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Isaev, Alexander V.
Sections 1 and 2
VIRTUAL Cosmology (worlds Isaeva)
ABSTRACT
This book covers mainly about beauty, harmony and perfection of the world of natural numbers 0, 1, 2, 3, 4, 5, 6, 7, ... Of course, that I - not the first one with the sacred awe relates to the world of numbers. Here, above all, it should be called Pythagoras (570 - 490 years. BC. Er.), Which according to legend the first called himself a philosopher, which means "lover of wisdom." He first called the universe of space, that is "perfect order". The subject of his teaching was the world as a harmonious whole, subordinate to the laws of harmony and number. World harmony, which is the law of the universe, there is unity in the set and the set in unity. How to think this truth? Direct response to this is the number: it combines multiple, it is the beginning of each measure. The so-called Pythagoreans, hand in mathematics, the first pushed them forward, fed on these sciences, they recognized the mathematical start of the outbreak of everything. From such beginnings, of course, are the first number. In numbers they saw as a set of analogies, or similarities with things. They further suggests a number of properties and relationships of musical harmony, and as all the other things in nature were named the likeness of numbers, the number of the same - first of all nature, and they acknowledged that the number of elements are the elements of all things, and that the whole sky there is harmony and number (Aristotle, Met., I, 5).
My "doctrine" can be called Pythagoreanism of the XXI century, since in the world of numbers, I find some "reflection" of the latest discoveries in physics, for example, string theory, which attempts to combine the microcosm of elementary particles with the laws of the vast cosmos. And my key know-how "- the identification of a number of positive integers with the flow of time quanta (Planck time) - and it was simply impossible at the time of Pythagoras. My "doctrine" seems to Pythagoreanism in the sense that part of it also falls under the definition of such concepts as ... "religion." And despite the "deathly silence" on the part of professionals (physicists and mathematicians), I continue to just believe in the power of the world of numbers, in that for me perfectly obvious fact that the world of numbers "reflect the" fundamentals of the universe.
I expounded the material available to so many, starting with high school students. Moreover, my book is useful even to those who allegedly "did not like" mathematics, since even the great English philosopher and scientist Roger Bacon (c. 1214-1294) said: "Whoever does not know mathematics can not learn any other science and can not even detect their ignorance. " But readers who are familiar with at least well-known program "Excel", may well not only to test my numerous hypotheses, but also they will be able to perform similar studies of the world of numbers on a computer. And if before the reader, for sure, evaluated my work with the words "All this - crap" (my first book was published in 1998), we now discern a more cautious assessment: "This is something there." It only remains to hope that in the near future, many will say about these studies, the world of numbers that, say, "That's all - so obvious!" ...
1. ANA GREAT
In this chapter, collected sayings of great men of the world of numbers (and mathematics in general). It is in these prophetic words, I draw strength for their next works, because the terms of my communication is practically zero, and even on the internet I have found almost no supporters. Well, professionals (Physics and Mathematics), alas, just ... keep silent regarding my research, hypotheses, texts ...
Pythagoras (c. 570-500 BC. Er.)
"God - this number." "The wisest - number". "The number of such things."
"Prototype and first principles can not be a clear exposition on the words because they are difficult to comprehend and difficult to make - that's why have to clear learning to resort to numbers."
"It is not because of numbers, but according to the number, because in those - the primary order ..."
Roger Bacon (c. 1214-1292)
"Anyone who does not know mathematics can not learn any other science and can not even detect their ignorance."
Leonardo da Vinci (1452-1519)
"Anyone who blames the supreme accuracy of Mathematics, fed by the confusion and never give up the tricks Sufi sciences, generating endless chatter." ...
"There is no credibility there in the sciences, where you can not make any of the Mathematical Sciences.
Leonhard Euler (1707-1783)
"Of all the issues dealt with in mathematics, not those that would be considered at the present time, more barren and devoid of proposals than the problems concerning the nature of numbers and their divisors .... In this regard, the current mathematics are very different from the ancients, which gave much more important studies of this kind. ... Mathematics, probably would never have reached such a degree of perfection, if the ancients did not have devoted so much energy development issues that today most neglected because of their imaginary futility. "
Carl Gauss (1777-1855)
"Mathematics - the queen of sciences, and number theory - the queen of mathematics".
Carl Jacobi (1804-1851)
"... The sole purpose of science is to exalt the human mind, and with such an approach the question of numbers is as significant as the question of the system of the world. "
Leopold Kronecker (1823 - 1891)
"Integers made the Lord God, and everything else - the case of human hands."
Henri Poincare (1854-1912)
"... What is mathematics? ...
I need to distinguish between people who asked a similar question. People practical demand of us only ways to gain money. These people do not deserve an answer. Rather, they should ask for what they are accumulating wealth, and whether you want to spend time on their purchase and neglected art and science, which only makes our spirit is able to enjoy. "
J. Littlewood (1885 -1977)
"It is now universally recognized that pure mathematics can lead to unexpected conclusions, and even have an impact on everyday life."
"The theory of numbers, more than any other mathematical discipline, vulnerable to reproach, that some of its problems arising issues, which generally should not be put. I personally think that the danger is severe, resulting in a concentrated reflection within a reasonable time or there are new ideas and methods, or have a problem just leave. "Perfect numbers" certainly never any benefit not brought, but they did not cause much harm. "
A. S. Besicovitch (1891 -1970)
"Reputation mathematics based on the number of bad evidence, which he invented.
(Works pioneers clumsy).
Richard Feynman (1918 - 1988)
"... The physical representation of the world ... is now the main part of the culture of our era."
2. MY KEY
Internet sites, in general, are not designed for text with mathematical symbols, formulas, graphs. Even the simplest characters different sites "know" differently or do not "understand". This is more than strange, given that truth is written clearly in the language of mathematics (see above sayings of great people). But all of this - a topic for a separate serious conversation. Here I will only add that my texts look strange to the eyes of professionals (physicists, mathematicians). For example, even the simplest mathematical expressions I write, in general, as is done in the formulas of the program ... "Excel":
10 ^ 3 - this is number 10 in grade 3 (when the number 10 multiplied by itself 3 times: 10 ∙ 10 ∙ 10 = 1000);
10 ^ -35 - the number 10 to the minus 35 (ie 1 / 10 ~ 35 - a unit divided by the number 10 ^ 35);
N ^ 0,5 - is the number N of 0.5 degree (or degree of 1 / 2, we have it - the square root of the number N).
These and similar mathematical "strange" just allow me to post their texts on a variety of sites on the Internet without additional modifications (editing). A careful reader will quite easily get used to such "oddities" in the text, especially if it is primarily interested in the meaning of the text ...
+, -, = - Arithmetic operations: "addition", "subtraction", "equal";
(or "x"), /, ^ - "multiplication", "division", "exponentiation" ("cap");
8E 60 = 8 10 ^ 60 the exponential number format (may be on the charts in my book);
(R ^ 2) - the value of the reliability of the approximation of the trend line on the chart (ideally tends to unity);
- infinity (as opposed to the notion of a finite);
N - means that the value of N tends to infinity;
, , , - sum, product, square root, integral;
, , , - "less", "more", "less (greater) than or equal to";
- "equal by definition, for example n! 1 2 3 ... n;
n - factorial, that is n! 1 2 3 ... n (formula in combinatorics);
- a symbol of the approximate equality, for example, ^ 2 3,14 ^ 2 10;
const - a numerical constant (when it matters to us is not essential);
N = f (X) - the value of N is some function (f) the value of X (it is - the argument of the function);
lgN - logarithm of the decimal number N [strictly speaking, should be written lg (N)] - an elementary function;
lnN - natural logarithm of the number N [strictly speaking, to write ln (N)] - an elementary function;
lnlnN - double the natural logarithm of N [strictly speaking, we should write ln (ln (N))];
exp (N) - exponential function with base e = 2, 718 ... in other words, exp (N) e ^ N (the number of "e" in the degree of N);
abs (N) - absolute value (modulus) of N, then N is the number without its sign (plus or minus);
max, min - maximum, minimum (for example, for certain values of T can be written: Tmin, Tmax);
lim - limit (function, sequence, and so on.)
A (N) - function "Antje" (integer part of x), for example, if x = 5,761, then A (x) = 5;
(A; b), [a; b], (a; b] - interval interval interval (where a and b - some integers with a <b);
~ (Tilde), this mathematical symbol in my book has two meanings:
- The equality of orders (eg, the number N ~ 10 ^ 3 could mean, say, N = 630, and N = 2050);
- A sign of the asymptotic equality (eg, the law K ~ N / lnN, which will be discussed below).
3. BASIC CONCEPTS AND DEFINITIONS
Age of the universe - the time elapsed from the time when there was a universe (time, matter, stars, planets, etc.). Modern science believes that our universe came about 13.75 billion years ago (plus or minus 0.11 billion years). This estimate is adopted on the basis of one of the common models of the universe - the so-called standard cosmological ΛCDM-model. Since even in the special theory of relativity time depends on the motion of the observer, and in general relativity - and even on the position, you will need to clarify what is meant in this case under the age of the universe. In today's presentation of the universe - is the maximum time that would have measured the clock since the Big Bang to the present time, getting them now in our hands. Obviously, the age of the universe can be easily expressed in seconds:
13,750,000,000 years 365 days ∙ ∙ ∙ 24 ∙ 60 minutes 60 seconds = 433.620.000.000.000.000 = 4,3 ∙ 10 ^ 17 seconds.
Radius of the universe - is the path traveled by photons (quanta) of light during the existence of the universe (for 4,3 ∙ 10 ^ 17 seconds). Because the speed of light in vacuum is known (299792458 km / s), then multiplying this rate by age of the universe - we get ... radius of the Universe: 1,3 ∙ 10 ^ 26 meters.
Big Bang - a cosmological theory of the expansion of the universe, to whom the universe was in a singular state. Modern conceptions of the theory of Big Bang theory of the hot universe are as follows. Universe came into 13.75 billion years ago from some initial "singular" condition and has since continuously expanded and cooled. According to certain restrictions on the applicability of modern physical theories, the earliest moment, allowing the description, is the moment of Planck era with a temperature of about 10 ^ 32 K (Planck temperature) and a density of about 10 ^ 93 g / cc (Planck density). Early Universe was a high homogeneous and isotropic medium with an unusually high energy density, temperature and pressure. As a result of expansion and cooling of the universe took place phase transitions analogous to the condensation of liquid from the gas, but applied to elementary particles.
Approximately 10 ^ -35 seconds (less than the moment!) After the Planck epoch a phase transition caused exponential expansion of the universe. This period is known as cosmic inflation. After the end of this period, the building material of the universe consisted of a quark-gluon plasma. As time passes, the temperature dropped to values for which was made possible following a phase transition, which is called baryogenesis. At this stage, the quarks and gluons combined into baryons such as protons and neutrons. At the same time going on the asymmetric formation of a matter that has prevailed and antimatter, which mutually annihilate, transforming into radiation.
A further drop in temperature resulted in the following phase transitions - the formation of physical forces and elementary particles in their modern form. After that came the era of nucleosynthesis, in which protons are teaming up with neutrons to form nuclei of deuterium, helium-4 and a few other light isotopes. After a further fall of temperature and the expansion of the universe came following a transitional moment in which gravity was the dominant force. After 380 thousand years after the Big Bang, the temperature had fallen so much that it became possible existence of hydrogen atoms (before ionization and recombination of protons and electrons are in equilibrium). After the recombination era matter became transparent to radiation, which propagates freely in space, came to us in the form of the CMB.
Extrapolation of the observed expansion of the universe back in time leads using the general theory of relativity and other alternative theories of gravity to an infinite density and temperature at the final time in the past. Moreover, the theory gives no opportunity to talk about anything that preceded this moment (because our mathematical model of space-time at the Big Bang loses its applicability: for this theory does not deny the possibility of the existence of anything before the Big Bang) and the size of the universe then vanish - it was compressed to a point. This condition is called the cosmological singularity and signals the inadequacy of describing the universe of classical general relativity. How close to the singularity is possible to extrapolate the known physics, is the subject of scientific debate, but almost universally accepted that the era of doplankovskuyu consider the known methods can not. Many scientists jokingly, half-seriously called the cosmological singularity of the "birth" (or "creation") of the universe. Inability to avoid the singularity in cosmological models of general relativity has been proved, among other theorems on the singularities of Roger Penrose and Stephen Hawking at the end of 1960. Its existence is one of the drivers of the construction of alternative and quantum gravity theories that try to solve this problem.
"Generally speaking" - the expression on the strict language of mathematics means that "there are times when it is not so." For example, one can say with certainty that people with liberal arts education, in general, will not read this book of mine (alas, is just the next paragraph may "scare off" those readers).
Mathematical constant - the value whose value does not change, in which it is opposed to a variable. Unlike physical constants (see below), the mathematical constant defined independently of any kind was physical measurements. Below are over 20 common mathematical constants (in increasing order of magnitude), with accuracy, in general, up to 9-th digit after the decimal point (for a 12-Constants know much more significant digits). We must assume that the number of mathematical constants can increase with time due to new research in mathematics.
-0.0000000027 Constant de Bruijn - Newman (number theory);
0.007874997 constant Chaitin (information theory);
0.261497212 constant Meysselya - Mertens (number theory);
0.577215664 constant Euler - Mascheroni (number theory);
0.660161815 constant prime twins (number theory);
0.70258 constant Embree - Treftena (number theory);
0.764223653 Landau constant - Ramanujan (number theory);
0.870588380 Bruna constant for prime quadruples (number theory);
0.915965594 constant Catalan (Combinatorics);
1.08366 constant Legendre (English) (number theory);
1.131988240 constant Visvanata (number theory);
1.414213562 constant Pythagoras, the square root of 2 (ordinary mathematics);
1.451369234 constant Ramanujan - Soldner (number theory);
1.606695152 constant Erdos - Borveyna (number theory);
1.618033988 golden ratio or the number of Phidias (ordinary mathematics);
1.732050807 Teodorusa constant, square root of 3 (ordinary mathematics);
1.902160582 constant Bruna for prime twins (number theory);
2.502907875 Feigenbaum constant (chaos theory);
2.718281828 number of "e" or a constant Napier, the base of natural logarithm (ordinary mathematics);
3.058198247 constants of-Gaussian (number theory);
3.141592653 number of pi, Archimedes constant (ordinary mathematics);
4.669201609 Feigenbaum constant (chaos theory);
6174 constant Kaprekara (number theory).
