Mathematical constants. Mathematical constant. Fine structure constant

Archimedes number

What is equal to: 3.1415926535…Today, 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 digits of the number. And July 22, or 7/22, is nothing more than a rough approximation of π as a fraction. At 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 vacation

What is pi? 3.14, number from school tasks 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 appears in every second equation - you can open the textbook theoretical physics randomly and choose any one. If you don't have a textbook, a world map will do. An ordinary river with all its kinks and bends is π times longer than the straight path from its mouth to its source.

The 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 the stones leave circles on the water. So π turns out to be 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 strongly curved Universe, π no longer plays such an important role. For example, in Lobachevsky’s geometry, a circle is four times longer than its diameter. Accordingly, rivers or explosions of “crooked space” would require other formulas.

The number π is as old as all mathematics: about 4 thousand. The oldest Sumerian tablets give it a figure of 25/8, or 3.125. The error is less than a percentage. The Babylonians were not particularly interested in abstract mathematics, so π was derived experimentally by simply measuring the length of circles. By the way, this is the first experiment in numerical modeling of the world.

The most elegant of the arithmetic formulas for π is more than 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 much more: π, for example, is part of the well-known “error function”, which works equally flawlessly in casinos and among sociologists.

There is even a “probabilistic” way to count the constant itself. First, you need to stock up on a bag of needles. Secondly, throw them, without aiming, onto the floor, lined with chalk into strips the width of an igloo. 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 it is used: In the theory of chaos and catastrophes, with the help of which you can describe any phenomena - from reproduction coli before the development of the Russian economy

Who opened it and when: American physicist Mitchell Feigenbaum in 1975. Unlike most other discoverers of constants (Archimedes, for example), he is alive and teaching at the prestigious Rockefeller University

When and how to celebrate δ day: Before general cleaning

What do broccoli, snowflakes and a Christmas tree 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. In 1975, mathematician Mitchell Feigenbaum became interested not in the patterns themselves, but in the chaotic processes that cause them to appear.

Feigenbaum studied demography. He proved that the birth and death of people can also be modeled according to fractal laws. That's when 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.

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

Time

Napier number

What is equal to: 2,718281828…

Who opened it and when: 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. What is known for certain is that the symbol e came from the last name

When and how to celebrate e-day: After repaying a bank loan

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

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

Like π, e- transcendental number. To put it simply, it cannot be expressed through fractions and roots. There is a hypothesis that such numbers in the infinite “tail” after the decimal point contain all possible combinations of numbers. For example, you can find the text of this article there, written in binary code.

Light

Constant fine structure

What is equal to: 1/137,0369990…

Who opened it and when: German physicist Arnold Sommerfeld, whose graduate students were two Nobel laureate- Heisenberg and Pauli. In 1916, even before the advent of real quantum mechanics, Sommerfeld introduced a constant in an ordinary article about 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 day α: 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, a gravitational wave or sound inside stars.

It seems clear that this is a law of universal importance. And yet the speed of light is not fundamental constant. The problem is that there is nothing to measure it with. Kilometers per hour will not do: a kilometer is defined as the distance that light travels in 1/299792.458 of a second, that is, itself expressed in terms of the speed of light. A platinum meter standard is also not a solution, 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 quietly throughout the Universe, humanity will not know about it.

This is where the quantity that connects the speed of light with atomic properties comes to the aid of physicists. 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 seconds, or 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 quality” of the world. The same problem is associated with both constants - there is nothing to compare them with. And together, in the form of α, they represent something like a guarantee of the constancy of the Universe.

One might wonder if α has not changed since the beginning of time. Physicists seriously admit a “defect” that once reached millionths of its current value. Had it reached 4%, humanity would not have existed, because inside the stars it would have ceased thermonuclear fusion carbon, the main element of living matter.

