Numeric constants. Formula of connection of fundamental physical constants and structure of time and space. Einstein's correction is the density of dark matter, which pushes the cosmos from the inside

Relationship formula for fundamental physical constants

and the structure of time and space.

(NIAT Research Fellow: Gravitational Constant(G) Measurement Group).

(This article is a continuation of the author's work on the formula for the connection of fundamental physical constants (FPC), which the author published in the article (1 *). A model for combining the main four interactions and a new look at time and space is proposed. The article is also supplemented with new data based on the values ​​of the FPC received by CODATA in 1998, 2002 and 2006.)

1. Introduction.

2) Derivation of the formula for the connection of fundamental physical constants:

3) Combining four main types of interaction:

4) Structure of time and space:

5) Practical proof of the formula:

6) Mathematical proofs of the formula and its structural analysis: etc.

8) Conclusion.

1. Introduction.

After the unsuccessful development of early models of the unification of gravity and electromagnetism, the opinion was established that there is no direct connection between the fundamental physical constants of these two interactions. However, this opinion has not been fully tested.

To find the formula for the connection between the fundamental physical constants of electromagnetic and gravitational interaction, the method of "successive logical selection" was used. (this is the choice of certain variants of the formula and constants for substitution, based on the established physical premises and criteria).

In our case, the following physical prerequisites and criteria for choosing constants and variants of the formula were taken.

Prerequisites.

1. The nature of the interaction of electromagnetic and gravitational forces is close enough to make the assumption that their constants are interconnected:

2. The intensity of the gravitational interaction is set by those particles that simultaneously participate in the electromagnetic interaction.

These are: electron, proton and neutron.

3. The above particles determine the structure of the main element in the Universe - hydrogen, which in turn determines the internal structure of space and time.

As can be seen from the above (paragraphs 2,3) - the interconnectedness of gravity and electromagnetism is inherent in the very structure of our Universe.

Criterias of choice.

1. Constants for substitution in the formula must be dimensionless.

2. Constants must satisfy physical prerequisites.

3..gif" width="36" height="24 src=">

4. Stable matter mainly consists of hydrogen, and its main mass is given by the proton mass. Therefore, all constants must be related to the mass of the proton, and the ratio of the masses of the electron and proton https://pandia.ru/text/78/455/images/image016_33.gif" width="215 height=25" height="25">

Where: - coefficient given by weak interaction;

https://pandia.ru/text/78/455/images/image019_28.gif" width="27" height="24 src=">- coefficient given by nuclear interaction.

In terms of its significance, the proposed formula for the connection of the constants of electromagnetic and gravitational interaction claims to unify gravitation and electromagnetism, and upon detailed consideration of the elements of the presented formula, to unify all four types of interactions.

Lack of theory of numerical values ​​of fundamental physical constants (FPC)

required to find mathematical and practical examples proving the truth of the formula for the connection of fundamental physical constants of electromagnetic and gravitational interaction.

The given mathematical conclusions claim to be a discovery in the field of FPC theory and lay the foundation for understanding their numerical values.

2) Derivation of the formula for the connection of fundamental physical constants .

To find the main link in the formula for the connection of constants, one must answer the question: “why are gravitational forces so weak compared to electromagnetic forces?” To do this, consider the most common element in the universe - hydrogen. It also determines its main visible mass, setting the intensity of gravitational interaction.

Electric charges of electron (-1) and proton (+1) forming hydrogen are equal in absolute value; at the same time, their "gravitational charges" differ by 1836 times. Such a different position of the electron and proton for the electromagnetic and gravitational interaction explains the weakness of the gravitational forces, and the ratio of their masses should be included in the desired formula for the connection of the constants.

We write the simplest version of the formula, taking into account the prerequisites (item 2.3.) and the selection criterion (item 1,2, 4):

Where: - characterizes the intensity of gravitational forces.

From data for 1976.gif" width="123" height="50 src=">

Let's find the module "x":

The found value is well rounded up to (12).

Substituting it, we get:

(1)

The discrepancy found between the left and right sides of the equation in formula (1):

For numbers with a degree of "39" there is practically no discrepancy. It should be noted that these numbers are dimensionless and do not depend on the chosen system of units.

Let's make a stand in formula (1), based on the premise (item 1) and selection criteria (items 1,3,5), which indicate the presence in the formula of a constant characterizing the intensity of electromagnetic interaction. To do this, we find the degrees of the following relation:

where: https://pandia.ru/text/78/455/images/image029_22.gif" width="222 height=53" height="53">

For x=2, y=3.0549 i.e. y rounds well to "3".

We write formula (1) with substitution:

(2)

Find the discrepancy in formula (2):

Using a fairly simple substitution, we obtained a decrease in the discrepancy. This speaks of its truth from the point of view of constructing a formula for the connection of constants.

From data for 1976, (2*):

Since , further refinement of formula (2) is necessary. This is also indicated by the prerequisites (items 2 and 3), as well as the selection criterion (item 5), which refers to the presence of a constant characterizing the neutron.

To substitute its mass into formula (2), it is necessary to find the degree of the following relationship:

Let's find the module z:

Rounding z up to "38", we can write formula (2) with a clarifying substitution:

(3)

Find the discrepancy in formula (3):

With error precision, valueequal to one.

From this we can conclude that formula (3) is the final version of the desired formula for the connection between the fundamental physical constants of electromagnetic and gravitational interaction.

We write this formula without reciprocals:

(4)

The found formula allows to expressfundamental physicalgravitational interaction constants through electromagnetic interaction constants.

3) Combining the four main types of interaction.