Physical constants (fundamental physical constants)
In contrast to the immutable (unchanging forever), mathematical constants, physical constants are likely to change ... (!) During the evolution of the universe - some scientific evidence of this (though still highly controversial) have emerged in recent years. However, even if the physical constants and vary with time, very slowly, and any significant changes should be expected only at scales of order the age of the Universe (about 13.75 billion years). In other words, the rate of change of physical constants are so minuscule that they are still beyond the technical capabilities of experimental science. It is difficult to overestimate science (including philosophy), the value of physical constants, since they characterize the properties of our world (Universe) in general and arise in mathematics (and the only true!) Describing the world through theoretical physics. The fundamental physical constants (they are many and various), in particular, include:
- Gravitational constant (G) [in parentheses - the drive letter in the book];
- Planck's constant (h) [the quantum of action - the basic constant of quantum theory, with its founders was Max Planck];
- Elementary charge (e) [the minimum portion (quantum) of electric charge];
- Planck charge (Qpl) [one of the basic units of the Planck units];
- The speed of light (c), equal to 299,792,458 km / s, and in nature (the universe), nothing can travel faster than photons (quanta) of light.
Planck length (the unit length)
With three physical constants (G, h, c) Physics, "constructed" (by a straightforward formula), a new physical constant, called Planck length (Lpl):
Lpl = [h / (2PI) ∙ G / c ^ 3] ^ (1 / 2) = 1,616 ∙ 10 ^ -35 meters. (3.1)
For its size the Planck length is far beyond human imagination. Judge for yourself. We can easily imagine one millimeter (mm) - is one thousandth (1 / 1000) of a meter (m), ie, in terms of mathematics, 1 mm - is 0.001 m, or, in other words, it's 10 in the "minus" third degree (10 ^ -3) of the length of a meter. Most "watchful" of us can probably imagine ... even one micron (um) - is one millionth (1 / 1000000) of a meter, that is 1 micron - is 10 minus 6-degree (by the way, the thickness a human hair - an average of 80 microns, and at the exit of the hair yourselfers can draw even ... the whole picture!). Now try to imagine a fraction of a meter, written as 10 minus 35-degree - it is this and is equal to the Planck length! American physicist Brian Greene (born 1963), one of the most renowned experts on string theory, in his famous non-fiction book "The Elegant Universe ..." gives the following comparison: if a tiny atom (cesium) to increase the diameter of the Universe (2, 6 ∙ 10 ^ 26 m), even if the Planck length becomes equal to the total, but the average tree height (9 meters). Note that among the atoms of all known chemical elements cesium atom has a maximum diameter of about 4,5 ∙ 10 ^ -10 m and the smallest atom - an atom of helium with a diameter of about 6,4 ∙ 10 ^ -11 m.
Planck time (elementary time interval)
Planck time (Tm) - this is the time at which photons (quanta) of light will overcome the Planck length:
Mp = Lpl / c = 5,391 ∙ 10 ^ -44 seconds. (3.2)
For brevity, instead of "Planck time" we often use the following notation: Evie - an elementary time interval. This is the minimum time interval that is required for the occurrence of any conceivable physical event. Moreover, some theories argue that at this level is already quantized, discrete character, though in everyday life, time seems to be something continuous (as a "river of time"). To date, the smallest experimentally observed time of the order of attoseconds (10 ^ -18 seconds), which corresponds to the order of 10 ^ 25 Evie.
All Planck values (length, time, mass, etc.) - it is also important physical constants.
According to the Big Bang theory, we can not say anything about the universe at the initial time, although it is assumed that it contains all the fundamental interactions, as well as all kinds of matter and energy. Space-time begins to expand from a single point. After one Evie (one Planck time) after the event, according to modern theoretical physics, gravitational forces are separated from the other forces.
The time elapsed since the Big Bang (13.75 billion years = 4,3 ∙ 10 ^ 17 seconds), approximately equal to 8 ∙ 10 ^ 60 Evie, or (in the first approximation), 10 ^ 61 Evie.
Space-time - is the main form of existence of matter, which are crucial for the construction of a physical picture of the world, our universe. In modern quantum theory of space and time play a central role, there is even a hypothesis, where the visible matter (consisting in 99,9% of the atoms of hydrogen and helium) is considered no more than a disturbance of this basic structure. The average density of visible matter in the universe is estimated as 1 hydrogen atom per cubic space with an edge of 2.6 m (this can be represented as a single hydrogen atom in a small room ...), that is our universe - it is almost "empty" space-time which possible (in terms of science), discrete (at the deepest consideration - at the Planck size) and expands.
Expansion of the Universe - a phenomenon predicted by general relativity, and consists in a homogeneous and isotropic expansion of space throughout the universe. Experimentally observed expansion of the universe in the form of execution Hubble law. Beginning of the expansion of the Universe science considers the so-called Big Bang. Expansion of the universe - just a hypothesis with a large number of assumptions. One of them - the calculation of rate of change of location of objects in the universe based on observations with various telescopes. However, this does not mean that the observed law of motion can be extrapolated to other time periods.
According to the theory of relativity, the universe has three spatial dimensions and one time dimension. The concept of space-time has historically played a key role in creating the geometric theory of gravitation. As part of the general theory of relativity, the gravitational field is reduced to manifestations of the geometry of four-dimensional space-time, which in this theory is not flat.
The number of measurements needed to describe the universe not completely defined. String theory (superstring), for example, required the presence of 10 (counting time), and now even the 11 measurements (in the M-theory). It is assumed that the additional (unobserved) 6 or 7 dimensions are collapsed (compactified) to the Planck size, so that experimentally they can not be detected. It is expected, however, that these measurements are somehow manifested in the macroscopic scale.
A large segment - a segment of the natural numbers [0; N *], containing as many integers as the Planck time (Evie) is contained at the age of the universe. That is a large segment of the numbers 0, 1, 2, 3, 4, 5, 6, 7, ..., N *, where N * = 8 10 ^ 60 - conventional (approximate) boundary of the Great segment, since the exact age of the universe known (see above). For the very rough estimates can be assumed that N * = 10 ^ 61.
Large segment is a discrete structure, because we consider the integers (0, 1, 2, 3, 4, ...), the latter, we can say ... "extended": 0, 1, 1 +1, 1 +1 +1, 1 +1 +1 +1, .... And if every one identified with the Planck time (Evie), then the natural series embodies a simple mathematical model of space-time - is a key hypothesis of my theory (virtual cosmology). Of course, this hypothesis is quite controversial, but is in itself an identification ("mathematical") units with a (physical) the Planck time (Evie) - the idea is quite fruitful, because thanks to a natural number and dry mathematical laws (number theory) like "come to life "in time and in our imagination ... It is, at least, may serve to promote the foundations of mathematics among the public at large.
Virtual cosmology - is one of the last names of my theory, which originated in 1997 was when I began to conduct various studies of the natural numbers 1, 2, 3, 4, 5, 6, 7, .... using a personal computer (PC), and the construction and analysis of graphs - have become the main tool for penetrating the mysteries of the world of numbers. That is why the material accumulated in this way, I called - graphic theory of natural numbers (GTNCH) or worlds Isayev (all positive integers i "divided" into different worlds, see below). In contrast to the well-known theory of numbers (rather complicated section of higher mathematics), GTNCH was popular for a wide audience. I note that the ability to work in a spreadsheet "Excel", and, especially, the ability to write a simple utility for PC - promises to every inquisitive reader a lot of "wonderful discoveries" in the endless and mysterious depths of the natural numbers. For example, I have such "discoveries" was typed into 9 books, which are published at my expense in the 1997 - 2006. (On the Internet, I came out only at the end of 2009).
In 2007-2008. I discovered an interesting analytical relation of real numbers from interval (0, 1) and (1, 2) with the remaining (large) real and natural numbers. Hence the name "GTNCH" has become "tight", and originated a new name for my theory - the "virtual Cosmology" (or again - the worlds Isaeva, but now, rather, in a philosophical sense: as my worldview as Pythagoreanism XXI century etc.).
Of course, that the virtual cosmology should not be contrary to accepted physics and mathematics, but, since the apparent lack of competence of the author, is, alas, may have a place in my books. Furthermore, it should remind the reader of the truism: any new theory is simply doomed to failure.
A characteristic feature of the virtual cosmology is that it very often appeals to the real physical world in a highly questionable and, at the same time, curious, intriguing reflektsiyah.
Reflektsiya (from pozdnelat. Reflexio - Reflection) - difficult to explain "reflection" of the world the numbers of real (physical) reality (the actual structure of space-time structure of the universe). I coined the term "reflektsiya" is intended to emphasize the difficulty of my analogy: no clear reflection, and "God knows that ..." - some reflektsii. Skeptics may feel that reflektsii - it's just ... reflection of the author, but it seems to me, not the worst use of our mind ...
My reflektsii - an attempt to prove that the abstract world of numbers and the real physical world - are isomorphic (at least partially, if at all possible). The concept of "isomorphism" can be explained as an example the following assertion: the number of partitions of a convex heptagon into triangles equals the number of options for placement of brackets to 6 letters. That is a triangulation of polygons is isomorphic (similar) problem of placement of the brackets (which leads to the Catalan numbers).
Reflektsii not form a unified picture, they may even contradict each other. But they have something attractive and certainly instructive for an inquisitive mind. In addition, reflektsii just curious scientific fact complement the main text of the book (if we ignore all of my "fantasy"). For example, many reflektsy devoted to the fine structure constant, which in itself - one of the mysteries of physics (see next chapter).
Singularity as a place where no action begins we know the laws of the natural numbers, of course, there is - it's the beginning. It was there that "do not work" almost all the formulas of the classical theory of numbers and my GTNCH: formulas out there either have no meaning (for example, we divide by zero), or give a huge relative error (see below).
Singularity in the world of natural numbers is, firstly, the interval between zero and one (0, 1), and secondly, it is a segment [1, Nc], where Nc - number is still causing me problems. Most likely, Nc is less than the number 10 ^ 17 (Evie), then there is a singularity in GTNCH clearly less attometra (10 ^ -18 meter or 10 ^ -27 seconds).
Curiously, that if physics in their experiments, dream fall below attometra, then GTNCH vice versa - can only dream about computers that can easily handle the numbers of the order of 10 ^ 17, and more. It's not yet available even the most powerful computer (IBM, 2004), performing 7,1 10 ^ 13 operations per second. But in a world of singular - "at a glance", so it is tempting to find in the beginning of the natural series of "reflection" of the physical world, its singularity.
Relative error (OP) approximation of B *. So we call the following expression:
OP = (B - B *) / B *, (3.3)
where B * - we found an approximate value of B. As a rule, the value of B is an unknown function f in the argument N, then there is B = f (N). But instead of the true function f (which may not even exist!), We can only find certain rough approximation - the function of B * = f * (N). After that, we commend the OP (usually in%) and make a decision on the fitness function f * for our assessments within a virtual cosmology. This engineering approach is absolutely unacceptable from the viewpoint of the classical theory of numbers, but remember that the goal of virtual cosmology in the first phase - is to get at least a certain quality results and assessments within the Great segment.
Evie conversion
Because of the Planck time (for 1 Evie) photons of light are equal to the path 1,6 10 ^ -35 m (the Planck length), then one of the natural numbers can be identified not only with the "quantum" of time, but with the "quantum" of length ( the Planck size). Then all the big segment will be equal to the characteristic size of the universe:
(1,6 10 ^ -35 m) (8 10 ^ 60 Evie) = 1,3 10 ^ 26 m.
Thus, working within GTNCH useful to remember that each ("mathematical") unit, forming a "natural number" is equal to (symbolize) 1 Evie = 5,4 10 ^ -44 seconds = 1,6 10 ^ -35 m.
It is obvious that, like the large segment of any other segment of the natural numbers (any length) by Evie can be translated into time intervals (in seconds) or in segments of length (in meters). Such a transfer, we shall call Evie conversion.
Small segment - a segment [1, 10 ^ 20]. After Evie converting its length will be equivalent to:
5,4 10 ^ -44 10 ^ 20 = 10 ^ -23 seconds,
that is a small segment can be identified with the nuclear time or with a typical size of a proton - particle of paramount importance to nuclear physics (eg, a proton is a part of the nucleus of any atom). Nuclear time - this time over which the light crosses the proton. Physically, it is the smallest interval of time required to proton was observed as a whole.
Central segment - this segment [1, 10 ^ 35]. After Evie conversion central segment will be equal to 1.616 m, which practically corresponds to the average growth on the planet and could be the characteristic size of the world's "human" scale. Segment be called "central" as in the logarithmic scale of the 10 ~ 35 is almost in the center of great length (in our modern age, symbolized by the number 10 ^ 61). Note that the overall picture of the universe allows us to see just a logarithmic scale (and no other), and the height of a man is almost in the center of the universe (in the center of the global scale of all scales of the universe).
It is easy to verify that a segment of 1.616 m corresponds to (after Evie conversion) while Tts = 5,4 ∙ 10 ^ -9 seconds. It is interesting that the time TN for a man is something of a second to the universe, so if we take the human lifespan of 92 years old, then this interval contains about 5,4 ∙ 10 ^ 17 TH, that is the same as found in seconds age of the universe (see above). To imagine a time TN is appropriate to say that: while the human response to discrete stimuli, independent - not less than 0.15 seconds, while a series of rhythmic discharge in the nerve cells reaches 10 ^ -4 seconds. For comparison: the time of chemical reactions in the explosion - about 10 ^ -5 seconds, the average life expectancy of quasiparticles in a solid and in liquid helium - from 10 ^ -2 to 10 ^ -8 seconds.
Limiting segment - a segment [1, 10 ^ 308], which, after Evie conversion can be interpreted as 10 ^ 257 years - for us this is a real eternity. Number greater than 10 ^ 308, just beyond the range of values works with an ordinary personal computer. When he reached the number 10 ^ 308, a computer just stops by and gives a special message, for example, "# NUM!".
The fact that my virtual cosmology is not always confined to great length, that is, the numbers of the order of 10 ^ 61 (Evie), symbolizing our modern age. Theoretical physics describes the future of the universe up to the age of 10 ^ 150 years - the so-called photonic century: the achievement of the Universe status of extremely low energy (?). Specified photon century equivalent to 10 ~ 200 (Evie) - and those numbers are up and sometimes I come within a virtual cosmology.
Important note. During the exposition virtual cosmology me introduce many of the notation (letters from Russian, English and other alphabets). Often these designations are valid only within a particular chapter or a specific section, and sometimes - just for a specific formula in the section. In other parts of the text, these signs may have a different meaning. Generally, because of the desire of the author to concise and widely proposed text symbols, names, and were themselves the arguments initially may "surprise", "puzzling" the reader, but in the sequel a lot, "falls into place."