Addition to reality

Imaginary unit

What is equal to: √-1

Who opened it and when: Italian mathematician Gerolamo Cardano, friend of Leonardo da Vinci, in 1545. The driveshaft is named after him. According to one version, Cardano stole his discovery from Niccolò 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, gives 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. The mathematician Gerolamo Cardano, while solving equations with cubes, introduced the imaginary unit. This was just an auxiliary trick - there was no i in the final answers: results that contained it were discarded. But later, having taken a closer look at their “garbage,” mathematicians tried to put it to work: multiplying and dividing ordinary numbers by an imaginary unit, adding the results to each other and substituting them into new formulas. This is how the theory of complex numbers was born.

The downside is that “real” cannot be compared with “unreal”: it won’t work to say that the greater is an imaginary unit or 1. On the other hand, unsolvable equations, if we use complex numbers, practically none remains. Therefore, with complex calculations, it is more convenient to work with them and only “clean up” the answers at the very end. For example, to decipher a brain tomogram, you cannot do without i.

This is exactly how physicists treat fields and waves. One can even consider that they all exist in a complex space, and that what we see is only a shadow of the “real” processes. Quantum mechanics, where both the atom and man are waves, makes this interpretation even more convincing.

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

Microworld

Proton mass

What is equal to: 1836,152…

Who opened it and when: Ernest Rutherford, a physicist originally from New Zealand, in 1918. 10 years earlier I received 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 Weight Loss Day, if one is introduced, this 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 quantity 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 to be approximately 1836. Without such a difference in the “weight categories” of charged particles, there would be neither molecules nor solids. However, the atoms would remain, but they would behave completely differently.

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 meaningsμ was 0.012%.

Dark matter

Cosmological constant

What is equal to: 110-²³ g/m3

Who opened it and when: Albert Einstein in 1915. Einstein himself called its discovery his “major blunder.”

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

The cosmological constant is the most nebulous of all the quantities that astronomers operate with. On the one hand, scientists are not entirely 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 comes 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 continuously accelerating growth. Einstein was the first to introduce it into the equations of general relativity when he suspected an error. His formulas indicated that space was either expanding or contracting, which was hard to believe. A new member was needed to eliminate conclusions that seemed implausible. After Hubble's discovery, Einstein abandoned his constant.

The constant owes its second birth, in the 90s of the last century, to the idea of ​​dark energy “hidden” in every cubic centimeter space. As follows from observations, energy of an unclear nature should “push” space from the inside. Roughly speaking, this is a microscopic Big Bang, happening every second and everywhere. The density of dark energy is Λ.

The hypothesis was confirmed by observations of the cosmic microwave background radiation. These are prehistoric waves born in the first seconds of the existence of space. Astronomers consider them to be something like X-rays, shining through the Universe. The “X-ray image” showed that there is 74% dark energy in the world - more than everything else. However, since it is “smeared” throughout space, it turns out to be only 110-²³ grams per cubic meter.

Big bang

Hubble constant

What is equal to: 77 km/s/mps

Who opened it and when: Edwin Hubble, the 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 where the Hubble constant is mentioned is a certain Milton Humason, a man without higher education, who worked at the observatory as a laboratory assistant. Humason owns the first photograph of Pluto, then an undiscovered planet, which was ignored due to a defect in the photographic plate.

When and how to celebrate H Day: January 0. From this non-existent number, astronomical calendars begin counting the New Year. As well as about the moment itself big bang, 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 expands as a result of the Big Bang. Both the idea itself and the constant H go back to the conclusions of Edwin Hubble. Galaxies anywhere in the Universe scatter from each other and do so the faster the greater the distance between them. The famous constant is simply the factor by which distance is multiplied to get speed. It changes over time, but rather slowly.

One divided by H gives 13.8 billion years, the time since the Big Bang. Hubble himself was the first to obtain this figure. As was later proven, Hubble's method was not entirely correct, but it was still less than a percent wrong when compared with modern data. The mistake of the founding father of cosmology was that he considered the number H constant since 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: as soon as you choose the correct coordinate system in curved space-time, the problem of exceeding the speed immediately disappears. Therefore, the visible Universe does not end beyond the Hubble sphere; its radius is approximately three times larger.