Consider formula (4) from the point of view of the selection criterion "5".

As expected, the desired formula consists of three coefficients:

Let's analyze each of the coefficients.

As seen, First coefficient determined by the fact that the weak interaction divided leptons and hadrons into two classes of particles different meanings masses:

Hadrons are heavy particles

Leptons are light particles

The tenth power in a fraction https://pandia.ru/text/78/455/images/image045_16.gif" width="21" height="21 src=">) reflects the intensity of electromagnetic interaction, and the degree "3" indicates three-dimensionality of the space-time in which leptons and hadrons exist as particles of electromagnetic interaction.In terms of significance, this coefficient takes the second place in the found formula.

Third coefficient Antiques" href="/text/category/antikvariat/" rel="bookmark">antiquarks)multiply by 3colors +1 gluon+1antigluon=38 states

As can be seen from the degree of "38", the dimension of the space in which quarks exist, as components of the proton and neutron, is thirty-eight. In terms of significance, this coefficient takes the third place in the found formula.

If we take orders of magnitude in the numerical values ​​of the coefficients, then we get:

Let us substitute these values ​​into formula (4):

Each of the coefficients, in order of magnitude, specifies the intensity of the interaction it represents. Hence, we can conclude that formula (4) allows us to combine all four types of interactions and is the main super-unification formula.

The found form of the formula and the values ​​of the degrees show that a single interaction for each interaction sets its own value for the dimension of space and time.

Unsuccessful attempts to combine all four interactions are explained by the fact that the same dimension of space was assumed for all types of interactions.

This assumption also led to a common erroneous join approach:

weak force + electromagnetic force + nuclear force + gravitational force = unified force.

And, as we see, a single interaction sets the dimensionality of space and time

for each type of interaction.

It also follows from this new approach» in combining interactions:

1st stage - weak interaction in ten-dimensional space:

Electromagnetic interaction in three-dimensional space-time:

Nuclear interaction in thirty-eight-dimensional space:

2nd stage - grav.1 + grav. 2 + grav. 3 = grav. = single interaction.

The found formula for the connection of constants reflects this "new approach", being the main formula of the 2nd stage, combining all four types of interactions into one single interaction.

The “new approach” also requires a different view of gravity, a view as a structure consisting of four “layers”:

Moreover, each “layer” has its own carrier of interaction: X Y Z G

(perhaps these carriers are associated with dark matter and dark energy).

Let's summarize the fundamental physical constants (FPC) connection formula:

https://pandia.ru/text/78/455/images/image003_129.gif" width="115" height="46"> the constant characterizes the gravitational interaction.

(the main mass of matter in the universe is given by the mass of the proton, so the gravitational constant is given by the interaction of protons with each other).

The constant characterizes the weak interaction.

(it is the weak interaction that sets the difference between the electron and the proton, and the ratio and difference of their masses makes the main contribution to the weakness of gravitational forces in comparison with other interactions).

The constant characterizes the electromagnetic interaction.

(electromagnetic interaction through the charge contributes to the formula).

the constant characterizes the nuclear interaction.

(nuclear interaction sets the difference between a neutron and a proton and reflects the specifics of this interaction: (6 quarks + 6 antiquarks) multiply by 3 colors + 1 gluon + 1 antigluon = 38 states

As can be seen from the power of "38", the dimension of the space in which quarks exist as components of the proton and neutron is thirty-eight).

4) The structure of time and space.

A new understanding of gravity gives a new understanding of time as a multidimensional quality. The existence of three types of energy (1 "potential energy 2" kinetic energy 3 "rest mass energy) indicates the three-dimensionality of time.

Looking at time as a three-dimensional vector overturns our understanding of time as a scalar and requires the replacement of all integral-differential algebra and physics, where time is represented by a scalar.

If earlier, in order to create a “time machine” (and this, in the language of mathematics, is to change the direction of time movement to the opposite, or give the value of time a minus sign), it was necessary to go through the “0” of time, now, approaching time as vector, - to change the direction to the opposite, you just need to rotate the time vector by 180 degrees, and this does not require operating with the uncertainty "0" of time. This means that after the creation of a time vector rotation device, the creation of a “time machine” becomes a reality.

All of the above makes it necessary to reconsider the law of causality, and, therefore, the law of conservation of energy, and, therefore, other fundamental laws of physics (all these laws “suffer” from one-dimensionality).

If formula (4) allows you to combine all four main types of interaction

then it should reflect the structure of time and space:

The degrees in formula (4) reflect the dimension of time and space in which there are four main interactions.

Let's rewrite (4): (4a)

that if time is a measure of system variability, then gravity (Newton's formula) and electromagnetism (Coulomb's formula) = carry the characteristics of time.

Weak and nuclear interactions are short-acting and therefore carry the properties of space.

Formula (4a) shows that:

A) there are two times: internal and external

(moreover, they are mutually looped on each other forming a single circle)

Gravity reflects external time

common dimension(+1) =

Electromagnetism reflects internal time

common dimension (+3)=

B) and there are two spaces: internal and external

(moreover, they mutually penetrate each other)

Weak interaction reflects outer spaces

common dimension(+10) =

Nuclear interaction reflects inner space

common dimension (+38)=

5) Practical proofs of the formula.