4. Fine structure constant
All the important physical constants (see above) have dimension. For example, the Planck length physics especially so "constructed" (combined the values of G, h, c as a formula) to the Planck length (as well as any other length) had a dimension in meters. However, the dimension of the constants in physics have a very interesting exception - the so-called fine structure constant (TCP) - a physical constant that dimension ... no! This constant physics is usually denoted the first letter of the Greek alphabet (alpha), but we denote it exactly as TCP (because of problems with the placement of "alpha" sites inernet).
TCP is a fundamental physical constant characterizing the strength of electromagnetic interaction. It was first described in 1916 by German physicist Arnold Sommerfeld, as a measure of relativistic corrections in the description of atomic spectral lines in the framework of the Bohr atom (which is why TCP is sometimes called the Sommerfeld constant).
Fine structure constant has a number of different definitions (formulas) and interpretations.
For example, PTAs can be defined as the square (second degree) of the ratio of the elementary electric charge (E) to the Planck charge (Qpl):
Title = (E / Qpl) ^ 2 = 0.007 297 352 537 6. (4.1)
Often use the value 1/PTS = 1 / 137, 035,999,679, or (in the first approximation) 1/PTS = 1 / 137.
The physical interpretation of the TCP.
TCP is the ratio of two energies:
1). Energy required to overcome the electrostatic repulsion between two electrons, brought them to the infinity to a distance s, and
2). Photon energy (a photon) with a wavelength of 2 ∙ "pi" ∙ s.
Historically, the first interpretation of the PTS was the ratio of the electron velocity in the first circular orbit in the Bohr model of the atom to the speed of light. This attitude has arisen in the work of Sommerfeld and determines the amount of the fine splitting of hydrogen spectral lines.
In quantum electrodynamics the title has a value of the coupling constant characterizing the interaction force between electric charges and photons. Its importance can not be predicted theoretically and is entered on the basis of experimental data. Fine structure constant is one of the twenty-odd "external parameters" of the standard model of particle physics.
The fact that TCP is much smaller than unity, can be used in quantum electrodynamics perturbation theory. The physical results of this theory are presented as a series in powers of PTAs, and the terms with increasing powers of PTAs are becoming less and less important. Conversely, a large coupling constant in quantum chromodynamics makes calculations taking into account the strong interaction is extremely complex.
In the theory of electroweak interactions shows that the value of PTS (strength of the electromagnetic interaction) depends on the characteristic energy of the process. Argued that PTAs increases logarithmically with increasing energy. The observed value of TCP is true at energies of the electron. The characteristic energy can not accept lower, since the electron (and positron) has the smallest weight among the charged particles. Therefore, say 1 / 137 - this value PTS at zero energy. In addition, the fact that as the characteristic energy of the electromagnetic interaction is approaching the strength of two other interactions, is important for theories of grand unification.
If the predictions of quantum electrodynamics were true, then the TCP would take an infinitely large value at the energy, known as the Landau pole. This limits the scope of quantum electrodynamics, only the domain of applicability of perturbation theory.
As far as TCP is constant?
Physicists have always wondered whether the TCP is constant, that is, it always had a value for the lifetime of the universe. Some theories say that it is not. The first experimental verification of this issue, among which the most interesting study of the spectral lines of distant stars and study the natural nuclear reactor in Oklo, did not reveal any changes in the TCP.
Improvements in the methods of astronomical observations give reason to believe that TCP may have changed its meaning over time. However, more detailed observations of quasars, made in April 2004 with the UVES spectrograph on Kueyen - a 8.2-meter telescopes of the telescope of the European Southern Observatory in Paranal (Chile), have shown that a possible change in TCP could not be more than 0.6 million shares (0,6 ∙ 10 ^ -6) over the past ten billion years. Since this restriction is contrary to earlier results, the question of whether the TCP is constant, it is considered open.
In 2010, with the VLT telescope to obtain new indications that this constant may decrease with time. Nevertheless, positive changes in TCP acknowledgments are still there.
Anthropocentric explanation
One explanation for the value of PTS includes the anthropic principle, which states that the value of PTS is precisely such a value, because otherwise it would be impossible the existence of stable matter and therefore, life and intelligent beings could not arise if the value of PTS was different. For example, we know that whether PTAs only 4% higher production of carbon inside the stars would have been impossible. If TCP was greater than 0.1, then within stars could not occur processes of thermonuclear fusion.
Numerological formulas
By the end of his life the famous English astrophysicist Arthur Eddington (1882-1944) designed the numerological "proof" that 1/PTS an exact integer, and even relates it to the number of Eddington, which estimates the number of baryons in the universe. However, experiments carried out later showed that 1/PTS is not an integer.
There's also the Association of TCP with the proposed space-time dimension: one of the most promising theories of recent times - the so-called M-theory, emerging as a generalization of superstring theory and claims to describe all physical interactions and elementary particles - the space-time is assumed 11 - dimensional. In this case, one dimension at the macro level is perceived as a time, three more - as macroscopic spatial dimensions, the remaining seven (see "The magic number 7") - so-called "rolled" (quantum) measurements that are felt only at the micro level. TCP at the same time brings the number of 1, 3 and 7 with the factors that are multiples of ten, and 10 can be interpreted as the total dimension of superstring theory.
In a recent article by Olchaka is compact and clear formula that approximates the PTS, which is associated with ... the key to the dynamics of chaos, Feigenbaum constant (F = 4.669 ...). This constant, in the most general terms, describes the rate of approximation of solutions to nonlinear dynamic systems to a state of "instability at any point" or "dynamic chaos".
The value of TCP is very accurately calculated as the root of the simple equation (where pi = 3.14 ...):
1/PTS = 137 + r / (1/PTS - F ∙ "pi / 2). (4.2)
It should also be noted that, from the viewpoint of modern quantum electrodynamics, TCP is the running coupling constant that is dependent on the energy scale of interaction. This fact deprives most of the physical meaning attempts to construct a numerological formula for a particular (in particular - zero, if we are talking about the meaning) of the momentum transfer.
TCP in the virtual cosmology
TCP, being a dimensionless quantity, which does not correspond to any known mathematical constants, has always been an object of admiration for physicists. Richard Feynman (1918-1988), a prominent American physicist (one of the founders of quantum electrodynamics), called TCP "one of the greatest damn mysteries of physics: a magic number that comes to us without any understanding of his person."
As part of a virtual cosmology, we shall frequently "discover" the TCP. In this case, most often we will get some number close to the numerical value of the PTC, according to the formulas virtual cosmology, where the argument (an integer N) in these formulas is taken at the end of the Great segment (N = 8 ∙ 10 ^ 60 Planck times or Evie) that is the point that symbolizes our modern age ("our time").
5. What are the numbers?
Natural numbers - is the most basic and fundamental form of numbers, one of the basic concepts of mathematics. The set of all positive integers N = 0, 1, 2, 3, 4, 5, 6, 7, ..., that is, positive integers endowed with the natural order, called the natural numbers. The fact that this series now begins with zero, confirms Roger Penrose in his excellent book "The New King of the mind ..." (p. 91). Natural numbers have appeared among the ancient people as a result of account objects, and these numbers - the first an abstract truth, which opened to man. This truth may well be the latest, available to man - is so complex and fundamental world of numbers. But with this, for sure until a few would agree ...
The total number of all natural numbers is an infinite number, which we call - "aleph-null" (the first letter of the Hebrew alphabet). German mathematician Georg Cantor (1845-1918) proved the two paradoxical statements that our imagination, alas, "refuses" to understand:
1). Number of integers (ie integers with a plus sign and a minus sign) as well ... "aleph-null" (instead of twice as much as us "suggests" intuition).
2). Number of fractional numbers (fractions) as well ... "aleph-null".
Primes. All natural numbers N of mathematics is divided into two groups. The first group includes the numbers having exactly two divisors (1 and the number is N) - these numbers are called prime, and they are also infinitely many: 2, 3, 5, 7, 11, 13, .... Prime numbers have ahead of us will still be special and a long conversation.
The second group includes all other numbers are called composite.
In mathematics there is reason to assume that the number N = 1 - a very special number (neither simple nor composite).
Real numbers (or real numbers) include the rational and irrational numbers.
Rational numbers, ie numbers that can be represented as a ratio of two integers m / n. Rational numbers as represented in the form of finite or infinite periodic decimal. Period begins immediately after the last point, if the irreducible fraction m / n denominator n is not divisible by 2 and 5.
Need to understand that numerically equal fractions such as, for example, 3 / 4 and 9 / 12 are included in this set as a single number. Since the division of the numerator and denominator by their greatest common divisor can be unique irreducible representation of rational numbers, then we can speak about them set as a set of irreducible fractions with relatively prime integer numerator and denominator of the natural.
Set of rational numbers is a natural extension of the set of integers. It is easy to see that if a rational number of denominator n = 1 then a = m / n is an integer. In this regard, there are some misleading assumptions. However, although it seems that the more rational numbers than integers, and those and other countable (ie, both can be enumerated by natural numbers, and clearly).
Irrational (irrational) numbers, ie numbers in the decimal expansion which there is no period. All of them fall into the algebraic numbers which are roots of an x ^ n + ... + a1 x ^ 1 + a0 ∙ x ^ 0 with integer coefficients, and transcendental ("otherworldly") numbers. French mathematician (and politician), Emile Borel (1871-1956) found that "almost all" real numbers - a transcendental number, because they are not "numbered" (as opposed to algebraic numbers). It is very easy to come up with the very set of irrational numbers, for example, a number like 0.1010010001 ..., but much harder to prove that some particular number is just irrational. So it was with the number pi = 3.14159 ..., the number of "e" = 2.71828 ..., and for the number of C = 0.57721 ... (the Euler - Mascheroni) - remains an open question. More Teetet Athens (ca. 410-369 BC) to justify the irrationality of all numbers of the form N ^ 0,5 (square root of N) and N ^ (1 / 3) (cubic root of the number N), where N - integer, not a perfect square (cube).
German mathematician Adolf Hurwitz (1859-1919) and Emile Borel proved that for any irrational number w there exist infinitely many approximations by rational numbers m / n, for which the inequality | w - m / n | <1 / (n ^ 2 ∙ 5 ^ 0,5), with the number 5 ^ 0,5 (square root of five) can not be increased.
Correct definition of irrational numbers using an infinite sequence of approximations by rational numbers belong to the highest achievements of human reason, but hardly corresponds to anything real in the physical world (where any measurement always fraught with errors).
Number of real numbers greater than natural. And this conclusion Cantor intuitive, because the two real numbers (regardless of their proximity), there is a third real number. In this case, it is not clear whether one can reasonably say the same about physical distances or time intervals (the real numbers seem to provide the quantities needed to measure them - hence the term "valid"). The real numbers should be viewed as a kind of mathematical idealization rather than a real measure of objective physical quantities. Actual numbers may embody a notion of "continuity" (which is obviously violated at very small spatial and temporal scales, ie at the Planck length and time).
If C (continuum) - is the amount of all real numbers, and aleph-one "- an infinite number following the" aleph-null, then the statement C = aleph-one is famous and still unsolved problem of mathematicians (the so-called continuum hypothesis).
Inverse of (R, reverse) - so we call all real numbers that are between zero and one, ie 0 <R <1. The term "reverse" prosper only correspond to the so-called aliquot fractions, ie fractions of the form 1 / N, where N - a positive integer. It is clear that an infinite number of aliquots of fractions (1 / 2, 1 / 3, 1 / 4, 1 / 5, 1 / 6, 1 / 7, ...) does not exhaust the set of all inverse numbers.
The sum of aliquot fractions (Sa) is using the Euler formula for the sum of the so-called harmonic series (compare this formula with Dirichlet, see below):
Sa = 1 + 1 / 2 + 1 / 3 + 1 / 4 + 1 / 5 + 1 / 6 + ... + 1 / N = lnN + C + "epsilon. (5.1)
where C = 0.577216 ... - the Euler - Mascheroni, and the parameter "epsilon" tends to zero when an integer N increases indefinitely. At the end of great length, we obtain Sa ≈ 140,81.
Any positive rational number can be represented as the sum of a finite number of aliquots of fractions with different denominators, for example: 2 / 43 = 1 / 22 + 1 / 946 = 1 / 30 + 1 / 86 + 1 / 645 = 1 / 42 + 1 / 86 + 1 / 129 + 1 / 301. Equally important fact we call alikvotizatsiey rational number. In contrast to the factorization of natural numbers (this will be discussed below), alikvotizatsiya - procedure multivalued (options, generally speaking, a lot), but its laws (they have been studied by someone?) May also have a deeper meaning (even if only in part Virtual cosmology).
In ancient Egypt there were tables that give the decomposition of fractions 2 / N in aliquot fractions (where N - all odd numbers, say up to 331, as in the Rhind Papyrus). Therefore, we call the fraction of type 2 / N - Egyptian fractions. Using these tables to solve many practical problems for thousands of years (until the middle ages!), Although it sometimes required considerable artifice of ancient mathematicians. American mathematician Dirk Jan Struik (b. 1894), deeply exploring the history of mathematics, came to this conclusion: "Egyptian mathematics was rather primitive character." But personally, I fully admit that aliquots of the fraction could be some "tip" from an unknown civilization, now that humanity, alas, has not been able to grasp so far.
Simple aliquot fractions - so we call the fraction of 1 / P, where P - a kind of prime number. Similarly, we call fraction of the form 2 / P - a simple Egyptian fraction. It is obvious that any Egyptian fraction 2 / N can be represented as the product of fractions and fractions 1/Rk 2 / P, and the last - is the sum of the two aliquots of fractions:
2 / P = 1 / [(P +1) / 2] + 1 / [P (P +1) / 2] = 1 / [(P-1) / 2] - 1 / [P (P-1) / 2]. (5.2)
Formula (5.2) partly explains the multi-variant alikvotizatsii. Probably, the inverse numbers R "internal structure, in a sense, more diverse than the natural numbers N.
Among the first 500 prime aliquot fractions, only two (1 / 2 and 1 / 5) - final, and probably only 11 fractions - Recurrent: 1 / 3, 1 / 7, 1 / 11, 1 / 13, 1 / 37, 1 / 41, 1 / 73, 1 / 101, 1 / 137, 1 / 239, 1 / 271.
