entranceHow constant is denoted. The beauty of numbers
Communication formula for fundamental physical constants
and the structure of time and space.
(NIAT Researcher: Gravitational Constant Measurement Group (G)).
(This article is a continuation of the author's work on the relationship formula for fundamental physical constants (FPC), which the author published in the article (1 *). A model for combining the main four interactions and a new view of time and space are proposed. The article is also supplemented with new data based on the values \u200b\u200bof FFK received by KODATA in 1998, 2002 and 2006.)
1. Introduction.
2) Derivation of the relationship formula for fundamental physical constants: 
3) Combining four main types of interaction:
4) The 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 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. Although this opinion has not been fully verified.
To find a formula for the connection between the fundamental physical constants of the electromagnetic and gravitational interactions, the method of "sequential logical selection" was used. (this is the choice of certain variants of the formula and constants for substitution, based on the established physical prerequisites and criteria).
In our case, the following physical prerequisites and criteria for the selection of constants and variants of the formula 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 the gravitational interaction is set by those particles that simultaneously participate in the electromagnetic interaction.
They are: electron, proton and neutron.
3. The above mentioned particles set 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 (p. 2.3) - the interconnection 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 \u003d" 36 "height \u003d" 24 src \u003d "\u003e
4. Stable matter mainly consists of hydrogen, and its bulk is given by the mass of the proton. Therefore, all constants should be related to the proton mass, and the ratio of the electron and proton masses https://pandia.ru/text/78/455/images/image016_33.gif "width \u003d" 215 height \u003d 25 "height \u003d" 25 "\u003e
Where: - coefficient given by weak interaction;
https://pandia.ru/text/78/455/images/image019_28.gif "width \u003d" 27 "height \u003d" 24 src \u003d "\u003e is the coefficient set by the nuclear interaction.
In terms of its significance, the proposed formula for the connection of the constants of electromagnetic and gravitational interactions claims to unite gravity and electromagnetism, and upon a detailed examination of the elements of the presented formula, also to unite all four types of interactions.
Lack of the theory of numerical values \u200b\u200bof fundamental physical constants (FPC)
demanded to find mathematical and practical examples proving the truth of the formula for the connection of the fundamental physical constants of electromagnetic and gravitational interactions.
The above mathematical conclusions claim to be a discovery in the field of FFK theory and lay the foundation for the understanding of their numerical values.
2) Derivation of the connection formula for fundamental physical constants .
To find the main link in the formula for the coupling of constants, it is necessary to answer the question: "why are gravitational forces so weak in comparison with electromagnetic forces?" To do this, consider the most common element in the Universe - hydrogen. He also determines its main apparent mass, setting the intensity of the gravitational interaction.
The electric charges of the electron (-1) and proton (+1), which form hydrogen, are equal in magnitude; at the same time, their "gravitational charges" differ 1836 times. Such a different position of the electron and proton for the electromagnetic and gravitational interactions explains the weakness of gravitational forces, and the ratio of their masses should be included in the sought-for formula for the coupling of constants.
Let's write down the simplest version of the formula, taking into account the prerequisites (p. 2.3.) And the selection criterion (p. 1,2, 4):
Where: - characterizes the intensity of gravitational forces.
From data for 1976..gif "width \u003d" 123 "height \u003d" 50 src \u003d "\u003e
Let's find the module "x": 
The found value is well rounded to (12).
Substituting it, we get:
(1)
Found the discrepancy between the left and right side of the equation in formula (1):
There is practically no discrepancy for numbers with the degree "39". It should be noted that these numbers are dimensionless and do not depend on the chosen system of units.
Let us make a substitute in formula (1), proceeding from the premise (item 1) and selection criteria (item 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 ratio:
where: https://pandia.ru/text/78/455/images/image029_22.gif "width \u003d" 222 height \u003d 53 "height \u003d" 53 "\u003e
For x \u003d 2, y \u003d 3.0549 ie y rounds well to "3"
We write formula (1) with substitution:
(2)
Find the discrepancy in formula (2):
Using a fairly simple substitution, we get a reduction 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, it is necessary to further refine the formula (2). This is indicated by the prerequisites (Sec. 2, 3), as well as the selection criterion (Sec. 5), which speaks of the presence of a constant characterizing the neutron.