The absence of an absolutely rigorous derivation of formula (4) requires case study her checks. An example is the calculation of the value of the gravitational constant:

(5)

In formula (5), the biggest error is in the gravitational constant: https://pandia.ru/text/78/455/images/image067_14.gif" width="62 height=24" height="24">. from this one can find G with greater precision than the tabular value

Estimated value

(CODATA data (FFK) for 1976):

As you can see, the found value is included in the interval + of the table value and improves it by 20 times. Based on the result obtained, it can be predicted that the tabular value is underestimated. This is confirmed by a new, more accurate value of G adopted in 1986 (3*)

CODATA data (FFK) for 1986: Tabular https://pandia.ru/text/78/455/images/image072_12.gif" width="332" height="51">

We got a value - 40 times more accurate and included in the interval + 2, 3

Estimated for more

Estimated for more

CODATA data (FFK) for 2006 Tabular

Estimated for more

Compare table values:

CODATA data (FFK) for 1976 Tabular https://pandia.ru/text/78/455/images/image082_12.gif" width="79" height="21 src=">

CODATA data (FFK) for 1986 Tabular https://pandia.ru/text/78/455/images/image083_13.gif" width="80" height="21 src=">

CODATA data (FFK) for 1998 Tabular https://pandia.ru/text/78/455/images/image084_12.gif" width="79" height="21 src=">

CODATA data (FFK) for 2002 Tabular

for 2006.gif" width="325" height="51">

Value since 1976 to 2006 why, is constantly increasing, and the accuracy has remained at the level, and in 1986 more 2006 This suggests that there is an unaccounted for hidden parameter in Newton's formula.

Let's compare the calculated values:

CODATA data (FFK) for 1976 Estimated

for 1986.gif" width="332" height="51">

for 1998.gif" width="340" height="51">

for 2002.gif" width="332" height="51">

for 2006.gif" width="328" height="51"> (6)

Self-consistency (in terms of statistics) with increasing accuracy

133 times (!!!) withto calculated valuesG

speaks about the suitability of the formulain further clarifying calculationsG. If the calculated value (6) is confirmed in the future, then this will be a proof of the truth of formula (4).

6) Mathematical proofs of the formula and its structural analysis.

Having written a mathematical equality, - expression (4), we must assume that the constants included in it must be rational numbers (this is our condition of strict algebraic equality): otherwise, if they are irrational or transcendental, - equalize the formula ( 4) it will not be possible, and, therefore, to write a mathematical equality.

The question of the transcendence of the values ​​of the constants is removed after, by replacing h with in formula (4), it is not possible to achieve equality (the use in physics was that fatal delusion that did not allow finding the formula for the connection of the constants (4; 5). Violation strict equality with the substitution of a transcendental number also proves the correctness of the chosen equality condition for formula (4), and hence the rationality of the FPC.)

Consider one of the numerical values ​​obtained when calculating formula (5):

CODATA data (FFK) for 1986

A random sequence of three zeros is unlikely, so this is the period of a simple rational fraction: (7)

The value of this fraction is included in the interval 0.99 of the calculated value. Since the presented fraction is taken entirely from formula (5), it can be predicted that the value of the ratio of the mass of the proton to the mass of the electron to the tenth power will converge to the value (7). This is confirmed by new data for 1998:

CODATA data (FFK) for 1998

The new calculated value is closer (and therefore converges) to the exact value: https://pandia.ru/text/78/455/images/image073_13.gif" width="25 height=22" height="22" >

The proven convergence indicates the exact equality of formula (4), which means that this formula is the final version and is not subject to further refinement, both in the physical and mathematical sense of the word.

Based on this, we can make a statement that claims to be a discovery:

THE VALUE OF FUNDAMENTAL PHYSICAL CONSTANTS (FFK) IN THE POWERS PRESENTED IN THE FORMULA , CONVERG TO SIMPLE RATIONAL FRACTIONS AND ARE EXPRESSED IN TERMS OF THE OTHER BY FORMULA (5).

This is also confirmed by the fact that the new values ​​of the ratio of the neutron and proton masses revealed the period in the following fraction:

CODATA data (FFK) for 1998

CODATA data (FFK) for 2002

There is a convergence to the number: (8)

Based on the first values ​​found (7; 8) and an intuitive idea of ​​the simple structure of constructions in nature, it can be assumed that the value of the prime numbers included in the fractions in formula (4) is of the order of "10000":

Another interesting convergence was found on the left side of formula (4): https://pandia.ru/text/78/455/images/image109_10.gif" width="422" height="46">

CODATA 1998 data:

CODATA 2002 data:

CODATA 2006 data:

There is a convergence to the number: (9)

You can find a more precise value:

It is included in the interval +0.28 of the CODATA value for 2006 and is 25 times more accurate:

We substitute the found numbers (7) and (8) into the formula :

On the right we have a large prime number 8363, it must be present and on the left in the upper part of the formula, therefore, we divide:

2006: https://pandia.ru/text/78/455/images/image114_9.gif" width="40 height=28" height="28">:

Formula data:

The limited accuracy of tabular values ​​does not allow direct calculation to find the exact numerical values ​​to which the FPC converge in formula (5); the exceptions are the values ​​of the constants (7; 8; 9). But this difficulty can be circumvented by using the mathematical properties of simple rational fractions in decimal notation - show periodicity in numbers of the last characters, for number () this is a period ... from here you can find: https://pandia.ru/text/78/455/images/image126_10.gif" width="361" height=" 41 src=">substitute

https://pandia.ru/text/78/455/images/image129_9.gif" width="586" height="44 src=">.gif" width="215" height="45">

You can find a more precise h :

It is included in the interval +0.61 of the CODATA value for 2006 and is 8.2 times more accurate:

7) Finding the exact values ​​of FFK in the formula (4 and 5).