Aliquot sum of simple fractions (Spa) can be found using the Gauss-Mertens [see Book V. Boro, etc. "Living the number", p. 16]:
Spa = 1 / 2 + 1 / 3 + 1 / 5 + 1 / 7 + ... + 1 / P = 0.261497 lnlnR + ... + "epsilon". (5.3)
where
where 0.261497212 ... - constant Meysselya - Mertens (number theory), and the parameter "epsilon" tends to zero when the prime number p increases indefinitely. In my opinion "Epsilon" <C / P ^ C, where C - where C = 0.577216 ... - Euler's constant - Mascheroni. At the end of the segment we get the Big Spa 5,2048 (see below "Magic number 7).
Important note. Above in formula (5.1) we have already met the parameter "epsilon" (btw, it's just 5-th letter of the Greek alphabet). And such an option, "epsilon" not yet time for us to meet a well-known formulas of number theory, and in all these formulas, "epsilon" tends to zero, in general, by their own laws (with different "speeds"). But we always denote this parameter is just that - "Epsilon".
Complex numbers - is of the form z = x + i y, where x and y - real numbers and i = (-1) ^ 0,5 - imaginary unit, ie the number whose square is equal ... minus 1. Real numbers - a special case of complex numbers with y = 0. For the first time, apparently imaginary quantities appeared in the work of G. Cardano in 1545, but their benefit is not immediately recognized, because, after more than a hundred years the great Newton did not even include them in the concept of number. In the late 19 th century, it was proved that any extension of the concept of number beyond the field of complex numbers is possible only in case of refusal on any properties of the usual action (hypercomplex number). "Imaginary" numbers are not less real than that which have already become familiar "real" numbers, a timeless reality of complex numbers goes far beyond the thought processes of any math, because, as, for example, argues Roger Penrose, the complex numbers there ... in the world of Plato (see below)!
Currently, the complex numbers are an integral part of the structure of quantum mechanics (this is one of the pillars of modern physics) and thus underlie the behavior of the world in which we live. Furthermore, complex numbers, of course, is a one of the great wonders of mathematics.
Sequences of integers - they now number in mathematics, probably about ... 200,000! Ironically, the first Handbook of Integer Sequences "was published only in 1973, and the first who thought of to do it was American and British mathematician Neil James Alexander Sloane (born Neil James Alexander Sloane; race. 1939). In his first reference Sloan collected and streamlined over 2300 sequences, each of which describes a recursive and (or) the exact formula, but also maintains a list of recommended literature. Currently, Neil Sloane (Sloane) is an author and curator of the site (an online Internet resource) The Encyclopedia of Integer Sequences "(English On-Line Encyclopedia of Integer Sequences, OEIS).
At the end of 2009 (18-20 October), I put in OEIS seven infinite integer sequences from my GTNCH (which was not! At Sloan among the 164,537 sequences): A166688, A166689, A166690, A166691, A166693, A1666721, A1666722. In this case, I have personally sent a letter to H. Sloan (in Russian and broken Eng. Language) about his "discovery", but this story did not matter to me any further. To this we must add that my theory (GTNCH, virtual cosmology) "generates" almost endless ... (!) A number of sequences. Moreover, the virtual cosmology, probably, "takes away" any philosophical significance of the project OEIS (if any meaning it had an original).
The most famous of all infinite sequences - this is the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, ...) in which each number sequence is the sum of the previous two. More could be mentioned in the Catalan numbers: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, .... These numbers are not as well known as the Fibonacci numbers, but they are no less significant, and arise in unexpected places, especially in solving combinatorial problems. By some expert estimates of the Catalan numbers - the most common sequence (!), But it is still not known, even among mathematicians.
Infinitely many so-called self-generation of numbers, which opened the Indian mathematician DR Kaprekar in 1949 (about these numbers as very few people know). There is a sequence, the continuance of which is at issue (not yet known), for example, the number of farm (Gaussian primes) 3, 5, 17, 257, 65537, ... (?); Perfect numbers (Mersenne numbers), friendly and sociable numbers; cyclic number (142857, 285714, ...). There are a number of fancy, nearly whole numbers, number of the beast "(666), etc.
And, certainly, there are many interesting sequences and a variety of numbers.
6. Magic Number 7
"Everywhere I pursued another sign ... - Wrote to an American psychologist George Miller. - This number should be just behind me, I'm constantly confronted with it in their own backyard, it gets me from the pages of our most popular magazines. It takes many guises, sometimes it is a little more, sometimes less, but it never changes so that it can not be recognized. "With these words, Miller began his famous article" The magic number 7 (plus or minus 2) "(ie 5 to 9). He came to the conclusion that man is able to once hold in memory at an average of seven "pieces" of information:
seven letters of the alphabet;
five (7 - 2 = 5) single-syllable words;
eight (7 + 1 = 8) decimal digits;
nine (7 + 2 = 9) binary digits, etc.
"In memory of this purse - said Miller, - placed a total of seven coins. Dollars and cents is, she is indifferent. She is interested in is not the meaning of information, and its purely external characteristics - color, shape, volume. Sense of interest long-term memory. It identifies and evaluates the contents of the purse. " Miller came across seven in experiments with visual perception. (However, within a few millionths of a second our visual analyzer is capable of holding much more "pieces" of information than can Millerovskiy purse. In those moments on the periphery of the visual system stores all the information requirements to man, no matter how much it Anyway.) Then in July came to light in studies of auditory perception: for example, it is difficult to grasp the whole phrase, if it contains more than 7 linguistic branches.
But why nature often "highlights" is the number 7? (Plus or minus 2 - this "tolerance" for the magic number 7, we will always mean, but to write "access" more generally will not).
Neither the above mentioned George Miller, no other recognized scientific authority, alas, no intelligible answer to this question. I came across only the following explanation (psychology and biology): the process of evolution, along with many mental and physical constants, such as the propagation velocity of nerve impulses by nerve fibers (0,2 - 180 m / s), a person to develop and is a constant value as the amount of operational memory. For millennia, this constant influences the production of everyday lifestyle, cultural traditions, religious and ethical beliefs. Person was most convenient to think of concurrent things if their number does not exceed seven.
You, dear reader, this explanation satisfied? Personally, I'm - obviously not happy. Therefore, all that you have read above about the magic number 7 - just a note, and focus on what is set forth below.
So, it seems to me, the mystery of magic sevens we can tell ... the world of numbers.
Anyway, if you began exploring the world of numbers (like what I'm doing in the virtual cosmology), then very soon they would see that in a world of numbers at the end of the Great segment (which is "equivalent" in our modern age), the magic number 7 absolutely certain - it will literally "make callous your eyes! That is why I believe that the apparent magic of our 7 (in the "internal" structure of the natural numbers) in some way reflects the fundamental (mathematical) features of the real structure of space-time, which generates a magic number 7 in the real (physical) world. Many examples of the latter below. Moreover, all these examples may constitute the beginning of a kind collection, which is useful to divide into two parts:
1). Basic magic number 7 - these are examples that are not dependent (or nearly independent) of man's will, and if somewhere (on an extrasolar planet), live "green men", then they will make a similar list of magic number 7, because the laws of physics are united in all corners of the universe in all its planets;
2). Haunted magic number 7 - these are examples, which are the result of human imagination, his preferences, traditions, tastes, habits, etc. (In other words, all of this - people just came up with himself, and in view of the fundamental ... the magic number 7, which is objectively there).
FUNDAMENTAL magic number 7 (numbers 5, 6, 7, 8, 9)
I remind you that such examples - is an objective reality independent of human will.
5 spins can be in particles in the microcosm (0, ½, 1, 2, 3 / 2), 5 electric charges can be a particle in the standard model (popular physical theory), 5 (or 6) different options includes the M-theory (see string theory), 5 types of axial symmetry in crystals; 5-fold symmetry in a substance, 5 how the location of points in the plane of Crystallography, 5 sequence stars (of tribes), 5 Lagrangian points in the Sun-Jupiter-asteroid, 5 main zones of the Earth (core, mantle, crust, ocean, atmosphere), 5 shells (Geosphere) Earth 5 functions of blood, 5 functions of epithelial tissue Rights;
6 types of quarks, 6 the topological properties of space-time plus-minus 6 - this is equal to the Euler number (the sum of the dimensions of the homology groups of varieties) for spaces of Calabi-Yau (in the superstring theory), 6 crystallographic systems (triclinic, ...) 6 levels in biological taxonomy (species, genus, ...) 6 Erato in the biography of the Earth (Geochronology);
7 extra dimensions in superstring theory; 7 "charges" of elementary particles; 7 basis cell arrays in crystals, 7 "accidentally" agreed FPC determine the chemical composition of our universe (and our very existence), 7 numbers "give" the universe of I. Rosenthal ; 7 periods and 8 groups in Table DI Mendeleev (up to 7 shells may be in an atom), the specific binding energy of all nuclei in an average of 7 times the rest mass of an electron-proton, on average there are 7 (?) Versions of any substance in the crystalline state;
7 spectral classes of stars, a factor of 7 different diameters of the white dwarf, 7 zones can be distinguished on the Sun (on the physics of the processes), 7 shells comprise scientists inside the Sun, 7 non-stationary structures in the solar atmosphere, 7 substances constitute 99.75% of the solar photosphere; ...
7 major component in the solar system: Sun, 9 planets, asteroids (meteoroids) mikrometeoridy (streams of dust), the body Kuiper-Edgeworth (Centaurs, comets), magnetic fields, cosmic rays, and 7 - is the average number of large satellites of the planets; 7 types of asteroids (the ratio of their densities 7:1), 7 hatches Kirkwood in the asteroid belt, and 7 rings of Saturn (9 Pluto), 7 layers of the Earth's atmosphere, and 7 continents on Earth, and 7 layers emit in the Earth's atmosphere, and 7 components ( shells) was isolated in the inner structure of the Earth and the Sun. ...
Man: 7 species (forms) of the cells, and 7 basic life manifestations of the cell (replication, ...), and 7 structural levels (atomic, molecular, cellular, tissue, individual organs, organ systems, the entire organism), 7 structures in the microstructure of bone man; 7 parts of the body (head, neck, torso, two arms and legs) 7 pairs of ribs a man reaches his belly, 7 twins (for once) gave birth to a woman.
On average, 7 "chunks" of information keeps people in memory of one time (Millerovskiy purse, see above). The most convenient way to think no more than about 7 things (simultaneously). 7 related to the visual and auditory perception of the person (hard to remember the phrase, in which more than 7 linguistic branches);
Seven sensory systems created by the nature of man: the visual system, auditory, vestibular, cutaneous (tactile, temperature, pain), motor, olfactory and gustatory.
Seven resuscitation points exist on the human body by acting on that can be "easy" kill or revive the person.
7 - this is the factor entsefalizatsii (EQ) rights;
Seven age periods in the psychosexual development of each person isolated scientists. Age-period end of these periods correspond to 1, 7, 12, 18, 26, 55, 70 years [an almost tilde-distribution (see below) with parameters S = 189; K = 7; A = 3,255; p = 0,521].
More than seven variables in the problem - it is virtually insoluble (joking statement, in which a large proportion of the truth).
Seven years left until retirement (which would be age does not equal) - and any employee begins to lose its grip (from the observation of witty people.)
Every seventh Frenchman has a cat or dog (60 million population of France has 8.9 million domestic cats and dogs, 8,2 million).
7 signal systems in plant cells;
8 gluons and 8 gluon fields in theoretical physics; 8 magic numbers (number of protons or neutrons) 8 ÷ 10 Fedorov groups for molecular crystals, 8 • R - the radius of a set of Delaunay, 8 times the density of the Earth (maximum for solid planets) greater than the density of Saturn (at least in part of solid planets), 8 stages of human origin (anthropogeny) 8 bones in the brain department of human skull, 8 structural units in the skeleton hands and feet of 8 main causes of die 99, 96% of Russians, 8 major sources energy (oil, coal, ...), 8 bits - length of a word in computer science;
9 planets of the Sun, 9 rings of Uranus, 10 times the different radii of main sequence stars;
Ephemeral magic number 7 (numbers 5, 6, 7, 8, 9)
I remind you that such examples of the magic number 7 - this is purely the fruits of human imagination
Here, in the first place should put a (very useful!) Approval: all sorts of useful classification should contain an average of 7 categories, and this applies absolutely to all areas of our knowledge. For example:
5 conditions of democracy in Mathematical Political Arrow;
7 temporary milestone in the history of the Earth (6 Erato and appearance of the planet);
7 comprise the world's civilizations in human history;
7 major reasons for the collapse of the Roman Empire;
9-point scale wind waves at sea, etc.
As part of the collapse of the Roman Empire can be added to the English historian Edward Gibbon (1737-1794) analyzed the decline of the Roman Empire in the 7 full-length volumes and identified 7 major reasons for its collapse:
- The struggle between the haves and have-nots;
- Huge spending on political campaigns (bribery of persons, theft of money);
- The burden of external debt (in Rome part romanization);
- Aversion to military service of soldiers from wealthy families;
- Lack of creative leadership;
- The death of the Roman principles of virtue, purity, simplicity;
- The high divorce rate.
The Fall of Rome took place 1700-1900 years ago, but the causes of the sad final look as if we are discussing the situation ... in modern Russia, is not it, dear reader? All this proves that fascinating "science" story of us ... teaches nothing!
As part of the conditions of democracy in theory, Arrow will also add the following.
At the beginning of perestroika in the Soviet Union, many of us hoped for democratic elections, but it turns out, and they initially present ... flawed. There are so-called netrazitivny paradox of voting in elections or Arrow paradox, named after CJ. Arrow, who played a crucial role in the proof of "Theorem on the impossibility of the ideal electoral system, for which he, among others in 1972 was awarded the Nobel Prize . Arrow identified five conditions that, according to popular belief, are essential for democracy in which social decisions are made by a vote of individuals. Arrow proved that these 5 conditions are logically inconsistent: there is no electoral system that would not be violated, at least one of the 5 conditions, that is an ideal democratic election system ... impossible in principle. In mathematical political science discovery Arrow occupies the same place, which is in mathematical logic theorem of Kurt Gödel (1906-1978) about the impossibility of constructing a consistent mathematical theory containing the axioms of arithmetic. (For example, when a man says: "I am a liar!", A paradox - true or false is this statement? Gödel proved that in mathematics there are also allegations are true, but the truth they can not be proven.)
Paradoxes of voting arise in situations where a decision is made based on the choice of two alternatives to select from a variety of three or more elements. The study of such paradoxes is beyond the scope of this book, so just give an example of a defect of the electoral system (the paradox to "smarter"). Let Yeltsin actively supported only 40% of voters, but the voices of its opponents were divided between Zyuganov (30%) and Yavlinsky (30%), and as a result of the election, Yeltsin won (although 60% of Russians were opposed to it), and Russia had happened what happened - the "zero" (lost to the country) years at the beginning of the XXI century ... By the way, when the real elections, there is a deadlock, then the population is normally elected "dictator", which just breaks the current situation.