Gravity

Planck mass

What is equal to: 21.76… µg

Where it works: Physics of the microworld

Who opened it and when: Max Planck, creator of quantum mechanics, in 1899. The Planck mass is just one of a set of quantities proposed by Planck as a “system of weights and measures” for the microcosm. The definition mentioning black holes—and the theory of gravity itself—appeared several decades later.

An ordinary river with all its kinks and bends is π times longer than the straight path 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 be created there

Jacob Bernoulli, a gambling expert and theorist, derived e by discussing how much money lenders earn

Matching theories to phenomena by size is a popular approach in the 20th century. If an elementary particle requires quantum mechanics, then a neutron star requires the theory of relativity. The harmfulness of such an attitude towards 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.

The Einstein correction is the density of dark matter, which pushes space from the inside

The Planck mass is the conventional boundary between “big” and “small”, that is, precisely 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 is that astrophysics treats 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” throughout space - and the barrier along with it.

The Planck mass is the mass of a mosquito larva. But as long as the mosquito is not threatened by gravitational collapse, quantum paradoxes will not affect it

mp is one of the few units in quantum mechanics that can 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 mosquito is not threatened by gravitational collapse, quantum paradoxes will not affect it.

Infinity

Graham number

What is equal to:

Who opened it and when: 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

When and how to celebrate G-Day: Not very soon, but for a very long time

The key operation for this design 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 are already the number 37625597484987, where the three in the ladder of power exponents is repeated exactly that many times - 7625597484987 - times. It's already more number There are only 3,168 atoms in the Universe. And in the formula for Graham’s number, it’s not even the result itself that 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 quantities associated with the present or future sizes of the Universe, planets, atoms and stars. Which, it seems, once again confirmed the frivolity of space against the backdrop of mathematics, by the means of which it can be comprehended.

Illustrations: Varvara Alyai-Akatyeva

Formula for the connection of fundamental physical constants

and the structure of time and space.

(NIAT researcher: group for measuring the gravitational constant (G)).

(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 KODATA 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 for unifying gravity and electromagnetism, it was believed that there was no direct connection between the fundamental physical constants of these two interactions. Although this opinion has not been fully verified.

To find the formula for the connection between the fundamental physical constants of electromagnetic and gravitational interaction, the method of “sequential logical selection” was used. (this is the selection of certain formula options and constants for substitution, based on established physical prerequisites and criteria).

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

Prerequisites.

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

2. The intensity of gravitational interaction is determined by those particles that simultaneously participate in 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 internal structure space and time.

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

Selection criteria.

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

2. Constants must satisfy physical premises.

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

4. Stable matter mainly consists of hydrogen, and its bulk is determined by the mass of the proton. 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 specified by weak interaction;

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

In terms of its significance, the proposed formula for the connection between the constants of electromagnetic and gravitational interaction claims to unite gravity and electromagnetism, and upon a detailed examination of the elements of the presented formula, claims to unite all four types of interactions.

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

required to find mathematical and practical examples that prove the truth of the formula for connecting the fundamental physical constants of electromagnetic and gravitational interaction.

The presented 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 connecting fundamental physical constants .

To find the main link in the formula for the connection of constants, it is necessary to 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.

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

Let's 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 to (12).

Substituting it, we get:

(1)

The discrepancy found between the left and right side equations in formula (1):

For numbers with degree “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 substitution 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 powers 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 is well rounded to “3”.

Let's write formula (1) with substitution:

(2)

Let's find the discrepancy in formula (2):

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

From data for 1976, (2*):

Since , further clarification of formula (2) is necessary. This is indicated by the prerequisites (items 2.3), 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 power of the following relationship:

Let's find the module z:

By rounding z to “38”, we can write formula (2) with a clarifying substitution:

(3)

Let's find the discrepancy in formula (3):

With accuracy errors, valueequals 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.