Let's write the exact values ​​of FFK that we have already found:

A=https://pandia.ru/text/78/455/images/image137_8.gif" width="147 height=57" height="57"> B=

G =https://pandia.ru/text/78/455/images/image140_8.gif" width="249" height="41">

E =https://pandia.ru/text/78/455/images/image142_8.gif" width="293" height="44">

In addition to https://pandia.ru/text/78/455/images/image144_9.gif" width="31" height="24">, the exact value of which we still do not know. Let's write "C" with the same accuracy as which we know her:

At first glance, there is no period, but it should be noted that, according to the formula (4) and according to the construction of the exact numbers E and W, it is a rational number, since it is represented in them in the first powers. This means that the period is hidden and in order for it to appear, it is necessary to multiply this constant by certain numbers. For this constant, these numbers are "primary divisors":

As you can see, the period (C) is "377". From here you can find the exact value to which the values ​​\u200b\u200bof this constant converge:

It is included in the interval +0.94 of the CODATA value for 1976.

After averaging we got:

(CODATA data (FFK) for 1976)

As you can see, the found value of the speed of light is in good agreement with the most accurate - the first value. This is proof of the correctness of the method of "search for rationality in the values ​​of FFK"

(To multiply the most accurate by "3": 8,. A clean period of "377" appeared).

It must be said that the presence of a direct connection between the fundamental physical constants (formula (4)) makes it impossible to arbitrarily choose the value of one of them, since this will lead to a shift in the values ​​of other constants.

The above also applies to the speed of light, the value of which was adopted in 1983.

exact integer value: https://pandia.ru/text/78/455/images/image154_8.gif" width="81" height="24"> and creates an unaccounted shift in FFC values)

This action is also mathematically incorrect, since no one has proven that the value

the speed of light is not an irrational or transcendental number.

Moreover, it is premature to take it whole.

(Most likely - no one dealt with this issue and "C" was taken "whole" by negligence).

Using the formula (4), it can be shown that the speed of light is a RATIONAL number, however, NOT A INTEGER.

Archimedes number

What is equal to: 3.1415926535… To date, up to 1.24 trillion decimal places have been calculated

When to celebrate pi day- the only constant that has its own holiday, and even two. March 14, or 3.14, corresponds to the first characters in the number entry. And July 22, or 22/7, is nothing more than a rough approximation of π by a fraction. In universities (for example, at the Faculty of Mechanics and Mathematics of Moscow State University), they prefer to celebrate the first date: unlike July 22, it does not fall on holidays

What is pi? 3.14, the number from school problems about circles. And at the same time - one of the main numbers in modern science. Physicists usually need π where there is no mention of circles - say, to model the solar wind or an explosion. The number π occurs in every second equation - you can open a textbook of theoretical physics at random and choose any. If there is no textbook, a world map will do. An ordinary river with all its breaks and bends is π times longer than the path straight from its mouth to its source.

Space itself is to blame for this: it is homogeneous and symmetrical. That is why the front of the blast wave is a ball, and circles remain from the stones on the water. So pi is quite appropriate here.

But all this applies only to the familiar Euclidean space in which we all live. If it were non-Euclidean, the symmetry would be different. And in a highly curved universe, π no longer plays such an important role. For example, in Lobachevsky's geometry, a circle is four times as long as its diameter. Accordingly, rivers or explosions of "curved space" would require other formulas.

The number pi is as old as all of mathematics: about 4,000. The oldest Sumerian tablets give him the figure 25/8, or 3.125. The error is less than a percent. The Babylonians were not particularly fond of abstract mathematics, so pi was derived empirically, simply by measuring the length of circles. By the way, this is the first experiment on numerical modeling of the world.

The most elegant of the arithmetic formulas for π is over 600 years old: π/4=1–1/3+1/5–1/7+… Simple arithmetic helps to calculate π, and π itself helps to understand the deep properties of arithmetic. Hence its connection with probabilities, prime numbers, and many others: π, for example, is included in the well-known "error function", which works equally well in casinos and sociologists.

There is even a "probabilistic" way to calculate the constant itself. First, you need to stock up on a bag of needles. Secondly, to throw them, without aiming, on the floor, lined with chalk into stripes as wide as a needle. Then, when the bag is empty, divide the number of those thrown by the number of those that crossed the chalk lines - and get π / 2.

Chaos

Feigenbaum constant

What is equal to: 4,66920016…

Where applied: In the theory of chaos and catastrophes, which can be used to describe any phenomena - from the reproduction of E. coli to the development of the Russian economy

Who and when discovered: American physicist Mitchell Feigenbaum in 1975. Unlike most other constant discoverers (Archimedes, for example), he is alive and teaches at the prestigious Rockefeller University.

When and how to celebrate δ day: Before general cleaning

What do broccoli, snowflakes, and Christmas trees have in common? The fact that their details in miniature repeat the whole. Such objects, arranged like a nesting doll, are called fractals.

Fractals emerge from disorder, like a picture in a kaleidoscope. Mathematician Mitchell Feigenbaum in 1975 was not interested in the patterns themselves, but in the chaotic processes that make them appear.

Feigenbaum was engaged in demography. He proved that the birth and death of people can also be modeled according to fractal laws. Then he got this δ. The constant turned out to be universal: it is found in the description of hundreds of other chaotic processes, from aerodynamics to biology.

With the Mandelbrot fractal (see fig.), the widespread fascination with these objects began. In chaos theory, it plays approximately the same role as the circle in ordinary geometry, and the number δ actually determines its shape. It turns out that this constant is the same π, only for chaos.