The magic number 7 in man's life
Seven types of people choose to give psychologists: a dominant, analytical, aesthetic, an independent, inert, stable, satellite. With respect to the innovation (change) in people's lives are divided into seven types: innovators, enthusiasts, rationalists, neutrals, skeptics, conservatives, reactionaries. Business quality person distinguish on 7 scales: creativity (in man-converter, prone to change everything around you), executive, contemplative, conservative, adventurous, efficiency and reliability.
Seven lines of psychological "defense" of man's inner world from invasion from without: biographical data; personality traits (strengths and weaknesses); personal orientation, driving force, the actions and activities; area of personal relationships with people, relationship to the world as a whole and its component systems and the resistance associated with the attitude to someone who intends to reveal the features and capabilities of man.
Seven of the rules has brainstorming method, proposed in 1965 by the American psychologist A. Osborne to efficiently generate ideas for solving the scientific, technical, administrative and other tasks (for brainstorming, six people for half an hour could push 150 ideas!).
Seven days contains our week, which first came into use in the Ancient East, in Rome it with 1 century BC. er. Seven periods often we select a day: twilight, morning, noon, afternoon, evening, midnight, night.
Seven kinds of tricks includes modern etiquette: "a glass of wine (" champagne ")," Breakfast, "a buffet (" cocktail ")," dinner "," dinner buffet (buffet) "dinner" meeting at the tea or coffee table.
Seven types of wine came up with a man: dry, sweet, strong and sweet, flavored, sparkling.
Seven-year school existed in the Soviet Union in 1920 and 50 th. Russia now has 3 million homeless children (mean level of education not more than 7 classes?).
Seven main levels were in the organizational structure of the Communist Party in the era of "developed" socialism (of the Congress of the CPSU to the primary party organizations). "Council of seven" - the Russian government in 1610-12-ies. Consisting of 7 members of the Boyar Duma. "Semibankirschina" and 7 was the oligarchs in Russia in August 1998, seven countries of the Group of Seven - the richest powers of our time (the "Big Eight" with Russia - as long as fiction). By 7 federal districts of Russia is divided (Spring 2002).
Seven digits in phone numbers - and that's enough for most cities (allows you to have up to 107 subscribers). Cities-millionaires than 220 worldwide, but even they often lack the 10 million phone numbers (ie the seven digit number).
Seven groups form the road signs (as guests) in Russia.
Seven (numbers from 5 to 9) appears in many games, for example: Tangram (consists of 7 parts - tanov), chess and checkers (on board 8x8) dice (6 faces of the cube), soccer (at least 7 people from the team in the field), water polo (7 persons from the team on the field), volleyball (6 players from the team on the field), basketball (5 players from the team on the field);
Languages of human communication and the magic number 7
It is assumed that the development of speech began no later than 20 thousand years ago. Neanderthals already had vocal cords are suitable for the simplest question. Currently, humanity uses to communicate at least 4000 languages (it is not known), which is divided into 8 major language families: Indo-European (the largest by number of speakers of her people, including Russian language), the Sino-Tibetan, Niger-Congo, Austronesian , Amerind, Altaic, Dravidian, the Ural-Altaic. Still emit 9 small language families, as well as a language isolate, ie, without any connections with other languages, for example: Japanese, Korean, Sumerian (the oldest of them), etc. Starting from the XVII century. invented several hundred artificial languages to facilitate international communication. The most popular of them - Esperanto (1887) has only 16 grammatical rules (and no exception), and the language solresol (beginning of XIX century.) Based on 7-note scales: do, re, mi, fa, salt, la , si, so solresole can not only speak but sing and or whiz.
Most carriers in the world of spoken Chinese language (guoyo) - 810 million people, followed by Hindi (364 million) and English (335 million, although they say it the world's best), the Russian language is in eighth place (156 million). All in all, can be identified 33 major languages of the world, carried by 66.1% of the population, and number of speakers of these languages are close to the tilde-distribution with parameters: S = 4096 (million); K = 33 (language); A = 4,22; p = 0,164.
7 (or 8?) Letters - this is the average length of words in Russian, with 7 letters (n, c, a, a, n, p) begins 60% of Russian words, 6 points in Braille (for blind).
7 groups of punctuation marks are distinguished in the Russian letter, and other contemporary Latin and Cyrillic alphabet charts;
A number of words ("family", the "seed", "all", "in this", etc.) are generated by the number seven.
Sevens full of our proverbs and sayings: "Seven times measure, one cut," "Seven of one does not expect", "Seven Fridays in the week," "genius," "Seventh on water jelly," "Seven miles jelly slurp "" Seven troubles - one answer (a "Reset" - a modern interpretation), etc. Seven is often encountered in fairy tales, epics, legends, myths, etc.
Up to 7 characters, images in each word (or sentence?) Has not yet deciphered (or is it decrypted?) Phaistos disk (c. 1600 BC. Er. Found in Fest at the beginning of the twentieth century)
Art and the magic number 7
In the Middle Ages to the schools they studied seven subjects, under the name "liberal arts: grammar, rhetoric, dialectic, arithmetic, geometry, astronomy, music.
In modern art - it is an integral part of the spiritual culture of mankind, a specific kind of learning about the world in which secrete 7 regions of human activity: plastic arts, music, literature, cinema (teleart), theater, choreography, variety-circus arts (including . am certain sports).
Distinguishes 7 types of plastic arts: sculpture, painting, drawing, architecture, decorative arts, industrial design, photography. Number of major styles in the plastic arts is close to 21 (Renaissance art, classicism, romanticism, etc.). In every form of art can distinguish about 7 sub-sections, such as painting: monumental paintings, decorative, decorative easel (painting), icons, miniature design.
Painting and the magic number 7
In painting, one can count seven genres: landscape, portrait, still life, interior, battle genre, historical, everyday. In classical painting can be identified 7 main types: oil painting (oil), water color, tempera, pastel, gouache, sauce (sanguine), pencil (pen, charcoal).
Painter in his work uses the 7 main items: Easel (sketchbook, tablet), oil on canvas (cardboard, paper); palette, paints, brushes (usually 8.7 size or palette knife); solvent (oil), lacquer (fixatives).
Technique of oil painting includes 7 basic concepts (methods):
- The use of texture of canvas (from coarse to fine);
- Underpainting (pre Contouring forms only 1-2 colors);
- Relief underpainting (almost sculptural molded shapes smear);
- Technology alla-prima (work "on a wet, usually in one session);
- The nature of the stroke (from "licked" technology to clutter paints);
- Multi-layered paintings (each layer of paint while dried);
- Glaze (a thin, transparent layers of paint on already dried paint).
Eye of the common man discerns a thousand shades of colors, and sharper than the artist's eye to the dozens of times. Meanwhile, the old masters of painting used by only 7-8 colors, creating the masterpieces of color. Usually an artist enough to have on your palette of about 21 colors, although the list of all colors could reach 200. On the panel recommended to mix no more than 3 colors (not counting the white paint that have a very special role), or obtained dirty "mixings. It is quite normal palette contains 8 colors: white, strontium yellow, light ocher, English red, kraplak, emerald green, ultramarine, burnt bone. Incidentally, in the heavenly rainbow has 7 basic colors: red, orange, yellow, green, blue, indigo, violet. (Their order is easy to remember a phrase: "Every hunter wants to know where to sit pheasants." On the screen color television full-color image is formed of only 3 colors: red, blue and green - these are the laws of physics and biology.)
Music and the magic number 7
Music - is "the language of the soul" of man. In a very thin (but not translated into the language of words) expressing emotions music is unparalleled. Everyone probably has heard about the seven-note diatonic scale: do, re, mi, fa, G, A, B. Who have studied music know 7 music symbols: treble clef, bass clef, B flat, C sharp, Bekar, a sign of recurrence, duration of notes, as well as six titles of music: a whole note, half, 4 th, 8 th, 16 th, 32 -I.
To indicate the pace in music are about 33 terms: adagio, allegro, piano, forte, etc. The standard of the orchestra consists of 21 group instruments: violins, violas, cellos, etc. The number of items of musical instruments is close to 200 , all of them can be divided into 5-7 major groups: strings, keyboards, woodwinds, brass, percussion, .... Number of musical forms (symphony, suite, overture, minuet, etc.) is also close to the number 200. The culmination of many musical works have a point of "golden section" of their total duration.
The magical number 7 in the myths and religions
Seven of the gods and goddesses, disposed of the fate of the Sumerians (the first civilization on Earth), every Oroch hunter knew that the founders of his family were seven. Seven honored the tribes and the farther into the past, the more we meet the sevens, including patterns and rock frescoes.
Seven Sages was the ancient Greeks.
The Pythagoreans number "7" was considered miropravyaschim (divine), requiring special reverence, it is "space" part. As vseopredelyayuschey force ("force accomplishments") and creator of all ("demiurge"), the number "7" is the fate of the name of kairos (critical time). The number "7", the fate and kairos for the Pythagoreans the same thing.
Seven wonders of the world knew the Hellenic world, from which all survived only pyramid.
Even my cursory review of the Bible (which consists of the Old Testament and New Testament, recognized only by Christianity) shows that it is not very original in terms of numerals (the rest and say not even worth it) and many scenes contain all the same magic number is 7.
The Old Testament is divided into 7 parts: the five books of exercise (in the Hebrew Torah - Genesis, Exodus, Leviticus, Numbers, Deuteronomy), the Prophets and the Writings. About 7 days of creation tells us the book of Genesis and the 7 obese and 7 lean cows that had a dream the Pharaoh. According to the Jewish myth of the Flood destroyed "all things to the ground, except for righteous Noah and his family pairs of all flesh" - 7 pairs of clean animals, 7 pairs and 7 pairs of unclean birds (they are all supposedly saved on the ark built by Noah, at the behest of God).
The New Testament and is divided into 7 parts: the four Gospels (Matthew, Mark, Luke, John), Acts of the Apostles, the Epistles (of 21) and the Apocalypse (Revelation). In the parables of the New Testament the number 7 refers to more than 25 cases (excluding repeats). For example, 7 bread that Jesus fed the people, and 7 Commandments (Thou shalt not kill, Do not commit adultery, not steal, Do not bear false witness, Honour thy father and thy mother, love thy neighbor, distribute their wealth to the poor); 7 signs of wisdom from above (it is clean, peaceful , modest, obediently, compassionate, impartial, without hypocrisy).
Religion, in my opinion, is on a par with art. This is all the more believable when you consider all the splendor of religious surroundings. And most importantly, in spite of centuries of evolution, like art, religion and accomplishing little in the knowledge of the essence of being and the growth of social well-being of mankind. And this helplessness of their main difference from the natural (exact) science. Another Engels (1820-1895) gave a profound definition: "... Every religion is nothing but a fantastic reflection in men's minds of those external forces that dominate them in their daily lives ...".
Religion arose, apparently, in the Stone Age around 40-50 thousand years ago. Today in the world can be identified 7 major religions: Catholicism, Protestantism, Orthodoxy, Islam, Hinduism, Buddhism, Judaism. If you go into more detail, for example, inside one of Christianity, one can count at least 15 religions, in addition, there are "new" religion (Scientologists, Hare Krishnas, and others), and then the total number of all religions to be about 33. The driving force, supporting a religion - a passionate desire to find definitive human meaning and purpose in life. However, the intelligence of the vast majority of people simply are not able to join the beauty and richness of complex scientific knowledge, and to assert itself in religion at all is enough of a weak mind - this is the main reason for religion to thrive (in terms of worldly goods, thrives, of course, only the elite of the church).
Religion in our day - this is a very reliable source of income, a kind of business a few "elite". But most importantly - religion has always helped the poor in the rich control the masses, to regulate their behavior, and therefore strongly support the power of religion. In modern Russia, religion has never had to "to the court." Most of the disadvantaged, to throw out to the margins of life of people turning to religion, as his last consolation. Probably, for this we must be grateful to those who invented a religion ...
7. GOLDEN SECTION
The notorious "Golden Section" is usually associated with one of two irrational numbers:
(5 ~ 0,5 - 1) / 2 = 0.618 ... - it is a positive root of the quadratic equation X ^ 2 + X - 1 = 0;
(5 ^ 0.5 + 1) / 2 = 1.618 ... - the number of Phidias (see below).
Golden section is traditionally ascribed to some mysterious significance. However, it seems to me, the golden section just close to certain values (-0.5 e = 0.606 ..., pi ^ 2 / 6 = 0.644 ..., e 0,5 = 1.648 ..., etc.) which are relatively frequently "generated" in the world of numbers (in the virtual cosmology, the world of Plato, see below). That is, a sense of harmony with us does not arise from the "use" the golden section, say, the architectural proportions, but because of the compatibility of these proportions deep natural laws that are "encrypted" in the world of numbers (this is what should be admired!). Probably just the nature of "taught" a person perceives a match as a perfect harmony. Of course, this is purely my point of view on the golden section, which will add a few more words (commonly known facts).
Golden section (golden proportion, the division in extreme and mean ratio) - this division is a continuous quantity into two parts in this respect, in which the smaller part so applies to most, as much to the entire value. Relationship of parts in this proportion is expressed by a quadratic irrationality (5 ^ 0.5 + 1) / 2 = 1.618 ....
In the extant ancient literature division of the segment in the extreme and mean ratio was first mentioned in the "Elements" of Euclid (c. 300 BC. Er.), Where it is used to construct a regular pentagon.
Luca Pacioli, a contemporary and friend of Leonardo da Vinci called this attitude "divine proportion". The term "golden section" (goldener Schnitt) was put into use by Martin Om in 1835.
The Golden Section has a number of remarkable properties, but even more fictional properties. Many people "try to find the" golden ratio throughout that between half and two.
Under the "rule of the golden section in architecture and art are usually understood asymmetrical compositions do not necessarily contain the golden ratio mathematically.
Many argue that the objects that contain a "golden section", are perceived by people as the most harmonious. Typically, these studies did not withstand strict scrutiny. In any case, all these allegations should be treated with caution, since in many cases, this may be the result of fit or match. There is reason to believe that the significance of the golden section in art, exaggerated and based on erroneous calculations. Some of these statements:
The proportions of the Cheops pyramid, temples, bas-reliefs, household objects and ornaments from the tomb of Tutankhamun allegedly indicate that the Egyptian master of the golden section ratios used in their creation.
According to Le Corbusier, the relief from the Temple of Pharaoh Seti I at Abydos and the statue of Pharaoh Ramses, the proportions of the figures correspond to the golden section. In the facade of an ancient Greek temple of the Parthenon also contains the golden ratio. In the compass of the ancient Roman city of Pompeii (Naples Museum) also laid the golden ratio division, etc., etc.