Let's write this formula without reciprocals:

(4)

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

3) Combining the four main types of interaction.

Let's consider formula (4) from the point of view of the selection criterion “5”.

As expected, the required formula consists of three coefficients:

Let's analyze each of the coefficients.

As you can see, First coefficient is determined by the fact that the weak interaction divided leptons and hadrons into two classes of particles with different masses:

Hadrons - heavy particles

Leptons are light particles

The tenth degree in the 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 the three-dimensionality of the space of time in which leptons and hadrons exist as particles of electromagnetic interaction. According to the significance of the formula found, this coefficient ranks second.

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

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

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

Let's substitute these values ​​into formula (4):

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

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.

The general erroneous approach of unification followed from this assumption:

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

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

for each type of interaction.

It follows from this that “ new approach» in combining interactions:

Stage 1 - weak interaction in ten-dimensional space:

Electromagnetic interaction in three-dimensional time space:

Nuclear interaction in thirty-eight-dimensional space:

Stage 2 – gr.1 + gr. 2 + engrav. 3 = gr. = unified 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” requires a different view of gravity, a view as a structure consisting of four “layers”:

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

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

Let us summarize the formula for the connection between fundamental physical constants (FPC):

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

(the bulk of matter in the Universe is determined by the mass of the proton, therefore the gravitational constant is determined by the interaction of protons with each other).

The constant characterizes the weak interaction.

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

The constant characterizes the electromagnetic interaction.

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

the constant characterizes nuclear interaction.

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

As can be seen from the degree “38”, the dimension of 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 also gives a new understanding of time as a multidimensional quality. Existence three types energy (1" potential energy 2" kinetic energy 3" rest mass energy) speaks of the three-dimensionality of time.

A look at time as a three-dimensional vector overturns our ideas about 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 mathematical terms, is to change the direction of movement of time to the opposite, or to give the value of time a minus sign), it was necessary to go through “0” time, now, approaching time as vector - to change direction to the opposite, you just need to rotate the time vector 180 degrees, and this does not require operating with the uncertainty of “0” time. This means that after creating a device for turning the time vector, the creation of a “time machine” becomes a reality.

All of the above forces us to reconsider the law of causality, and, therefore, the law of conservation of energy, and therefore others fundamental laws physicists (all these laws “suffer” from one-dimensionality).

If formula (4) allows us 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 the variability of a system, then gravity (Newton’s formula) and electromagnetism (Coulomb’s formula) = bear 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

(and they are mutually fixated on each other, forming a single circle)

Gravity reflects external time

total dimension(+1) =

Electromagnetism reflects internal time

overall dimension (+3)=

B) and there are two spaces: internal and external

(and they mutually penetrate each other)

Weak interaction reflects external spaces

total dimension (+10) =

Nuclear interaction reflects internal space

overall dimension (+38)=

5) Practical proof of the formula.

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

(5)

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

Estimated value

(data from KODATA (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 table value is underestimated. This is confirmed by the new, more accurate, value of G, adopted in 1986 (3*)

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

We got a value that is 40 times more accurate and included in the interval + 2, 3https://pandia.ru/text/78/455/images/image074_13.gif" width="307" height="51 src=">

Estimated for more

Estimated for more

KODATA (FFK) data for 2006 Tabular

Estimated for more

Let's compare the table values:

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

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

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

KODATA (FFK) data for 2002 Tabular

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

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

Let's compare the calculated values:

KODATA 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 (from a statistical point of view) with increasing accuracy

133 times (!!!) sto 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 proof of the truth of formula (4).

6) Mathematical proof 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 for 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 replacing h with in formula (4), it is not possible to achieve equality (use in physics was the fatal error that did not allow one to find the formula for connecting the constants (4; 5). Violation strict equality when substituting a transcendental number also proves the correctness of the chosen condition of equality to formula (4), and therefore the rationality of FFC.)