Time

Napier number

What is equal to: 2,718281828…

Who and when discovered: John Napier, Scottish mathematician, in 1618. He did not mention the number itself, but he built his tables of logarithms on its basis. At the same time, Jacob Bernoulli, Leibniz, Huygens and Euler are considered candidates for the authors of the constant. It is only known for certain that the symbol e taken from last name

When and how to celebrate e day: After the return of the bank loan

The number e is also a kind of twin of π. If π is responsible for space, then e is for time, and also manifests itself almost everywhere. Let's say the radioactivity of polonium-210 decreases by a factor of e per average term the life of one atom, and the shell of the Nautilus mollusk is a graph of powers of e wrapped around an axis.

The number e is also found where nature obviously has nothing to do with it. A bank that promises 1% per year will increase the deposit by about e times in 100 years. For 0.1% and 1000 years, the result will be even closer to a constant. Jacob Bernoulli, a connoisseur and theorist of gambling, deduced it exactly like this - arguing about how much moneylenders earn.

Like pi, e is a transcendental number. Simply put, it cannot be expressed in terms of fractions and roots. There is a hypothesis that in such numbers in an infinite "tail" after the decimal point there are all combinations of numbers that are possible. For example, there you can also find the text of this article, written in binary code.

Light

Fine structure constant

What is equal to: 1/137,0369990…

Who and when discovered: German physicist Arnold Sommerfeld, whose graduate students were two Nobel laureates at once - Heisenberg and Pauli. In 1916, before the advent of true quantum mechanics, Sommerfeld introduced the constant in a routine paper on the "fine structure" of the spectrum of the hydrogen atom. The role of the constant was soon rethought, but the name remained the same

When to celebrate α day: On Electrician's Day

The speed of light is an exceptional value. Einstein showed that neither a body nor a signal can move faster - be it a particle, gravity wave or the sound within the stars.

It seems to be clear that this is a law of universal importance. And yet the speed of light is not a fundamental constant. The problem is that there is nothing to measure it. Kilometers per hour are no good: a kilometer is defined as the distance that light travels in 1/299792.458 of a second, which is itself expressed in terms of the speed of light. The platinum standard of the meter is also not an option, because the speed of light is also included in the equations that describe platinum at the micro level. In a word, if the speed of light changes without unnecessary noise throughout the Universe, humanity will not know about it.

This is where the physicists come to the aid of a quantity that connects the speed of light with atomic properties. The constant α is the "speed" of an electron in a hydrogen atom divided by the speed of light. It is dimensionless, that is, it is not tied to meters, or to seconds, or to any other units.

In addition to the speed of light, the formula for α also includes the electron charge and Planck's constant, a measure of the "quantum" nature of the world. Both constants have the same problem - there is nothing to compare them with. And together, in the form of α, they are something like a guarantee of the constancy of the Universe.

One might wonder if α has changed since the beginning of time. Physicists seriously admit a “defect”, which once reached millionths of the current value. If it reached 4%, there would be no humanity, because thermonuclear fusion of carbon, the main element of living matter, would stop inside the stars.

Addition to reality

imaginary unit

What is equal to: √-1

Who and when discovered: Italian mathematician Gerolamo Cardano, friend of Leonardo da Vinci, in 1545. The cardan shaft is named after him. According to one version, Cardano stole his discovery from Niccolo Tartaglia, a cartographer and court librarian.

When to celebrate day i: March 86th

The number i cannot be called a constant or even a real number. Textbooks describe it as a quantity that, when squared, is minus one. In other words, it is the side of the square with negative area. In reality, this does not happen. But sometimes you can also benefit from the unreal.

The history of the discovery of this constant is as follows. Mathematician Gerolamo Cardano, solving equations with cubes, introduced an imaginary unit. This was just an auxiliary trick - there was no i in the final answers: the results that contained it were rejected. But later, looking closely at their "garbage", mathematicians tried to put it into action: multiply and divide ordinary numbers by an imaginary unit, add the results to each other and substitute them into new formulas. Thus was born the theory of complex numbers.

The downside is that “real” cannot be compared with “unreal”: to say that more - an imaginary unit or 1 - will not work. On the other hand, there are practically no unsolvable equations, if we use complex numbers. Therefore, with complex calculations, it is more convenient to work with them and only at the very end “clean out” the answers. For example, to decipher a tomogram of the brain, you cannot do without i.

This is how physicists treat fields and waves. It can even be considered that they all exist in a complex space, and what we see is only a shadow of "real" processes. Quantum mechanics, where both the atom and the person are waves, makes this interpretation even more convincing.

The number i allows you to reduce the main mathematical constants and actions in one formula. The formula looks like this: e πi +1 = 0, and some say that such a compressed set of rules of mathematics can be sent to aliens to convince them of our reasonableness.

Microworld

proton mass

What is equal to: 1836,152…

Who and when discovered: Ernest Rutherford, New Zealand-born physicist, in 1918. 10 years earlier, he received the Nobel Prize in Chemistry for the study of radioactivity: Rutherford owns the concept of "half-life" and the equations themselves that describe the decay of isotopes

When and how to celebrate μ day: On the day of the fight against overweight, if one is introduced, it is the ratio of the masses of two basic elementary particles, the proton and the electron. A proton is nothing more than the nucleus of a hydrogen atom, the most abundant element in the universe.

As in the case of the speed of light, it is not the value itself that is important, but its dimensionless equivalent, not tied to any units, that is, how many times the mass of a proton is greater than the mass of an electron. It turns out approximately 1836. Without such a difference in " weight categories There would be no charged particles, no molecules, no solids. However, the atoms would remain, but they would behave in a completely different way.