To substitute its mass in formula (2), it is necessary to find the degree of the following ratio:
Find the module z:
Having rounded z to "38", you can write formula (2) with a refining substitution:
(3)
Find the discrepancy in formula (3):
With precision error, meaning is equal to one.
Hence, we can conclude that formula (3) is the final version of the sought-for formula for the connection between the fundamental physical constants of the electromagnetic and gravitational interactions.
Let's write this formula without reciprocal values:
(4)
The found formula allows one to expressfundamental physical constants of gravitational interaction through constants of electromagnetic interaction.
3) Combining four main types of interaction.
Let's consider the formula (4) from the point of view of the selection criterion "5".
As expected, the sought formula consists of three coefficients:
Let's analyze each of the coefficients.
As seen, First coefficient is determined by the fact that the weak interaction has divided leptons and hadrons into two classes of particles with different mass values:
Hadrons are heavy particles
Leptons are light particles
The tenth degree in the fraction https://pandia.ru/text/78/455/images/image045_16.gif "width \u003d" 21 "height \u003d" 21 src \u003d "\u003e) reflects the intensity of electromagnetic interaction, and the degree" 3 "speaks about the three-dimensionality of the space of time, in which leptons and hadrons exist as particles of electromagnetic interaction.In terms of the importance of the formula found, this coefficient takes the second place.
Third coefficient Antique "href \u003d" / text / category / antikvariat / "rel \u003d" bookmark "\u003e antiquarks) multiply by 3 colors +1 gluon + 1 antigluon \u003d 38 states
As can be seen from the degree "38", the dimensionality of the space in which quarks exist, as components of a proton and a neutron, is equal to thirty-eight. In terms of importance, in the found formula, this coefficient takes the third place.
If we take orders of magnitude in the numerical values \u200b\u200bof the coefficients, then we get:
Substitute these values \u200b\u200binto formula (4):
Each of the coefficients, in order of magnitude, sets the intensity of the interaction it represents. Hence, we can conclude that formula (4) allows combining all four types of interactions and is the main super-unification formula.
The found form of the formula and the values \u200b\u200bof the degrees show that a single interaction for each interaction sets its own value for the dimensionality of space and time.
Unsuccessful attempts to combine all four interactions are explained by the fact that the same dimensionality of space was assumed for all types of interactions.
This assumption also led to a common erroneous unification approach:
weak interaction + electromagnetic interaction + nuclear interaction + gravitational interaction \u003d unified interaction.
And, as we can see, a single interaction sets the dimensionality of space and time
for each type of interaction.
This leads to a "new approach" in combining interactions:
1st stage - weak interaction in ten-dimensional space:
Electromagnetic interaction in three-dimensional space of time:
Nuclear interaction in thirty-eight-dimensional space:
2nd stage - grav. 1 + grav. 2 + grav. 3 \u003d grav. \u003d one 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 formula for the relationship of fundamental physical constants (FPC):
https://pandia.ru/text/78/455/images/image003_129.gif "width \u003d" 115 "height \u003d" 46 "\u003e the constant characterizes the gravitational interaction.
(the bulk of matter in the Universe is set by the proton mass, therefore the gravitational constant is set 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 in their masses gives the main contribution to the weakness of gravitational forces in comparison with other interactions).
The constant characterizes the electromagnetic interaction.
(electromagnetic interaction through charge gives its contribution 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 \u003d 38 states
As can be seen from the degree "38", the dimensionality of space in which quarks exist, as components of a proton and a neutron, is equal to 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 "energy of rest mass) speaks of the three-dimensionality of time.
Looking at time as a three-dimensional vector inverts our understanding of time as a scalar and requires replacing the entire integral-differential algebra and physics, where time is represented by a scalar.
If earlier to create a "time machine" (and this, in the language of mathematics, - to change the direction of the movement of time to the opposite, or to give the value of time a minus sign), it was necessary to go through "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 device for turning the vector of time, the creation of a "time machine" becomes a reality.
All of the above makes it necessary to revise the law of causality, and, therefore, the law of conservation of energy, and therefore - and 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 dimensionality 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 the system, then gravity (Newton's formula) and electromagnetism (Coulomb's formula) \u003d carry the characteristics of time.