Let's consider one of the numerical values ​​obtained when calculating formula (5):

KODATA (FFK) data 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 of 0.99 of the calculated value. Since the presented fraction is taken entirely from formula (5), we can predict that the value of the ratio of the proton mass to the electron mass to the tenth power will converge to the value (7). This is confirmed by new data for 1998:

KODATA (FFK) data 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 clarification, 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 (FPC) IN THE POWERS PRESENTED IN THE FORMULA , CONVERGENCE TO SIMPLE RATIONAL FRACTIONS AND ARE EXPRESSED THROUGH EACH OTHER BY FORMULA (5).

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

KODATA (FFK) data for 1998

KODATA (FFK) data for 2002

There is convergence to the number: (8)

Based on the first found values ​​(7; 8) and the intuitive idea of ​​​​the simple structure of constructions in nature, we can assume that the value prime numbers, included in the fractions in formula (4) - 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">

KODATA data 1998:

KODATA data 2002:

KODATA data 2006:

There is convergence to the number: (9)

You can find a more precise value:

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

Let's substitute the found numbers (7) and (8) into the formula :

On the right we have a large prime number 8363, it should be present and on the left in the upper part of the formula, so 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 the tabular values ​​does not allow direct calculation to find the exact numerical values ​​to which the FFCs converge in formula (5); The exception is 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 - to show periodicity in the numbers of the last digits, for number() this is the period ... you can find it from here: 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">

A more precise h can be found:

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

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

Let's write the exact values ​​of the 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 meaning of which we do not yet know. Let’s write “C” with the same precision as which we know it to be:

At first glance, there is no period, but it should be noted that this, according to formula (4) and the construction of the exact numbers E and F, 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 manifest itself, this constant must be multiplied by certain numbers. For this constant, these numbers are the "primary divisors":

As you can see, the period (C) is “377”. From here you can find the exact value to which the values ​​of this constant converge:

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

After averaging we got:

(data from KODATA (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 “searching for rationality in the values ​​of FFK”

(We multiply the most accurate one by “3”: 8,. A pure period of “377” appears).

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 the FFK 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, accepting it whole is premature.

(Most likely, no one dealt with this issue and “C” was accepted as “whole” out of negligence).

Using formula (4), we can show that the speed of light is a RATIONAL number, however, it is NOT A WHOLE number.

Natural sciences

Physical and mathematical sciences Mathematics

Mathematical analysis

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

EXACT RELATIONSHIPS BETWEEN FUNDAMENTAL MATHEMATICAL CONSTANTS

Problems of finding and interpreting exact relationships between fundamental mathematical constants (FMC), primarily P, e, zo-

lot proportion φ = (-1 + V5)/2 □ 0.618, φ = φ + 1 = (1 + “s/5)/2, Eile constant

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

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

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

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

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

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

φ = φ + 1 = 1 +--(3)

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

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

gp/gp+1, Ф = A

f= lim Fn /Fn+1, Ф = 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) = 1 / 2, w(l /2 - Ni f) = (f + f)/2 (6)

φ = ^ 1+ W1 + (Ф + iW1 + (Ф + 2)Vi+T7

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

n = 4 - arctg[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 = 2 - sin(n /10) = tan (9)

Ф = 2 - cos(n / 5) = tan[(n - arctan(2)) / 2] (10)

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

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

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

that \/ 2 + v 2 + 2 +----= 2 (!) with infinite number radicals.