Results of the study the golden section in the music for the first time in the report Rozenova Emilia (1903) and later developed in his article "Law of the golden section in poetry and music" (1925). Rosens shows the effect of the proportions in the musical forms of the Baroque and classical works on the example of Bach, Mozart, Beethoven.
When discussing the optimal aspect ratio of rectangles (size A0 sheet of paper and fold, the size of plates (6:9, 9:12), or frames of film (often 2:3), the size of cinema and television screens - for example, 3:4 or 9: 16) were tested by a variety of options. It turned out that most people do not perceive the golden section as the best and regards it as the proportion of "very stretched".
Examples of conscious use of the golden section
Beginning with Leonardo da Vinci, many artists consciously used the proportion of "golden section". The golden section was the basis for constructing a composite of many works of world art, mainly paintings and architecture of antiquity and the Renaissance. The culmination of many pieces of music often falls on the golden section point of their total duration. Do artists in most landscapes the horizon line divides the canvas height for close to 0.618, while choosing the size of the picture itself (the ratio of height to width of the canvas), artists were often close to the number .618. Russian architect Zholtovsky also used the golden section in their projects.
It is known that Sergei Eisenstein artificially constructed film "Battleship Potemkin" on the rules of the golden section. He broke the tape into five parts. In the first three action takes place on the ship. In the last two - in Odessa, where the rebellion unfolds. This shift in the city is the exact point of the golden section. And in every part has a fracture occurring under the law the golden section. In the scene, scene, episode occurs a jump in the development of themes: the plot and mood. Eisenstein believed that, since such a transition is close to the point of golden section, it is perceived as the most logical and natural.
Another example of using the rules of the "golden section" in the cinema is the location of the main components of the frame at the singular points - the visual centers. Often, four points located at a distance of 3 / 8 and 5 / 8 from the respective edges of the plane.
On the golden section has written the great German astronomer - Johannes Kepler (1571-1630).
"Golden Section" and the Fibonacci series
Everyone probably has heard about the "Fibonacci series" or "Fibonacci numbers". So called the number sequence 1, 1, 2, 3, 5, 8, 13, 21, ..., in which each member, beginning with the third is the sum of the previous two. It is believed that first formulated the problems leading to such a number of Italian mathematician Leonardo of Pisa (1180-1240), also known as Fibonacci (son of Bonacci "-" good-natured son ") and which was not equal, not only among his contemporaries, but also three next century. A statement of the problem, which leads to Fibonacci numbers, as follows: How many pairs of rabbits will be born from one pair for a year, if: a) each pair of each month generates a new pair which from the second month becomes a producer, and b) the rabbits do not die.
It is easy to prove that any n-th Fibonacci number (Fn, where n = 1, 2, 3, 4, ...) is determined by the exact formula, which gives every n-th term of the sequence without referring to previous members
Fn = [Fs ^ n - (-Fi) ^ n] / 5 ^ 0,5 (7.1)
where Fs and Fi - the number of Phidias, which are the roots of the quadratic equation x ^ 2 - X - 1 = 0 and are respectively:
Fs = (1 + 5 ^ 0,5) / 2 = 1,618 ..., Fi = (5 ^ 0,5 - 1) / 2 = 0.618 ... . (7.2)
The number of Fs and Fi differ by exactly one: Fi = Fs - 1. These numbers are named after the Greek sculptor Phidias, who lived in the V. BC. er. He supervised the construction of the Parthenon in Athens, in proportions which repeatedly present the number 0.618: Phidias shared segment L into two parts so that a large part of A is a mean proportional between the whole segment and a smaller part of it (A = 0,618 L).
Five-pointed star has always attracted people's attention perfection of form. Not without reason the Pythagoreans it was her chosen symbol of their union. She was considered an amulet of health. Nowadays star flaunts on flags and emblems of many countries. And the thing is that in this figure, "incorporated" the number of Fs = 1,618 (in a surprising constancy of the ratio of segments that make up the five-pointed star).
Formula (7.1) suggest that the ratio of two adjacent Fibonacci numbers (3 / 2, 5 / 3, 8 / 5, 13 / 8, ...) quickly rushes to the number of Fs, that is, the Fibonacci series is well "is superimposed on an exponential function c argument n = 1, 2, 3, 4, ... (And an intensity equal to ln (Fs) = 0,4812 ...):
N G. exp [n ln (Fs)], where G = Fs / (1 + Fs ^ 2) = 0.4472 ... . (7.3)
8. WORLD OF PLATO
Plato's world - so we call the world of fundamental mathematical truths that supposedly can exist outside of time (forever) and independently of us mortals. The famous ancient Greek philosopher Plato (428 - 347 years. BC. Er.) Obviously the first to express this idea, however, in Plato we are talking about all the truths, the ideas (in art, poetry, literature, philosophy, politics, etc . etc., which, in my opinion, is debatable). Below is set out looking at the world of Plato one of the best minds of our time - Roger Penrose (b. 1931) - an outstanding scientist of our time, actively working in various fields of mathematics, general relativity and quantum theory (the author of twistor theory). On Plato's world of Penrose writes, in particular, in his best selling book "The New King of the mind ..." - from this very popular science books are taken Penrose thought given me below.
Plato's world available to us solely through the intellect (with the help of mathematical reasoning and intuitive guesses), this is the reality with which researchers are dealing with in moments of creativity. This is the realm of pure mathematics (its objects), this "divine book, which contains all the best evidence. And mathematicians sometimes revealed to one or another of its page: in moments of epiphany mind just in contact with the objective truth (coming in the head with "sky"). As part of the personal "revelations" Penrose said that they were always preceded by long persistent conscious thought, although the desired solution appears suddenly like a "flash" (when he thought about the issue in "background", not deliberately), and with full confidence in the correctness and beauty solutions. It is also noteworthy that many of the ideas born in moments of inspiration is inherent in the scale, that is, the idea covers a very broad area of mathematical thought.
Plato, in particular, taught: our soul existed before we were born [but when and where there was our soul?] Soul of the deceased continues to exist in Hades (the realm of the dead) and has the ability to think the soul is immortal and indestructible. That is why mathematical discovery, perhaps - only one form of memories! In any case, Penrose said: "... I am often struck by the similarity between the two states, when you're painfully trying to remember someone's name - and when you're trying to find an adequate mathematical concept.
The great Albert Einstein (1879-1955) once wrote in a letter: "The words or language, both orally and in writing, apparently play no role in the mechanism of my thinking." The same is said and Penrose: "... I find words useless for mathematical thinking ... There is no doubt that everyone thinks differently ... most polar styles of mathematical thinking are, as seems to be an analytical / geometric.
Many people think that a mathematical proof is constructed as a chain of successive statements, where each step follows from the previous one. However, only a general and intuitive conceptual content - that's what is really necessary for the construction of a mathematical proof. Another interesting observation is the Penrose: "In the dream, unusual ideas arise easily and in large numbers - but only in very rare cases, they are a critical control of waking consciousness. (As for me, I have a dream never came into fruitful scientific ideas ...). "
All the most accurate theory (general theory of relativity, quantum mechanics, string theory, ...) and extremely fruitful in terms of mathematics, reflecting the deep connections between the real (physical) world and the world of Plato. It may be that these worlds are identical? Functioning of the real world, ultimately, can only be understood in terms of mathematics, that is, in terms of the Platonic world. The very precision of the general theory of relativity and quantum mechanics provides an almost mathematical level of the existence of our physical reality (and it seems not so obvious as to create these profound theories).
The notion of mathematical truth goes beyond the creation of man. True mathematical discoveries should normally be regarded as achieving greater than the "simple" inventions - the essence of the works of man. " In mathematics, it often takes very real opening - is when a structure (object) gives you much more than what it was originally laid down (say, by the author, who proposed to consider the object). Examples of such objects: complex numbers, the Mandelbrot set, etc. In connection with such objects, even scientists are atheists think about the possibility of "creations" Overmind, the existence of a certain higher mental activity. Mathematical discovery consists in expanding the area of direct contact with the world of Plato. No meaningful "information" in the conventional sense of mathematical research facility does not receive, because all the information is already there originally. All that is required of researchers - is to connect the different parts and "see" the answer. "The independence of the investigator" mathematical object and gives it platonic existence.
We emphasize that the mathematical structures (even the most exotic, such as fractal structures), there are no less "real" than Mount Everest, and can be investigated just as we study the jungle (this applies to the world of numbers.) But Plato's world does not consist of tangible things, but from the "mathematical objects". Objects, say, a pure geometry - lines, circles, triangles, planes, etc. - May be only approximately realized in the real world of physical things.
In communication (conversation), say, two mathematicians their individual sentences (phrases, facts, images, concepts) are most likely remain ... are not understood. Nevertheless, the two men still able to understand each other, for interesting or profound mathematical truths solutions (with low density) in the mass of all possible mathematical truths. During the conversation, each of Mathematicians in direct contact with one and the same world of Plato, which leads to mutual understanding at the level of intuition.
One of the most striking properties of mathematics lies in the fact that the truth of mathematical statements could be established by abstract reasoning (which are transferable)! Mathematical truth is constructed from the simple and obvious components, and when they become clear and understandable to us, their truth can agree without exception. We must "see" the truth of mathematical reasoning to verify their validity. This "vision" - the very essence of consciousness. Absolutely accurate, correct wording sometimes are a hindrance during the first presentation of mathematical ideas, so that at first may require less clear narrative form (a characteristic, for example, the popular scientific literature).
Computability property - not the same as that mathematical precision. How much mystery and beauty in the world of Plato - and yet most unknown part of this world is connected not with the algorithms and calculations. Penrose said: "... I can not escape the feeling that in the case of Mathematics, a belief in a supreme eternal existence - at least for the most profound mathematical concepts - is under a lot more reasons than in other areas of human activity. Undoubted uniqueness and universality of this kind of mathematical ideas by their nature substantially different from anything with which to face in the field of art and technology. "
9. MATHEMATICS AND THE REAL WORLD
Despite the declaration of independence of mathematics, no one would deny that mathematics and the physical world are connected with each other. Of course, it remains in effect a mathematical approach to solving the problems of classical physics. It is true that a very important area of mathematics, namely the theory of differential equations, ordinary and partial cross-fertilization of mathematics and physics rather fruitful.
Mathematics is clearly useful in interpreting the phenomena of the microworld, described by quantum mechanics. However, the new "applications" of mathematics are very different from the classical ones. One of the most important tools of physics was the theory of probability, which previously applied mainly in the theory of gambling and insurance. Mathematical objects that physics assigns to "nuclear states", "transition", the space of Calabi-Yau manifolds, etc., are very abstract in nature and have been introduced and studied by mathematicians long before the advent of quantum mechanics. We should add that after the first successes of serious difficulties. This happened at a time when physicists tried to apply mathematical ideas to more subtle aspects of quantum theory, however, many physicists are still looked upon with hope for new mathematical theories, believing that they will help them in solving new problems (including string theory).
Even if we include in the "pure" mathematics probability theory and mathematical logic, it turns out that at present the other natural sciences are used less than 50% of the known mathematical results. What should we think about the other half? What are the motives behind those branches of mathematics, which are irrelevant to the solution of physical problems?
We have already mentioned the irrational as a typical representative of such theorems. Another example is the theorem proved by Lagrange [Joseph Louis Lagrange (1736-1813) - French mathematician and engineer of Italian origin. Along with the Euler - the best mathematician of the XVIII century.] "Important" and "beautiful" in terms of any mathematics, this theorem states that every positive integer can be represented as the sum of the squares of no more than four numbers (eg, 23 = 32 + 32 + 22 + 12). Currently, however, it is inconceivable that this result could be useful theoretical physicist, and even more so the experimenter [incidentally, from the standpoint of virtual cosmology Lagrange's theorem may be the most fundamental: for example, she "explains" ... the number of observables (namely, four!) Measurements in the real physical world]. It is true that physicists have to deal with whole numbers today are much more common than in the past, but the integers, which they operate, are always limited (they rarely exceed a few hundred [dear reader, remember this scientific conclusion (!) Community within the virtual cosmology we him come back and not one time!]), hence this theorem, as Lagrange's theorem, can be "useful" only if applying it to an integer that does not go to some of the border. But if we restrict the formulation of Lagrange's theorem, it immediately ceases to be interesting for a mathematician, since all the fascination of this theorem lies in its applicability to all integers. There are so many allegations of integers that can be checked by computers for very large numbers, but as long as the total (analytical) evidence has been found, they are hypothetical and are not interested in professional mathematicians [the latter is therefore not interested in my virtual cosmology!] .
Focusing on topics far removed from immediate application, is not uncommon for scientists working in any area, whether it be astronomy or biology. However, while the experimental result can be refined and improved by a mathematical proof always is final. That is why it is difficult to resist the temptation to consider the math, or at least the part that has no relation to "reality" as an art (not science).
Mathematical problems are not imposed from outside, and if we take the modern point of view, we are completely free to choose the material. In assessing some mathematical work by mathematicians there is no "objective" criteria, and they are forced to rely on their own "taste". Tastes also vary greatly depending on the time, countries, traditions and personalities. In modern mathematics, there are fashion and the three "schools": "classicists", "modernists" and "abstractionists. To better understand the differences between them, we analyze the four criteria used by mathematicians when evaluated theorem or group of theories:
1). "Beautiful" mathematical result must not be trivial. This is not a consequence of the axioms or the well-known theorems, should be a new idea or clever apply old ideas. That is, for mathematics is important not the result itself, but the process of overcoming the difficulties it faced when it is received.
2). An essential element of "beauty" of the theorem is its simplicity. Search for simplicity is characteristic of all scientific thought, dating back to the ancient Greek philosopher Epicurus (341 - 270 years. BC. Er.) First proposed the idea that the apparent complexity and infinite variety of the world around us may be hiding inside the simplicity of the structure.
3). Mathematician must solve a new problem by any means possible. However, since the 19 th century mathematics is clearly divided into "tacticians" who are seeking a purely military solution to the problem (the classical tools of mathematics), and "strategists" who are prone to detour (the more "abstract" structures), giving them an opportunity to break the problem small forces.
4). In any mathematical problem has its own history (pedigree). When the solution is obtained (eg, through 356 years like Fermat's Last Theorem), the story does not end there, for start-known process of expansion and generalization. Thus, Lagrange's theorem leads to the question of the representation of any integer as a sum of cubes, fourth powers, etc. (Waring's problem "is still not completely solved). Even if the original theory, eventually "dies", it usually leaves behind numerous live shoots.