Weak and nuclear interactions, 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 looped on each other, forming a single circle)
Gravity reflects outside time
general dimension (+1) \u003d
Electromagnetism reflects internal time
general dimension (+3) \u003d 
B) and there are two spaces: internal and external
(and they mutually penetrate each other)
Weak interaction reflects outside spaces
general dimension (+10) \u003d 
Nuclear Interaction Reflects Internal Space
general dimension (+38) \u003d
5) Practical proof of the formula.
The absence of an absolutely rigorous derivation of formula (4) requires a practical example of its verification. An example is the calculation of the value of the constant of gravitation:
(5)
In formula (5), the greatest error is in the constant of gravitation: https://pandia.ru/text/78/455/images/image067_14.gif "width \u003d" 62 height \u003d 24 "height \u003d" 24 "\u003e. from this one can find G with greater accuracy than the tabular value
Calculated value
(data of 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 a new, more accurate, - the value of G, adopted in 1986 (3 *)
data of KODATA (FFK) for 1986: Tabular https://pandia.ru/text/78/455/images/image072_12.gif "width \u003d" 332 "height \u003d" 51 "\u003e
We got the value - 40 times more accurate and included in the interval + 2, 3 https://pandia.ru/text/78/455/images/image074_13.gif "width \u003d" 307 "height \u003d" 51 src \u003d "\u003e
Estimated for more
Estimated for more
data of KODATA (FFK) for 2006 Tabular
Estimated for more
Compare table values:
data of KODATA (FFK) for 1976 Tabular https://pandia.ru/text/78/455/images/image082_12.gif "width \u003d" 79 "height \u003d" 21 src \u003d "\u003e
data of KODATA (FFK) for 1986 Tabular https://pandia.ru/text/78/455/images/image083_13.gif "width \u003d" 80 "height \u003d" 21 src \u003d "\u003e
data of KODATA (FFK) for 1998 Tabular https://pandia.ru/text/78/455/images/image084_12.gif "width \u003d" 79 "height \u003d" 21 src \u003d "\u003e
data of KODATA (FFK) for 2002 Tabular
for 2006..gif "width \u003d" 325 "height \u003d" 51 "\u003e
Value since 1976 to 2006 why, it is constantly increasing, and the accuracy has remained at the level, and in 1986 atmore 2006 This suggests that there is an unaccounted for hidden parameter in Newton's formula.
Let's compare the calculated values:
data of KODATA (FFK) for 1976 Estimated
for 1986..gif "width \u003d" 332 "height \u003d" 51 "\u003e
for 1998. gif "width \u003d" 340 "height \u003d" 51 "\u003e
for 2002..gif "width \u003d" 332 "height \u003d" 51 "\u003e
for 2006..gif "width \u003d" 328 "height \u003d" 51 "\u003e (6)
Self-consistency (in terms of statistics) with increasing accuracy
in133 times (!!!) s to calculated valuesG
talks about the suitability of the formula in 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 proofs of the formula and its structural analysis.
Having written the 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 - and write a mathematical equality.
The question of the transcendence of the values \u200b\u200bof the constants is removed after, having replaced h by in formula (4), equality cannot be achieved (the use in physics was that fatal error that did not allow finding the formula for the connection of constants (4; 5). strict equality under the substitution of the transcendental number also proves the correctness of the chosen equality condition for formula (4), and hence the rationality of the FFK.)
Consider one of the numerical values \u200b\u200bobtained when calculating formula (5):
data of KODATA (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 range of 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 proton mass to the electron mass in the tenth power will converge to the value (7). This is confirmed by new data for 1998:
data of KODATA (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 \u003d" 25 height \u003d 22 "height \u003d" 22 " \u003e
The proven convergence speaks of the exact equality of formula (4), which means that this formula is the final version and cannot be further refined, both in the physical and mathematical sense of the word.
Based on this, a statement can be made that claims to be a discovery:
THE VALUE OF FUNDAMENTAL PHYSICAL CONSTANTS (FPC) IN DEGREES PRESENTED IN THE FORMULA , Converge to simple rational fractions and are expressed through each other according to the formula (5).