Like α, μ is suspected of slow evolution. Physicists studied the light of quasars, which reached us after 12 billion years, and found that protons become heavier over time: the difference between prehistoric and modern valuesμ was 0.012%.

Dark matter

Cosmological constant

What is equal to: 110-²³ g/m3

Who and when discovered: Albert Einstein in 1915. Einstein himself called her discovery his "major blunder"

When and how to celebrate Λ day: Every second: Λ, by definition, is always and everywhere

The cosmological constant is the most obscure of all the quantities that astronomers operate on. On the one hand, scientists are not completely sure of its existence, on the other hand, they are ready to use it to explain where most of the mass-energy in the Universe came from.

We can say that Λ complements the Hubble constant. They are related as speed and acceleration. If H describes the uniform expansion of the Universe, then Λ is a continuously accelerating growth. Einstein was the first to introduce it into the equations of the general theory of relativity when he suspected a mistake in himself. His formulas indicated that the cosmos was either expanding or contracting, which was hard to believe. A new term was needed to eliminate conclusions that seemed implausible. After the discovery of Hubble, Einstein abandoned his constant.

The second birth, in the 90s of the last century, the constant is due to the idea of ​​dark energy, "hidden" in every cubic centimeter of space. As follows from observations, the energy of an obscure nature should "push" the space from the inside. Roughly speaking, this is a microscopic Big Bang that happens every second and everywhere. The density of dark energy - this is Λ.

The hypothesis was confirmed by observations of relic radiation. These are prehistoric waves born in the first seconds of the existence of the cosmos. Astronomers consider them to be something like an X-ray that shines through the Universe through and through. "X-ray" and showed that there is 74% of dark energy in the world - more than everything else. However, since it is "smeared" throughout the cosmos, only 110-²³ grams per cubic meter is obtained.

Big Bang

Hubble constant

What is equal to: 77 km/s /MPs

Who and when discovered: Edwin Hubble, founding father of all modern cosmology, in 1929. A little earlier, in 1925, he was the first to prove the existence of other galaxies outside the Milky Way. The co-author of the first article that mentions the Hubble constant is a certain Milton Humason, a man without higher education who worked at the observatory as a laboratory assistant. Humason owns the first picture of Pluto, then not yet open planet, due to defective photographic plate left unattended

When and how to celebrate H day: January 0 From this non-existent number, astronomical calendars begin counting the New Year. Like the moment of the Big Bang itself, little is known about the events of January 0, which makes the holiday doubly appropriate.

The main constant of cosmology is a measure of the rate at which the universe is expanding as a result of the Big Bang. Both the idea itself and the constant H go back to the findings of Edwin Hubble. Galaxies in any place of the Universe scatter from each other and do it the faster, the greater the distance between them. The famous constant is simply a factor by which distance is multiplied to get speed. Over time, it changes, but rather slowly.

The unit divided by H gives 13.8 billion years, the time since the Big Bang. This figure was first obtained by Hubble himself. As later proved, the Hubble method was not entirely correct, but still he was wrong by less than a percentage when compared with modern data. The mistake of the founding father of cosmology was that he considered the number H to be constant from the beginning of time.

A sphere around the Earth with a radius of 13.8 billion light years - the speed of light divided by the Hubble constant - is called the Hubble sphere. Galaxies beyond its border should "run away" from us at superluminal speed. There is no contradiction with the theory of relativity here: it is enough to choose the correct coordinate system in a curved space-time, and the problem of exceeding the speed immediately disappears. Therefore, the visible Universe does not end behind the Hubble sphere, its radius is approximately three times larger.

gravity

Planck mass

What is equal to: 21.76 ... mcg

Where does it work: Physics of the microworld

Who and when discovered: Max Planck, creator of quantum mechanics, in 1899. The Planck mass is just one of the set of quantities proposed by Planck as a "system of measures and weights" for the microcosm. The definition referring to black holes - and the theory of gravity itself - appeared a few decades later.

An ordinary river with all its breaks and bends is π times longer than the path straight from its mouth to its source

When and how to celebrate the daymp: On the opening day of the Large Hadron Collider: microscopic black holes are going to get there

Jacob Bernoulli, an expert and theorist of gambling, deduced e, arguing about how much moneylenders earn

Fitting a theory to phenomena is a popular approach in the 20th century. If an elementary particle requires quantum mechanics, then a neutron star - already the theory of relativity. The disadvantage of such an attitude to the world was clear from the very beginning, but a unified theory of everything was never created. So far, only three of the four fundamental types of interaction have been reconciled - electromagnetic, strong and weak. Gravity is still on the sidelines.

Einstein's correction is the density of dark matter, which pushes the cosmos from the inside

The Planck mass is a conditional boundary between "large" and "small", that is, just between the theory of gravity and quantum mechanics. This is how much a black hole should weigh, the dimensions of which coincide with the wavelength corresponding to it as a micro-object. The paradox lies in the fact that astrophysics interprets the boundary of a black hole as a strict barrier beyond which neither information, nor light, nor matter can penetrate. And from a quantum point of view, the wave object will be evenly "smeared" over space - and the barrier along with it.

Planck mass is the mass of a mosquito larva. But as long as the gravitational collapse does not threaten the mosquito, quantum paradoxes will not touch it.

mp is one of the few units in quantum mechanics that should be used to measure objects in our world. This is how much a mosquito larva can weigh. Another thing is that as long as the gravitational collapse does not threaten the mosquito, quantum paradoxes will not touch it.

Infinity

Graham number

What is equal to:

Who and when discovered: Ronald Graham and Bruce Rothschild
in 1971. The article was published under two names, but the popularizers decided to save paper and left only the first one.