Mathematicians are already faced with such immense bulk of problems that, even if it broke off all contact with experimental science, their decision would take more ... a few centuries!
However, experimenters are ready to accept the "ugly decisions", as long as the problem was solved. Likewise, in mathematics and classicists abstractionists not very concerned about the appearance of "abnormal" results. (For example, in 1890, obtained the following "abnormal" result: he managed to construct an example of a continuous curve, which completely fills a square, that is, through all its points (Peano curve). Since then, have been invented hundreds of mathematical "monsters" contradicts the "common sense" and mathematicians plunged into shock.) On the other hand, the modernists have gone so far as to see in the emergence of "pathologies" in the new theory - a symptom, indicating the inadequacy of basic concepts.
10. Classical number theory
Number theory - it is infinitely vast field of mathematics, which can be studied from various points of view. This is evidenced by the fact that the words "number theory" often specify the section name of this theory: the classical (analytical) ... ... additive, multiplicative ... and, finally, my graphics theory of natural numbers (GTNCH), and even distinguish between a theory algebraic numbers and transcendental numbers.
The classical theory of numbers - an extremely complex branch of mathematics. We illustrate this in a brief excursion into history, associated with three bright, talented individuals who made notable contributions to mathematics (including number theory).
Major works of English mathematician GH Hardy (1877-1947) are devoted to number theory and the theory of functions. Most of his works performed in conjunction with the English mathematician - James I. Littlewood (1885-1977), but some of the work of Hardy been implemented together with distinctive Indian mathematician S. Ramanujan (1887-1920), which Hardy saw his discovery and "the only romantic event "in life. Hardy says of him (see the book of Hardy's' Twelve lectures on Ramanujan).
Ramanujan was the most romantic figure in the modern history of mathematics. Without special mathematical education, he is for his short life has done a lot: his published works form a book of 400 pages, and left the mass of unpublished papers. It can be considered the greatest formalist of his time since his formula (Hardy showed them a half-dozen nearly every day) - simply amazing.
Exceptional ability in mathematics in Ramanujan's evident to 10 years. However, critical for the career of any math period (18-25 years, when the mind is the most elastic) was unfortunately lost in the struggle with the difficulties of living a poor Hindu family (in 1922, Ramanujan was married). Thus, in the best years of his genius was directed in the wrong direction, relegated to the sidings and to some extent distorted. Only in the age of 26, Ramanujan wrote to Hardy, after which he was able to take him from India to England. But just three years later, Ramanujan fell ill with tuberculosis, and so more and not recovered.
Hardy, Ramanujan said that the goddess Namakkal inspired him with the formula in my dreams (see above about the dreams of Penrose). Often, rising from his bed, he could record the results and quickly check them, though not always able to give a rigorous proof. But Ramanujan was not a mystic, and religion was not an important part of his life, he believed that all religions are more or less equally true, that is not singled out the Hindu religion, but only served his ceremonies [by the way, Hardy does not believe in the ancient wisdom of the East] . Ramanujan was not a convinced atheist, he was typical of an agnostic, and an orthodox Hindu of high (but very poor) castes, was a strict vegetarian, and he was preparing something to eat (pre-dressed in his pajamas).
In independent work of Ramanujan is no simple and inevitable characteristic of the greatest works of other mathematicians. His work has been rather strange. Most of my life [before coming to England], he worked, while remaining ignorant of the discoveries of modern European mathematicians (about 2 / 3 of his works - is reopening). Hence, it is a challenge almost all the canons: his formulas contain virtually no evidence, he did not fully understand what an analytic function, and he never used the deep theorem of Cauchy. Ramanujan was far from understanding the true complexities of the analytic theory of numbers, which was his only real defeat: he alone is almost nothing to prove and a lot of what he imagined was false. But at the same time, he had a passion for numbers themselves: his ability to remember the characteristics of numbers was almost supernatural. According to Littlewood, every positive number has been one of Ramanujan's personal friends ...
In the words of Hardy is important to us that even such a genius like Ramanujan number theory has not opened its secrets. Here is how it explains itself Hardy [emphasis mine]: "Analytic number theory is one of those exceptional areas of mathematics, in which evidence is everything and nothing, devoid of absolute rigor, is not accepted. ... You can not achieve any real understanding of the structure and meaning of the theory [of numbers], or to obtain a common instinct that leads you into further research, until you examine the evidence. It is comparatively easy to make clever guesses, really, there are theorems, like 'Goldbach's Theorem ", which has never been proved and which any fool can guess." "A mathematician usually gets a theorem by intuition, he finds a plausible conclusion, and starts working on the creation of evidence. Sometimes it is a routine action, and any well-trained professional can provide the desired result, but more often imagination is a very unreliable guide. In particular, what happens in the analytic theory of numbers, where even the imagination of Ramanujan led him down the wrong path. " "No one can convince yourself that 2 ^ 127-1 is prime, if not to examine the evidence. No one is so lively and comprehensive imagination. "
Great delusion? As a rule, still in school for lessons in arithmetic, we first learned the word genius Karl Gauss: Mathematics - the queen of sciences, and number theory - the queen of mathematics ". However, not all mathematicians share that view. For example, Mr Hardy agreed with Gauss, but in the sense that the theory of numbers as well ... useless, as the Queen of England. This witty saying Hardy is perhaps an example of the great fallacies (it comes from the great minds). In the history of science, such examples are few, so genius, Euler believed that the human mind will never penetrate the mystery of the distribution of prime numbers (many of the secrets of these numbers were later revealed!), And the great Einstein never acknowledged quantum mechanics (which became soon one of the pillars modern physics!).
Everyone knows that the sacred awe attitude toward the natural numbers famous Pythagoras (see "Maxims of the great"). However, the worship of the ancient Pythagoreans before the numbers eventually become entirely explain their limited knowledge (which may be controversial, thought so, for example, even Euler).
In our time, the value of well-known theory of numbers is clearly underestimated, even among professional mathematicians. For example, when in 2005 I first came to the St. Petersburg Mathematical Institute. VA Steklov Institute, I identified only one ... the only (!) Expert on the theory of numbers - by the distinguished Veniaminovich Boris Lurie (alas, my "theory" has left him completely indifferent). Autumn 2010 I was first sent (by email) to the specified institute their articles on a virtual Cosmology (many specific specialists in relevant areas of mathematics), but no answer was not forthcoming. As for the "general public", for her theory of numbers - it is something like getting the secret ("science" of numbers), astrology, etc. obscurantism. And even my virtual cosmology (rather curious game of publicly available "in numbers" rather than science) for readers of the Internet is, in general, poorly understood ... delusions (but sadly aware of it).
The main argument of "usefulness" of the theory of numbers in real life was the use of prime numbers in cryptography (since 1977) dealing with is known coding secret messages. It turned out that a convenient encryption key can be a huge composite number derived by multiplying two large prime numbers (say, about 10 ~ 80). These two numbers - a reliable deciphering key, to search for which it is necessary to factorize the encryption key into two prime factors that make it practically impossible, since even the most powerful computers in the world, it would take several years.
Another prime numbers allegedly involved in the world of nature, however, prove this look so far is weak [eg, see the book by S. Singh's "Fermat's Last Theorem, p. 100]. On the background of a very modest role played by official science removes the theory of numbers, my virtual cosmology seems complete madness, because it reflektsii "perceive" the world's deepest bond numbers with the fundamentals of the universe (space-time). We can say that the virtual cosmology - is Pythagoreanism XXI century, appealing to the discoveries of modern science. Incidentally, the lion's share of these discoveries accomplished over the past 300 years (just a tiny moment in the long history of mankind!), And such knowledge is, in principle, could have the previous "wave" of human civilization vanished in some global catastrophe (like the infamous Atlantis ). Perhaps the motto of the Pythagoreans "All is number" - a kind of "echo" lost civilization, as any oral tradition quickly lose credible information, "diluting" its almost zero ...
1. Types of numbers (WORLDS NUMBER)
Consider the number of positive integers N, ie the numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ... (and so - to infinity). Among these numbers are called prime numbers (2, 3, 5, 7, 11, 13, ... - is also up to infinity), which are divided (integer division) only by one and themselves. Speaking of prime numbers, we denote them by the letter P (and not by the letter N). I note that the prime number later we will have a very special conversation.
The fundamental theorem of arithmetic states that every positive integer N, a superior one (N> 1), it seems the only way in the form:
N = P1 ^ a P2 ^ b P3 ^ c P4 ^ d ... Pn ^ m, (1.1)
where P1, P2, P3, P4, ..., Pn - some primes (arrange them in ascending order, they do not exceed N);
a, b, c, d, ..., m - exponent (any positive integers greater than zero: 1, 2, 3, 4, 5, 6, 7, ...).
Representation (writing) a natural number N in the form (1.1) is called the canonical decomposition of N (the prime factors) or the factorization of N. For example: 20 = 2 ^ 2 ∙ 5 (note that 5 ^ 1 = 5, that is, any number of degree 1 is equal to the number itself) 36 = 2 ^ 2 ∙ 3 ^ 2 42 = 2 ∙ 3 ∙ 7 , 84 = 2 ^ 2 ∙ 3 ∙ 7 132 = 2 ^ 2 ∙ 3 ∙ 11. Because, for example, 261,360 = 2 ^ 4 ∙ 3 ^ 3 ∙ 1 ∙ 5 ^ 11 ^ 2, then no other set of primes P will never give us the number 261360, that is to be clearly understood that the factorization of any number N has its outcome the only possible result!
Search the exponents a, b, c, ..., d in the factorization - a difficult task away. In a serious computer program Mathcad 2000, focused on solving mathematical problems, there is even a special team (Factor) to automatically search these indexes, that is, for the factorization of N (up to the number N of the order of 10 ^ 15?). We arbitrarily take a sufficiently large number N is not easy to find its canonical decomposition, even with a computer. And even to determine whether a given large number N is prime or composite - it is sometimes very difficult.
Thus, starting with N = 2, any integer N, can be represented (and the only way) in the form of a certain product of primes (eg, 261360 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3 ∙ 3 ∙ 5 ∙ 11 ∙ 11). That is the world's natural numbers are prime numbers are the fundamental building blocks ("atoms"), of which construct an infinite set of all other numbers (similar to how a small number of atoms of "periodic table" constructed a huge amount of various molecules of the real world).
One of the key concepts of virtual cosmology is the concept of type integer.
Type the numbers - the number of integer divisors of the integer N (including 1 and the number is N).
Type the numbers denoted by the letter T. For example, the number N = 20 a total of six subgroups: 1, 2, 4, 5, 10, 20, so its type T = 6. Note that the first positive integer N = 0 has infinitely large type (T tends to the number of "infinity"), because zero divided by any number (except zero). We assume that the number of "infinity" (strictly speaking, this is not a number), "divided" on all the natural numbers N (starting with one) so the first natural number N = 0, in a sense, "closes" (in terms of its type T) with the number "infinity".
Number N = 1 (one) is the only specific number, which type is equal to one (T = 1).
It is clear that any prime T = 2 (and these numbers are infinitely many).
Canonical decomposition of N - is a kind of "record" ("Passport"), from which we can extract a lot of interesting information on the number N. So, knowing the canonical decomposition of N [its expansion according to the formula (1.1)], we can calculate the type (T) given number N:
T = (a +1) (b +1) (c +1) (d +1) ... (m +1). (1.2)
Proverbial formula (1.2) we shall call the canonical type of N, thereby emphasizing that the type T of the number N, we have found in its canonical decomposition. For example, for the number N = 261 360 = 2 ^ 4 ∙ 3 ^ 3 ∙ 1 ∙ 5 ^ 11 ^ 2, we obtain a canonical type T = (4 +1) (3 +1) (1 +1) (2 +1) = 120, that is a specified number N has exactly 120 divisors of integers (see for yourself, finding a computer all the divisors of this number.)
Small Dividers
If all the divisors of any number of N arranged in ascending order, then exhausting their first half (small divisors), we find that the remaining (large divisors) are equal to the quotient of the number N of one of the small divisors. So, the number N = 20, there are three small divider - 1, 2, 4, and three great divider - 5, 10, 20 (which is equal to the relations: 20 / 4, 20 / 2, 20 / 1). Check: 20 = 2 ^ 2 ∙ 5 ^ 1, then T = (2 +1) (1 +1) = 6 - number of factors.
Thus, the definition of the type of N "forehead (when we say we do not know its canonical decomposition) is reduced to finding small divisors of N, and on [1; N ^ 0,5], where the expression N ^ 0, 5 means the square root of the number N (this root is not time for us to meet in the virtual cosmology). After all, if the number N> 1 and equal to the product of two positive integers, then at least one of them no more than N ^ 0,5 - noticed it yet Fibonacci. Thus, small dividers - it's also a "passport" number N, with crucial information about him (along with the canonical decomposition of N).
The average number of type (Dirichlet's formula)
In the physical appearance of a number of different types (numbers, with different types) is a pseudo-random in nature, that is, in practice, generally speaking, impossible to predict which type will have the following number N. Obviously only one thing - the further we move away from unity (N = 1), the more types may occur, and along with them are bound to appear and the number with the smallest types of T = 2, 3, 4, ... .
Any natural number N in addition to type T, we will also assign an average type (Ts), which is equal to the arithmetic mean of all types of numbers from 1 to N, inclusive:
Ts = (T1 + T2 + T3 + T4 + ... + Tn) / N. (1.3)
Despite the erratic fluctuations of type T, the average Ts type of behavior is surprisingly quiet. Growth law parameter Ts first established German mathematician Peter Gustav Lejeune Dirichlet (1805-1859), so this critical law for us, we shall call the formula of Dirichlet:
Ts = lnN + (2 C -1) + "epsilon, (1.4)
where C = 0.577 215 664 901 532 ... - the Euler-Mascheroni (or Euler's constant), and the parameter "epsilon" tends to zero when the number N increases indefinitely. At the beginning of the natural series average real type (Tsr) exceeds the (theoretical) value of Ts, but since the number N = 47, some of the real value of the average type may be less than Ts, that is, begin the chaotic oscillations of the real average type "around" the values of Ts . The relative error (AP) Dirichlet formula decreases rapidly, so, assuming "epsilon" is zero, I appreciated the module OP = (Tsr - Ts) / Ts Dirichlet formula (1.4) the following empirical expression:
abs (OP) 1 / N ^ w, where w = "pi ^ 2 / 12 = 0.8225. (1.5)
Mathematical constant C is defined as the limit of the difference between a partial sum of the harmonic series and the natural logarithm of the number N (we assume that the number N tends to infinity):
C = lim [(1 / 1 + 1 / 2 + 1 / 3 + 1 / 4 + 1 / 5 + 1 / 6 + 1 / 7 + ... + 1 / N) - lnN]. (1.6)
The constant C introduced in 1735 by Leonhard Euler. Italian mathematician Lorenzo Mascheroni in 1790 calculated the 32 characters constant and offered her a modern notation (Greek letter "gamma"). However, the arithmetic nature of C has not been studied: not yet revealed whether this is the rational number. However, the theory of continued fractions shows that if the constant - a rational fraction, then its denominator is greater than 10 ^ 242 080.