This is confirmed by the fact that the new values \u200b\u200bof the ratio of the masses of the neutron and proton revealed the period in the following fraction:
data of KODATA (FFK) for 1998 
data of KODATA (FFK) for 2002 
Convergence to a number is evident: (8)
Based on the first found values \u200b\u200b(7; 8) and an intuitive idea of \u200b\u200bthe simple structure of constructions in nature, we can assume that the value of the primes 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 \u003d" 422 "height \u003d" 46 "\u003e
data of KODATA 1998:
data of KODATA 2002:
data of KODATA 2006:
Convergence to a number is evident: (9)
A more precise meaning can be found:
It is included in the range + 0.28 of the KODATA value for 2006 and is 25 times more accurate:
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 \u003d" 40 height \u003d 28 "height \u003d" 28 "\u003e:
Formula data:
The limited accuracy of tabular values \u200b\u200bdoes not allow direct calculation to find the exact numerical values \u200b\u200bto which the FFK converge in formula (5); the exception is the values \u200b\u200bof 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 the number () this is a period ... from here you can find: https://pandia.ru/text/78/455/images /image126_10.gif "width \u003d" 361 "height \u003d" 41 src \u003d "\u003e substitute 
https://pandia.ru/text/78/455/images/image129_9.gif "width \u003d" 586 "height \u003d" 44 src \u003d "\u003e. gif" width \u003d "215" height \u003d "45"\u003e
A more precise h can be found:
It is included in the interval + 0.61 of the KODATA value for 2006 and is 8.2 times more accurate:
7) Finding the exact values \u200b\u200bof FFK in the formula (4 and 5).
Let's write the exact values \u200b\u200bof FFK that we have already found:
A \u003d https: //pandia.ru/text/78/455/images/image137_8.gif "width \u003d" 147 height \u003d 57 "height \u003d" 57 "\u003e B \u003d 
Г \u003d https: //pandia.ru/text/78/455/images/image140_8.gif "width \u003d" 249 "height \u003d" 41 "\u003e
Е \u003d https: //pandia.ru/text/78/455/images/image142_8.gif "width \u003d" 293 "height \u003d" 44 "\u003e
Except https://pandia.ru/text/78/455/images/image144_9.gif "width \u003d" 31 "height \u003d" 24 "\u003e, the exact value of which we do not know yet. Let's write down" C "with the same accuracy which we know of:
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 Ж, is a rational number, since it is represented in them in the first degrees. 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 the "main 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 range + 0.94 of the KODATA value for 1976.
After averaging, we got:
(data of 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 \u200b\u200bof FFK"
(We multiply the most accurate one by "3": 8, the net period "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 \u200b\u200bof 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 \u003d" 81 "height \u003d" 24 "\u003e and creates an unaccounted shift in 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, it is premature to accept it as a whole.
(Most likely - no one dealt with this issue and "C" was taken "whole" out of negligence).
Using formula (4), it is possible to show that the speed of light is a RATIONAL number, however - NOT INTEGRAL.
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Shelaev A.N., Doctor of Physical and Mathematical Sciences, Professor, N.N. D.V. Skobeltsyn, Moscow State University M.V. Lomonosov
EXACT RELATIONSHIP BETWEEN FUNDAMENTAL MATHEMATICAL CONSTANTS
Problems of finding and interpreting exact relationships between fundamental mathematical constants (FMC), first of all P, e, constants
by the lot of the proportion φ \u003d (-1 + V5) / 2 □ 0.618, φ \u003d φ + 1 \u003d (1 + «s / 5) / 2, the Eyle
1 / k _lnn) \u003d _l e lnxdx □ 0.577, Catalan constant n ^ yes k \u003d J 0
G \u003d Z "\u003d o (_1) n / (2n +1) 2 \u003d | oX-1 arctan X dx □ 0.915, imaginary unit i \u003d 1
This article reports on finding various types of exact relationships between FMC, including between algebraic and transcendental ones.