For the constant P, a whole series was also obtained trigonometric ratios, connecting it with other constants, for example:

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

n = 10 - arcsin(f /2) = 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 = lim n/^n! = yj(A + 1)/(A-1), where A = 1 +-Ц- (18)

x -n -yes 3 + 1

The connection of the constant e with other FMCs can be achieved, 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, Ф, C, G (20)

e = p1/L, where L = lim n (p1/n -1), p = n, f, Ф, С^ (21)

e = 1/p, p = p, Ф, Ф, С, G (22)

eip = cos(p) + i sin(p), i = V-Y, p = p, f, Ф, С, G (23)

A large number of exact relationships between FMCs can be obtained using integral relationships, for example, the following:

l/p = 2^2p j cos(px2)dx = 2^/2p j sin(px2)dx, p = e^, f,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)

С = -ln4 -4п 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 important 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 relationship connecting P with other FMCs,

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

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

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

During the research it was also found large number interesting approximate relationships between FMCs. For example, these:

C□ 0.5772□ 1§(p/6) = (ф2 +ф2)-1/2 □ 0.5773□ p/2е□ 0.5778 (31) arctg(e) □ 1.218 □ arctg(ph) + agC^(^f) □ 1.219 (32)

p□ 3.1416□ e + f3 /10□ 3.1418□ e + f-f-C□ 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)2C/p□ 0.918 (35)

Significantly more accurate relationships (with an accuracy of more than 10 14) were obtained by computer search of even “simple” types of approximating expressions. Thus, for fractional-linear approximation of the FMC by functions of the type (u φ + m φ) / (k φ + B φ),

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

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

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

p □ (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 MKs are possible

be included in the FMC system as the range of mathematical problems under consideration expands. At the same time, MK can be combined into an MK system precisely due to the establishment of precise relationships between them.

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In 2013, a group of molecular biologists from the United States studied a very interesting form of endoplasmic reticulum - an organelle inside a eukaryotic cell. The membrane of this organelle 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, the biologists’ work was noticed by astrophysicists. They were amazed: after all, exactly such structures are present inside neutron stars. The so-called “nuclear paste” consists of parallel sheets connected in spiral shapes.

Amazing structural similarity between 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 helical connections between flat sheets of membrane in a eukaryotic cell

There is an assumption that the laws of nature act on all objects of the micro- and macroworld in such a way that some of the most optimal forms and configurations appear as if by themselves. In other words, objects 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 when essentially different material objects exhibit similar properties.

For example, acoustic black holes, first observed in 2011, exhibit the same properties that real black holes are theoretically expected to 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, but neighboring parts did not. The boundary of these parts of the condensate simulated the event horizon of a black hole, where the flow speed is exactly equal to the speed of sound. At temperatures near absolute zero, sound begins to behave like quantum particles - phonons (the fictional quasiparticle personifies a quantum oscillatory motion atoms of the crystal). It turned out that the “sonic” black hole absorbs particles in the same way as a real black hole absorbs photons. Thus, the flow of liquid affects sound in the same way as a real black hole acts on light. Basically, sound black hole with phonons can be considered as a kind of model of real curvature in space-time.

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

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

The Barnsley fern is constructed 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, and on the sea coast. 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 written 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 shapes, without which it is impossible for a person to understand a single word of hers; without them he is like one wandering in the dark.”

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 a population, the growth rate of which dynamically decreases as it approaches 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 shapes are the most rational and suitable for many natural objects. In particular, the number 2π became a fundamental physical constant. It shows the angle of rotation in radians contained in one full revolution when rotating the body. Accordingly, this constant is found everywhere 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 oscillations of a mathematical pendulum of length L, motionlessly suspended in a uniform gravitational field with free fall acceleration g, is equal to

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

Pi is integral part Planck's constant- main constant quantum physics, which connects two systems of units - quantum and traditional. It connects the magnitude 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 force of electromagnetic interaction, as well as in the formulas of fluid mechanics, etc.

There are other mathematical constants found in the natural world. For example, number e, the base of the natural logarithm. This constant is included in the formula normal distribution probabilities, which is given by the probability density function:

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

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 the entire living and inanimate world around. As Galileo Galilei rightly noted, mathematics is the language in which nature speaks to us.