When and how to celebrate G-Day: Very soon, but very long

The key operation for this construction is Knuth's arrows. 33 is three to the third power. 33 is three raised to three, which in turn is raised to the third power, that is, 3 27, or 7625597484987. Three arrows is already the number 37625597484987, where the triple in the ladder of power exponents is repeated exactly as many - 7625597484987 - times. It's already more number atoms in the universe: there are only 3,168 of them. And in the formula for the Graham number, not even the result itself grows at the same rate, but the number of arrows at each stage of its calculation.

The constant appeared in an abstract combinatorial problem and left behind all the quantities associated with the present or future size of the universe, planets, atoms and stars. Which, it seems, once again confirmed the frivolity of the cosmos against the background of mathematics, by means of which it can be comprehended.

Illustrations: Varvara Alyai-Akatyeva

3D model of the endoplasmic reticulum of a eukaryotic cell with Terasaki ramps that connect flat sheets of 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 Terasaki ramps. Three years later, astrophysicists noticed the work of biologists. They were amazed: after all, precisely such structures are present inside neutron stars. The so-called "nuclear paste" consists of parallel sheets connected by spiral shapes.

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

Model of helical connections between flat membrane sheets in a eukaryotic cell

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 that underlie the entire universe.

Let's look at a few more examples that support this theory. These are examples of essentially different material objects exhibiting similar properties.

For example, first observed in 2011, acoustic black holes exhibit the same properties that real black holes should theoretically have. In the first experimental acoustic black hole, a Bose-Einstein condensate of 100 thousand rubidium atoms was spun up to supersonic speed in such a way that individual parts of the condensate broke the sound barrier, while neighboring parts did not. The boundary of these parts of the condensate modeled the event horizon of a black hole, where the flow velocity is exactly equal to the speed of sound. At temperatures near absolute zero, sound begins to behave like quantum particles - phonons (a fictitious quasi-particle represents a quantum of vibrational motion of crystal atoms). It turned out that a "sonic" black hole absorbs particles in the same way as a real black hole absorbs photons. Thus, the fluid flow affects sound in the same way that a real black hole affects light. In principle, a sonic black hole with phonons can be considered as a kind of model of a real curvature in space-time.

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

Take fractals. These are self-similar geometric shapes 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.

The Barnsley fern is built using four affine transformations of the form:

This particular sheet is generated with the following coefficients:

In the nature around us, such mathematical formulas are found everywhere - in clouds, trees, mountain ranges, ice crystals, flickering flames, on the sea coast. These are examples of fractals whose structure 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 to us, but which cannot be understood without learning 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 others. geometric figures, without which it is impossible for a person to make out a single word of hers; without them, he is like one who wanders 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 non-linear dynamics of the population size, the growth rate of which dynamically decreases when approaching the natural limit of the ecological niche. Or in quantum physics.

As for the most famous mathematical constants - for example, the number 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 number 2π has become a fundamental physical constant. It shows what the angle of rotation in radians is, contained in one complete revolution during the rotation of the body. Accordingly, this constant is ubiquitous in the description of the rotational form of motion and the angle of rotation, as well as in the mathematical interpretation of oscillations and waves.

For example, the period of small natural vibrations mathematical pendulum of length L, motionlessly suspended in a uniform gravitational field with free fall acceleration g is equal to

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 plane of oscillation of the pendulum depends on its geographical latitude.

The number pi is integral part Planck's constant - the main constant of quantum physics, which connects two systems of units - quantum and traditional. It connects the value of the energy quantum 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 for the fine structure constant - another fundamental physical constant that characterizes the strength of the electromagnetic interaction, as well as in the formulas of hydromechanics, etc.

Other mathematical constants can also be found in the natural world. For example, 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:

Many natural phenomena are subject to the normal distribution, including many characteristics of living organisms in a population. For example, the size distribution of organisms in a population: length, height, surface area, weight, arterial pressure people and much more.

A close observation of the world around us shows that mathematics is not at all a dry abstract science, as it might seem at first glance. Quite the opposite. Mathematics is the basis of all living and non-living world around. As Galileo Galilei correctly noted, mathematics is the language that nature speaks to us.

Natural

Physical and mathematical sciences Mathematics

Mathematical analysis

Shelaev A.N., Doctor of Physical and Mathematical Sciences, Professor, N.N. D.V. Skobeltsyn, Moscow State University. M.V. Lomonosov

EXACT RELATIONSHIPS BETWEEN FUNDAMENTAL MATHEMATICAL CONSTANTS

The problems of finding and interpreting the exact relationships between the fundamental mathematical constants (FMC), primarily P, e, the constants

lot proportion f \u003d (-1 + V5) / 2 □ 0.618, f \u003d f + 1 \u003d (1 + "s / 5) / 2, the Eule constant

1/k _lnn) = _l e lnxdx □ 0.577, Catalan's constant n^yes k= J 0

G = Z"=o(_1)n / (2n +1)2 = |oX-1 arctg X dx □ 0.915, imaginary unit i = 1

This article reports on finding various types exact relations between FMC, including between algebraic and transcendental.