Let's start with the constants of the golden ratio f, f. In addition to the above initial expressions for them, you can get other definitions, for example, as the limit of a sequence, continued fraction, the sum of nested radicals:
φ \u003d lim xn, where xn \u003d 1 / (1 + xn_1), x0 \u003d 1, n \u003d 1,2,3, ... (1)
φ \u003d 1/2 + lim xn, where xn \u003d 1 / 8_x2_1 / 2, x0 \u003d 1/8, n \u003d 1,2,3, ... (2)
φ \u003d φ + 1 \u003d 1 + - (3)
φ \u003d φ +1 \u003d 1 + 1 + yf [+ yl 1 + ... (4)
Note that in (1), (3) Xn and final fractions are expressed through the ratio of 2 consecutive Fibonacci numbers Bn \u003d 1,1,2,3,5,8, .... As a result, we get:
rn / rn + 1, Ф \u003d А
φ \u003d lim Fn / Fn + 1, Φ \u003d XG \u003d 1 (_1) P + 1 / (Pn-Fn + 1) (5)
ratios:
The relationship between the constants φ, φ, П and 1 \u003d is determined
b1p (1 1p f) \u003d 1/2, w (l / 2 - Ni f) \u003d (f + f) / 2 (6)
φ \u003d ^ 1+ W1 + (Ф + iW1 + (Ф + 2) Vi + T7
Taking into account that φ-φ \u003d 1 we obtain the following expression for n (φ):
n \u003d 4 - arctan [φ - ^ 1 + φ ^ / 1 + (φ +1) ^ 1 + (Ф + 2 ^ л / Г + TGT]
For the constants φ, φ, final expressions were also obtained in transcendental form, which, naturally, are reduced to algebraic expressions, for example:
φ \u003d 2 - sin (n / 10) \u003d tg (9)
Ф \u003d 2 - cos (n / 5) \u003d tg [(n - arctan (2)) / 2] (10)
The constant P can be determined and, for example, by the following ratios:
П \u003d 4-X °° \u003d 0 (-1) n / (2n +1) \u003d lim 2n 22+\u003e / 2 + V2 + --- V2 (11)
In this case, in (11), the number of radicals within the limit is equal to n. Also, it should be noted
that \\ / 2 + v 2 + 2 + ---- \u003d 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 \u003d 6 - arcsin \u003d 3 - arccos (12)
n \u003d 10 - arcsin (φ / 2) \u003d 10 - arccos ^ 5 - φ / 2) (13)
n \u003d 4 - (14)
n \u003d 4 - (15)
n \u003d 4 - (16)
n \u003d 4 - (17)
The constant e can also be defined by various expressions, for example:
e \u003d lim (1 + x) 1 / x \u003d lim n / ^ n! \u003d yj (A + 1) / (A-1), where A \u003d 1 + -Ц- (18)
x -n -yes 3 + 1
The connection of the constant e with other FMC can be realized, first of all, through the second remarkable limit, the Taylor and Euler formulas:
e \u003d lim [(2 / n) arctgx] -nx / 2 \u003d lim (tgx) -tg2x \u003d lim (2 - x) (n / 2\u003e tgnx / 2 (19) x-yes x-n / 4 x- one
e \u003d lim (1 + p / n) n / p, p \u003d п, ф, Ф, C, G (20)
e \u003d p1 / L, where L \u003d lim n (p1 / n -1), p \u003d п, ф, Ф, С ^ (21)
e \u003d 1 / p, p \u003d n, Ф, Ф, С, G (22)
eip \u003d cos (p) + i sin (p), i \u003d V-Y, p \u003d п, ф, Ф, С, G (23)
A large number of exact relationships between FMC can be obtained using integral relationships, for example, such as:
l / n \u003d 2 ^ 2p j cos (px2) dx \u003d 2 ^ / 2p j sin (px2) dx, p \u003d e ^, φ, C, G (24) J 0 »0
n \u003d Vp j0dx / (1 ± p cosx), p \u003d e, φ, φ, C, G (25)
G \u003d nln2 / 2-j 0ln (1 + x2) / (1 + x2) dx \u003d -nln2 / 2-j0 / 4ln (sinx) dx (26)
С \u003d -ln4 -4п 1/2 j 0 exp (-x2) lnxdx (27)
С \u003d j yes / x dx - ln (b / p), p, b \u003d n, e, φ, φ, G (28) 0
It is essential that in relation (28) Euler's constant C can be expressed not in terms of one, but in terms of two FMC p, b.
