the description of the rotational form of motion and angle of rotation, as well as in the mathematical interpretation of vibrations and waves.

For example, the period of small natural oscillations of a mathematical pendulum of length L, fixedly suspended in a uniform gravitational field with an acceleration of free fall g is



Under the conditions of the Earth's rotation, the plane of oscillation of the pendulum will slowly turn in the direction opposite to the direction of the Earth's rotation. The speed of rotation of the oscillation plane of the pendulum depends on its geographic latitude .



The number pi is a component of the Dirac constant — the reduced Planck constant, the basic constant of quantum physics, which connects two systems of units — the quantum and the traditional. It connects the quantum of energy of any linear oscillatory physical system with its frequency.



Accordingly, the number pi is included in the fundamental postulate of quantum mechanics, the Heisenberg uncertainty principle.



The number pi is used in the formula of the fine structure constant — another fundamental physical constant characterizing the strength of the electromagnetic interaction, as well as in the formulas of hydromechanics, etc.

In the natural world you can find other mathematical constants. For example, the number e , the base of the natural logarithm. This constant is included in the formula for the normal probability distribution, which is given by the probability density function:



The normal distribution is subject to many natural phenomena, including many characteristics of living organisms in the population. For example, the size distribution of organisms in a population: length, height, surface area, weight, blood pressure in humans, and more.

Close observation of the outside world shows that mathematics is not at all a dry abstract science, as it may seem at first glance. Just the opposite. Mathematics is the basis of the whole living and non-living world around. As Galileo Galilei correctly remarked, mathematics is the language in which nature speaks to us.