Let's start with the golden ratio constants φ, φ. In addition to the above initial expressions, other definitions can be obtained for them, for example, as the limit of a sequence, a continued fraction, the sum of nested radicals:

φ= lim xn, where xn = 1/(1 + xn_1), x0 = 1, n = 1,2,3,... (1)

φ = 1/2 + lim xn, where xn = 1/8_x2_1 /2, x0 = 1/8, n = 1,2,3,... (2)

f = f + 1 = 1 +--(3)

f = f +1 = 1 + 1 + yf[ + yl 1 +... (4)

Note that in (1), (3) Xp and final fractions are expressed through the ratio of 2 consecutive Fibonacci numbers Bp = 1,1,2,3,5,8,.... As a result, we get:

gp/gp+1, F = A

φ= lim Fn /Fn+1, Φ=ХГ=1(_1)П+1/(Рп-Fn+1) (5)

ratios:

The relationship between the constants φ, φ, P and 1 = is determined

b1p (1 1p f) \u003d 1 / 2, w (l / 2 - Ni f) \u003d (f + f) / 2 (6)

f = ^ 1+ W1 + (f + iW1 + (f + 2)Vi+T7

Given that f-f = 1, we obtain the following expression for p(f) :

n \u003d 4 - arctan[f - ^ 1 + f^/ 1 + (f + 1)^1 + (f + 2^l / G + TGG ]

For the constants φ, φ, finite expressions were also obtained in transcendental form, which naturally lead to algebraic expressions, for example:

f \u003d 2 - sin (n / 10) \u003d tg (9)

Ф = 2 - cos(n / 5) = tg[(n - arctg(2)) / 2] (10)

The constant P can also be determined, for example, by the following relations:

П = 4-X°°=0(-1)n/(2n +1) = lim 2n 22+ >/2 + V2 + ---V2 (11)

In this case, in (11) the number of radicals inside the limit is equal to n . In addition, it should be noted

that \/ 2 + v 2 + 2 +----= 2 (!) for an infinite number of radicals.

For the constant P, a number of trigonometric relations were also obtained, connecting it with other constants, for example:

n = 6 - arcsin = 3 - arccos(12)

n \u003d 10 - arcsin (f / 2) \u003d 10 - arccos ^ 5 - f / 2) (13)

n = 4 - (14)

n = 4 - (15)

n = 4 - (16)

n = 4 - (17)

The constant e can also be defined by various expressions, for example:

e = lim(1 + x)1/x = limn/^n! = yj(A + 1)/(A-1), where A = 1 +-Ts- (18)

x -n -yes 3 + 1

The connection of the constant e with other FMCs can be carried out, first of all, through the 2nd remarkable limit, the Taylor and Euler formulas:

e = lim [(2/ n) arctgx]-nx/2 = lim (tgx)-tg2x = lim(2 - x)(n/2>tgnx/2 (19) x-yes x-n/4 x- 1

e = lim (1 + p/n)n/p, p = p, f, f, C, G (20)

e = p1/L, where L = lim n (p1/n -1), p = n, φ, Φ, C^ (21)

e = 1/p, p = p, F, F, S, G (22)

eip = cos(p) + i sin(p), i = V-Y, p = p, f, f, s, g (23)

A large number of exact relationships between FMC can be obtained using integral relationships, for example, such:

l/n = 2^2p j cos(px2)dx = 2^/2p j sin(px2)dx, p = e^, φ, C, G (24) J 0 » 0

p = Vp j0dx/(1 ±p cosx), p = e, f, f, C, G (25)

G = nln2/2-j 0ln(1 + x2)/(1 + x2)dx = -nln2/2-j0/4ln(sinx) dx (26)

C \u003d -ln4 -4p 1/2 j 0 exp (-x2)lnxdx (27)

C = jda / x dx - ln(b / p), p, b = n,e, f, f, G (28) 0

It is essential that in relation (28) the Euler constant C can be expressed not in terms of one, but in terms of two FMCs p, b.

It is also interesting that from the ratio linking P with other FMCs,

(n/p)/sin(n/p) = j0 dx/(1 + xp), p = e,f,f,C,G (29)

we can get a new definition of the 1st remarkable limit:

lim(n/p)/sin(n/p)= lim j dx/(1 + x) = 1 (30)

In the course of the research, a large number of interesting approximate relationships between FMC were also found. For example, such:

S□ 0.5772□ 1§(p/6) = (f2 + f2)-1/2 □ 0.5773□ p/2e□ 0.5778 (31) arctg(e) □ 1.218 □ arctg(f) + arC^(^f) □ 1.219 (32)

p□ 3.1416□ e + f3 /10□ 3.1418□ e + f-f-S□ 3.1411 □ 4^/f p 3.144 (33)

l/pe□ 2.922□ (f + f)4/3 □ 2.924, 1ip□ 1.144□ f4 +f-f□ 1.145 (34)

O □ 0.9159 □ 4(f^l/f)/2 □ 0.9154□ (f + f)2S/p□ 0.918 (35)

Significantly more accurate ratios (with an accuracy of more than 10 14) were obtained by computer enumeration of even "simple" types of approximating expressions. Thus, for a linear-fractional approximation of the FMC by functions of the type

(where I, t, k, B are integers, usually changing in a cycle from -1000 to +1000), ratios were obtained that are correct with an accuracy of more than 11-12 decimal places, for example:

P □ (809-ft +130 ft) / (-80-ft + 925 ft) (36)

e □ (92 ^f + 295 ^f)/(340 f-693 f) (37)

n □ (660 e + 235 l/e) / (-214 e + 774 Te) (38)

C □ (635 e - 660 >/e)/ (389 e + 29 Te) (39)

O □ (732 e + 899 e)/(888 e + 835 Te) (40)

In conclusion, we point out that the question of the number of FMCs remains open. The FMC system, naturally, must first of all include the constants P, e, 1, φ(φ). Other MK can be

include in the FMK system as the range of considered mathematical problems expands. At the same time, MC can be combined into a MC system precisely due to the establishment of exact relationships between them.

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