It is also interesting that from the relation linking P with other FMC,
(p / p) / sin (n / p) \u003d j0 dx / (1 + xp), p \u003d e, φ, φ, C, G (29)
you can get a new definition of the 1st wonderful limit:
lim (n / p) / sin (n / p) \u003d lim j dx / (1 + x) \u003d 1 (30)
In the course of research, a large number of interesting approximate relationships between PMC were also found. For example, such:
C □ 0.5772 □ 1§ (n / 6) \u003d (f2 + f2) -1/2 □ 0.5773 □ p / 2e □ 0.5778 (31) arctan (e) □ 1.218 □ arctan (f) + arC ^ (^ f) □ 1.219 (32)
n □ 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 relations (with an accuracy of more than 10 14) were obtained by computer enumeration of even "simple" types of approximating expressions. Thus, for linear-fractional approximation of the FMK 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), relations were obtained that are correct with an accuracy of more than 11-12 decimal places, for example:
P □ (809-f +130 f) / (-80-f + 925 f) (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\u003e / 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 PMC remains open. The FMC system, naturally, must first of all include the constants P, e, 1, φ (φ). Other MK can be
to include in the PMK system as the range of mathematical problems under consideration expands. In this case, MC can be combined into an MC system precisely due to the establishment of exact relationships between them.
Archimedes number
What is equal to: 3.1415926535 ... Up to 1.24 trillion decimal places have been counted today
When to celebrate π - the only constant that has its own holiday, and even two. March 14, or 3.14, corresponds to the first characters in the number record. And July 22, or 7/22, 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 mark the first date: it, unlike July 22, does not go on vacation
What is π? 3.14, the number from school circle problems. And at the same time - one of the main numbers in modern science. Physicists usually need π where there is no word about circles, say, to simulate a solar wind or explosion. The number π occurs in every second equation - you can open a theoretical physics textbook at random and choose any. If there is no textbook, a map of the world will do. An ordinary river with all its kinks and bends is π times longer than the path straight from its mouth to its source.
This is the fault of the space itself: it is homogeneous and symmetrical. That is why the front of the blast wave is a ball, and circles remain from stones 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 "curved space" would require other formulas.
The number π is as old as all mathematics: about 4 thousand. The oldest Sumerian tablets give him the number 25/8, or 3.125. The error is less than a percent. The Babylonians were not particularly fond of abstract mathematics, so π was derived empirically, simply by measuring the length of the circles. Incidentally, this is the first numerical simulation of the world.
The most elegant of arithmetic formulas for π is more than 600 years old: π / 4 \u003d 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 both in casinos and among sociologists.
There is even a "probabilistic" way to calculate the constant itself. First, you need to stock up on a bag of needles. Secondly, throw them, without aiming, on the floor, lined with chalk into strips of needle width. Then, when the bag is empty, divide the number of 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 applies: In the theory of chaos and catastrophes, which can be used to describe any phenomenon - from the reproduction of E. coli to 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 teaches at the prestigious Rockefeller University
When and how to celebrate day δ: 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 confusion, like a picture in a kaleidoscope. Mitchell Feigenbaum's mathematics in 1975 was not interested in the patterns themselves, but in the chaotic processes that make them appear.
Feigenbaum dealt with demography. He proved that the birth and death of people can also be modeled according to fractal laws. And 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 about 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 π, only for chaos.
Time
Napier's 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 based on it he built his tables of logarithms. 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 took from the last name
When and how to celebrate e day: After returning the bank loan
The number e is also a kind of counterpart to π. If π is responsible for space, then e - for time, and also manifests itself almost everywhere. For example, the radioactivity of polonium-210 decreases by a factor of e over the average lifetime of one atom, and the shell of the mollusk Nautilus 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 promising 1% per year will increase its deposit by about e times over 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 this way - arguing about how much money lenders earn.
Like π, e - transcendental number. Simply put, it cannot be expressed in terms of fractions and roots. There is a hypothesis that such numbers have all possible combinations of numbers in the infinite "tail" after the decimal point. For example, you can find there the text of this article written in binary code.
Shine
Fine structure constant
What is equal to: 1/137,0369990…
Who opened it and when: German physicist Arnold Sommerfeld, whose graduate students were two Nobel laureates at once - Heisenberg and Pauli. In 1916, even before the advent of real quantum mechanics, Sommerfeld introduced the 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 α day: Electrician's Day
The speed of light is an exceptional value. Faster, Einstein showed, neither a body nor a signal can move - 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 a fundamental constant. The problem is that there is nothing to measure it with. Kilometers per hour are not good: a kilometer is defined as the distance that light travels in 1 / 299,792.458 seconds, that is, it 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 without unnecessary noise changes throughout the entire Universe, humanity will not know about it.
This is where physicists come to the rescue of the value 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 either 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 "quantumness" 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 may wonder if α has not changed since the beginning of time. Physicists seriously admit a "defect" that once reached millionths of the present value. If it had reached 4%, there would be no humanity, because the thermonuclear fusion of carbon, the main element of living matter, would stop inside the stars.
Additive 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 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 86
The number i cannot be called a constant or even a real number. Textbooks describe it as a value that, when squared, gives minus one. In other words, it is the negative area side of the square. In reality, this does not happen. But sometimes the unreal can be useful too.
The history of the discovery of this constant is as follows. Mathematician Gerolamo Cardano, solving equations with cubes, introduced the imaginary unit. It was just an auxiliary trick - there was no i in the final answers: the results that contained it were discarded. But later, having looked closely at their "garbage", mathematicians tried to put it into action: multiply and divide ordinary numbers by an imaginary unit, add the results together and substitute them in new formulas. This is how the theory of complex numbers was born.
The downside is that “real” and “unreal” cannot be compared: it will not work to say that there is more - an imaginary unit or 1. On the other hand, there are practically no unsolvable equations, if we use complex numbers. Therefore, in complex calculations, it is more convenient to work with them and only at the very end "clean up" the answers. For example, in order to decipher a tomogram of the brain, one cannot do without i.
This is how physicists deal with fields and waves. We can even assume 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 combine the main mathematical constants and actions in one formula. The formula looks like this: e πi +1 \u003d 0, and some say that such a succinct 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 he received the Nobel Prize in Chemistry for the study of radioactivity: Rutherford owns the concept of "half-life" and the equations themselves describing the decay of isotopes
When and how to celebrate μ day: On the Day of Fighting Overweight, if such is introduced, this is the ratio of the masses of the two basic elementary particles, a proton and an electron. The proton is nothing more than the nucleus of the 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 the proton is greater than the mass of the electron. It turns out about 1836. Without such a difference in the "weight categories" of charged particles, there would be no molecules or 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 that came down to us 12 billion years later, and found that protons get heavier over time: the difference between prehistoric and modern values \u200b\u200bof μ 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 her 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 obscure of all the quantities that astronomers operate on. On the one hand, scientists are not completely sure of its existence, on the other, they are ready to explain with its help 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 a uniform expansion of the Universe, then Λ is a continuously accelerating growth. Einstein was the first to introduce it into the equations of general relativity when he suspected he was mistaken. His formulas indicated that space was either expanding or contracting, and it was hard to believe in that. A new member was needed to eliminate the conclusions that seemed implausible. After the discovery of Hubble, Einstein abandoned his constant.
The second birth, in the 90s of the last century, is due to the idea of \u200b\u200bdark energy "hidden" in every cubic centimeter of space. As follows from observations, the energy of an obscure nature must "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 relic radiation. These are prehistoric waves born in the first seconds of the existence of space. Astronomers consider them to be something like an X-ray that shines through the Universe. The "X-ray" showed that the dark energy in the world is 74% - more than anything else. However, since it is "smeared" throughout space, it turns out 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. 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, which 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 photograph of Pluto, the then not yet discovered planet, due to a defect in the photographic plate, left without attention
When and how to celebrate H day:January 0. Astronomical calendars start counting the New Year from this nonexistent date. 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 expands as a result of the Big Bang. Both the idea itself and the constant H go back to the findings of Edwin Hubble. Galaxies anywhere in the Universe scatter from each other and do this the faster the greater the distance between them. The famous constant is simply the factor by which the distance is multiplied to get the speed. It changes over time, but rather slowly.
The unit divided by H is 13.8 billion years, the time since the Big Bang. Hubble himself was the first to receive this figure. As proved later, Hubble's method was not entirely correct, but still it was wrong by less than a percent 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.
The 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 boundary must "run away" from us at superluminal speed. There is no contradiction with the theory of relativity: it is worth choosing the correct coordinate system in curved space-time, and the problem of speeding 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 ... μg
Where does it work:Physics of the microworld
Who opened it and when:Max Planck, creator of quantum mechanics, in 1899. Planck mass is just one of the set of quantities proposed by Planck as a "system of measures and weights" for the microworld. The definition of 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 path straight from its mouth to the source
When and how to celebrate the daymp: On the day of the opening of the Large Hadron Collider: microscopic black holes are going to get there
Jacob Bernoulli, gambling connoisseur and theorist, deduced e when discussing how much money lenders earn
Fitting a theory by size 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 flaw in this 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 out of the way.
Einstein's correction is the density of dark matter, which pushes the cosmos from the inside
Planck mass is a conditional border 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 size of which coincides 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 uniformly "smeared" over space - and the barrier along with it.
The Planck mass is the mass of the mosquito larva. But until gravitational collapse threatens the mosquito, quantum paradoxes won't affect 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 affect it.
Infinity
Graham's 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: Very soon, but very long
The key operation for this construction is Knuth's arrows. 33 is three to the third degree. 33 is three raised to three, which in turn is raised to the third degree, that is 3 27, or 7625597484987. Three arrows are already the number 37625597484987, where the three in the ladder of exponential exponents is repeated exactly as many times - 7625597484987 - times. This is already more than the number of atoms in the Universe: there are only 3,168. 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 quantities associated with the present or future dimensions 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-Akatieva
3D model of the endoplasmic reticulum of a eukaryotic cell with Terasaki ramps that connect flat membrane sheets
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, exactly such structures are present inside neutron stars. The so-called "nuclear paste" consists of parallel sheets connected by spiral shapes.
The amazing structural similarity of living cells and neutron stars - where did it come from? It is obvious that there is no direct connection between living cells and neutron stars. Just a coincidence?
Model of spiral 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, 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 when essentially different material objects exhibit similar properties.
For example, acoustic black holes observed for the first time in 2011 exhibit the same properties that real black holes should have in theory. In the first experimental acoustic black hole, a Bose-Einstein condensate of 100,000 rubidium atoms was spun up to supersonic speed in such a way that some parts of the condensate overcame the sound barrier, while the neighboring ones did not. The boundary of these parts of the condensate simulated 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 (the fictional quasiparticle personifies the quantum of the vibrational motion of crystal atoms). 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 fluid acts on sound in the same way as a real black hole acts on light. In principle, a sound black hole with phonons can be viewed as a kind of model for real curvature in spacetime.
If you look more broadly at the structural similarities in various physical phenomena, you can see an amazing order in natural chaos. All of 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 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 ridges, ice crystals, flickering flame, in 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 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 other geometric shapes, without which a person cannot make out a single word of it; 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 nonlinear 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 became the fundamental physical constant. It shows what is the angle of rotation in radians, contained in one full revolution when the body rotates. Accordingly, this constant is ubiquitous in the description of the rotational form of motion and 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, which is motionlessly suspended in a uniform gravity field with gravitational acceleration g, is
Under the conditions of the Earth's rotation, the plane of oscillation of the pendulum will slowly turn in the direction opposite to the direction of the Earth's rotation. The rotation speed of the oscillation plane of the pendulum depends on its geographic latitude.
Pi is an integral part of Planck's constant, the basic constant of quantum physics, which connects two systems of units - quantum and traditional. It connects the magnitude of the energy quantum of any linear vibrational 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 electromagnetic interaction, as well as in the formulas of hydromechanics, etc.
Other mathematical constants can be found in the natural world. For example, the number e, the base of the natural logarithm. This constant is included in the formula for the normal probability distribution, which is given by the probability density function:
Many natural phenomena are subject to normal distribution, including many characteristics of living organisms in a population. For example, the distribution of the size 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 pointed out, mathematics is the language that nature speaks to us.