Reflektsiya 1.1. [Note that the constant C = 0.577 ... (very "in demand" in the theory of numbers!) Is close to the notorious "golden section" (.618), and at the end of the Great segment (ie, when N = 8 ∙ 10 ^ 60 Evie), the average type of is Ts = 140,39, which is close to the reciprocal value of the fine structure constant: 1/PTS = 137.
Looking ahead to the part of some parameters of great length, I will say that at its end with the individual numbers N type T can reach nearly a trillion (integer divisors)! Therefore, a small middle style Ts = 140,39 at the end of the Great segment can have only one explanation - the natural series of preferred numbers ... small type T: the vast majority of the numbers of small divisors. For example, prime numbers (with the smallest type of T = 2) in large stretches of about 7 10 ^ 58 (pieces), which represents approximately 0.7%. And this is a very respectable share, for an average of one world must account for 10 ^ 61/807430 = 1,2 10 ^ 55 numbers or 0.00012% of all positive integers great length.
Thus, the world of numbers as it copies (reflects) one of the fundamental laws of the real (physical) world: in nature, small specimens are more common than large ones, where the word "individual" should be understood not only as biological entities, but also a variety of inanimate objects matter.]
Worlds of numbers (the worlds Isaeva)
Fig. 1.1 points shows the types of T (their numerical values) in the first 400 natural numbers N, and these types, like sparrows on the wires, "sit" on different levels, ie, belong to different types. But if a sparrow can fly to the other wire (level), the number N of each type of form a closed world with its unique laws (which is easy to see through your computer). Therefore, in a virtual cosmology is sometimes convenient to talk about worlds that combine the numbers with the same type. Thus, all prime numbers (of type T = 2) form a world of "Number 2", and the world "number 3" form of a relatively rare type of T = 3; worlds' number 4 and number 8 "- rather" heavily populated " (numbers of T 4 and T = 8 quite a lot), etc.
Worlds there is not in ascending order of their numbers, we present the very first Worlds (type the first 17 natural numbers): 1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 7, 16, 15, 18 , 14, 20, 24.
Reflektsiya 1.2. [If you remember the Evi-conversion, the first 10 ^ 43 N of natural numbers can be interpreted as the time interval of one second. During this time, the natural numbers time to "shoot" about 105 different worlds - about as many different types of T is contained in the interval [1, 10 ^ 43]. On the scale of our (human) time at any moment, as there would be both "parallel" worlds - these are some of the worlds (how much - is not entirely clear) that are repeated often, and the high frequency of "flicker" makes it "affordable" for us ( other worlds, we simply do not have time to notice - so are born "hidden" worlds). Perhaps the essence of this "crazy" ideas are now clear to the reader. It seems that there is a place for the imagination ...]
So, we say that all the natural numbers with the same type of T form the world number-T ". Simply, if only because that is sometimes easier (of course, the taste of the author) to talk specifically about the worlds, not types.
All the worlds are divided into two significantly different groups:
- Rare (odd) worlds - they have an odd number N of type T.
- Frequent (even) worlds - they have an even number N of type T.
Rare worlds form a number of the form N = i ^ 2, where i = 1, 2, 3, 4, ... (natural number). The first numbers of the rare worlds: N = 1, 4, 9, 16, 25, 36, 49, 64, .... It is clear that the proportion of such numbers in the natural number decreases rapidly (according to the law of the square of the parabola). All of these numbers N the last small divisor is the first large divider, so the number of large subgroups will always be one less than the small - so there is an odd type T. For example, the number N = 36 is small divisors: 1, 2, 3, 4, 6, hence its large subgroups are as follows: 36 / 6 = 6 (which repeats the last small subgroup), 36 / 4 = 9, 36 / 3 = 12, 36 / 2 = 18, 36 / 1 = 36. Thus, we find that the number N = 36 = 6 ^ 2 type is an odd number of T = 7.
Leaders of the Worlds
In moving along the natural numbers have a certain number N, first appears type T. This number N will be called the leader of the world T. We can say that the leader of the "open" this world of T. It is clear that the leaders of the world (LFM) and the leaders of the worlds rare ( LRM) - this infinite series of natural numbers, here are the first of them:
LFM: N = 2, 6, 12, 24, 48, 60, 120, 180, 192, 240, 360, 720, 840, 960, ...
LRM: N = 4, 16, 36, 64, 144, 576, 900, 1024, 1296, 3600, 4096, 5184, 9216, ...
Fig. 1.2 shows the types of (T), leaders (N) the first 82 worlds, appears on the interval [1, 520000].
Back in early 2002, I managed to find in large stretches of (presumably) all 120,000 LRM. But LFM I could find only in the interval [1, 10 ^ 32], which was 22,164 LFM. Extrapolating this result, I received such an assessment: in large stretches probably are about 687 430 LFM (ie 5.7 times larger than the LRM, where 5,7 - the embodiment of "magic" number 7?).
An interesting question: in what range of values most likely to be the types of T leaders (LFM and LRM) within the Greater segment? Probably, these bands are as follows:
- To chirp - a T = 10 ^ 8 ... 10 ^ 9
- For LRM - is T = 10 ^ 7 ... 10 ^ 9.
Thus, we have introduced the concept of "leader" - it's kind of special number N (with a special type of T). If all found the leaders to plot T = f (N) in logarithmic axes (see Fig. 1.2), then such a graph all the points (all values of T among leaders N) will be in expanding (from the origin) corridor, reminiscent of your outline ... the tail of a comet (in the frequent and rare worlds - a "tail" of the leaders, that is, we have two "tail", which almost overlap each other on the graph). Said the "tail" on the graph T = f (N) can be mentally outline some of the upper and lower line (overseas) - so we come to the notion of upper and lower leaders (in the frequent and rare worlds).
Lower leaders - are the leaders with these types of N T is less than that in the future can not be. On the above schedule T = f (N) is a type of T (the point on the graph) of the lower leaders is lower imaginary line "tail." It is easy to see that the lower leaders in rare worlds - are the leaders of the so-called simple worlds, which type is equal to a prime T = 2, 3, 5, 7, 11, 13, 17, 19, .... A lower leaders in frequent worlds - are the leaders of the so-called doubled worlds, which type is twice a prime T = 4, 6, 10, 14, 22, 26, 34, ... . Lower N leader in the frequent and rare is the following (respectively):
N = 2 ^ (T / 2 - 1) 3 and N = 2 ^ (T - 1). (1.7)
Note that the world is the "number 8" (its leader, N = 24) features a sort of an anomaly: type T = 8 is not a double world, but the first of formulas (1.7) - still works (for T = 8 is shown ... "magic" numbers 7?).
The lower boundary of an imaginary "tail" on the schedule T = f (N) can be described by the following formulas (for common and rare worlds, respectively):
Tmin = 2 ln (N / 3) / ln2 + 2 and Tmin = lnN/ln2 + 1. (1.8)
The lower boundary of the "tail" is characterized by the fact that as the number N of any new LFM (LRM), the type T will be no less than Tmin.
By the end of the Great segment typed quite a bit lower Leaders: Part worlds - 46 pieces (Tmin = 398), and in rare worlds - 47 pieces (Tmin = 199). This means that in large stretches of frequent worlds will all (without spaces!) Are even types of T = 2, 4, 6, ..., 398, and in rare worlds - the odd types T = 1, 3, 5, ..., 199. And all the great styles of, say, in rare worlds will emerge with an average probability of only about 10 ^ -6, since the upper limit of "fluctuations" by the end of LRM Great segment will grow up to Tmax = 8 10 ^ 10 (see below), and all will be 120,000 odd worlds.
The top leaders of an imaginary form an upper border of the "tail" on the schedule T = f (N) for the leaders, which is well-guessed in Fig. 1.2. And parts, and for rare worlds can formulate the following definition: if the leader's type N T previously appeared more types, the kind of leader we call the top leader. Are not (see above series of LFM and LRM):
- Among the LFM - is N = 192, 960, ...;
- Among the LRM - is N = 64, 1024, 1296, 4096, 5184, 9216, ....
It is obvious that with increasing N top leaders will meet less frequently (Figure 1.2. This is not seen as a logarithmic scale). In my assessment in large stretches of the worlds rare recruited 270 top leader, and frequent worlds, probably about 748 top leaders.
Must be clearly understood that the senior (highest) top leader in the interval [1; N] - is a natural number that has the maximum number of subgroups (of all numbers in this segment). For example, a top leader N = 3600 type T = 45, and this means that on the interval [1, 3600], no number (of the rare of the Worlds), except for N = 3600, has so many subgroups (45 subgroups). Thus, top leaders - this is a very interesting number in the virtual cosmology (an infinite sequence of which is probably impossible to specify any precise formula ").
The maximum possible type (Tmax) on the interval [1; N]. Taking as a basis for the types T in the top leaders of N, we would like to build some kind of line Tmax = f (N), "the envelope" above all types of T (all points in Fig. 1.2). However, further we will be able to specify only the function, "around" ("along"), which are the real values of Tmax.
Tmax in rare worlds
As I said earlier, back in early 2002, I managed to find in large stretches of (presumably) all 120,000 of the worlds leading rare (LRM). Of these top leaders were only 270 numbers. For example, here are the canonical decomposition of the three largest (latest) on a large segment of the top leaders of the N (270-TI):
N = 2 ^ 12 3 ^ 4 5 ^ 4 7 ^ 4 (1911 1913 17 19 23 29 31 37 41 43 47 53 59 61 67 71 ) ^ 2 = 3,51 10 ^ 60;
N = 2 ^ 12 3 ^ 8 5 ^ 4 7 ^ 2 (1911 1913 17 19 23 29 31 37 41 43 47 53 59 61 67 71 ) ^ 2 = 5,81 10 ^ 60;
N = 2 ^ 10 3 ^ 6 5 ^ 4 7 ^ 4 (1911 1913 17 19 23 29 31 37 41 43 47 53 59 61 67 71 ) ^ 2 = 7,91 10 ^ 60;
Knowing the canonical decompositions of the numbers N, we find their type T (respectively) by (1.2):
T = (12 + 1) (4 + 1) (4 + 1) (4 + 1) (2 + 1) (2 + 1) (2 + 1) ... (2 + 1) = 69950921625 = 7 , 0 10 ^ 10;
T = 75,546,995,355 = 7,6 10 ^ 10;
T = 82,864,937,925 = 8,3 10 ^ 10.
As part of a virtual cosmology, we agreed to call the function Y = f (X) - a pseudo-function if it is (close to the mean) Y = exp [a (lnX) ^ b], where a and b - some empirical coefficients, say, "shot" with a trend line on the graph Y = f (X). Moreover, this schedule (consisting of discrete points-values) in fact may not have as its description of any of the well-known in the mathematics of continuous functions - that's why I chose the name "pseudo-function" (to immediately confess to the reader in a sort of " fraud ").
Thus, knowledge of the 270 top leaders of the worlds rare in N T, as is a schedule T = f (N), consisting of a 270-point T-values (type T) to determine the pseudo-function of the form:
Tmax / Ts = exp [a (lnN) ^ b], (1.9)
where Ts - medium type (found by the formula of Dirichlet) for the number N, for which we find Tmax, where:
a = 0,2465; b = 0,9028 and abs (OP) = 12% on the interval [24, 10 ^ 20] (a small segment - see "Initial concepts ...");
a = 0,3031; b = 0,8493 and abs (OP) = 9% in the interval [10 ^ 20, 10 ^ 61] (that is, until the end of the Great segment).
Tmax in the worlds of frequent (i-trillion)
As I said earlier, back in early 2002, I found a (presumably) all the leaders of the world (LFM), but only on the interval [1, 10 ^ 32], which was 22,164 LFM. Extrapolating this result, I received such an assessment: in large stretches probably are about 687 430 LFM.
I also managed to find some number N, located at the end of the Great segment, and having a very large number of divisors (with big type T). We assume that this is the top leaders of the world, and their canonical decomposition and the type T (respectively) are given below:
N = 2 ^ 10 3 ^ 6 ∙ ∙ ∙ 5 ^ 4 7 ^ 3 ∙ 2 ∙ 11 ^ 13 ^ 17 ^ 2 ∙ 2 ∙ M = 7,44 ∙ 10 ^ 60;
N = 2 ^ 10 3 ^ 5 ∙ ∙ ∙ 5 ^ 3 7 ^ 11 ^ 2 ∙ 2 ∙ 2 ∙ 13 ^ 17 ^ 2 ∙ M ∙ 137 = 9,28 ∙ 10 ^ 60;
where M = 19 ∙ 23 ∙ 29 ∙ 31 ∙ 37 ∙ 41 ∙ 43 ∙ 47 ∙ 53 ∙ 59 ∙ 61 ∙ 67 ∙ 71 ∙ 73 ∙ 79 ∙ 83 ∙ 89 ∙ 97 ∙ 101 ∙ 103 ∙ 107 ∙ 109 ∙ 113 ∙ 127 ∙ 131;
T = 697 596 641 280 = 7,0 10 ^ 11;
T = 717 527 973 888 = 7,17 10 ^ 11;
The most common (linear) interpolation between these two numbers N shows that at the end of the Great segment (ie, when N = 8 ∙ 10 ^ 60) we obtain T = 7,03 ∙ 10 ^ 11.
Reflektsiya 1.3. [As we see, at the end of the Greater largest segment (last) top leader N "built" using the first 32-33 prime numbers (2, 3, 5, 7, ..., 131, 137), which are consecutive (no gaps) in the canonical decomposition of the number N (and each of the primes was erected in some degree: 10, 6, 4, 3, 2, 2, 2, 1, 1, 1, ..., 1).
Specified number (32-33) of prime numbers, probably finds its "reflection" in the real world, where these numbers (or close-to-many) play a fundamental role. Here are some examples:
32 ∙ 2 codon (and 46 chromosomes) in the structure of DNA, 32 teeth in humans, 27 bones in the hand man (and this is the perfect body!) 26 bones in his foot man, 33-34 vertebrae in the spine rights; to 33 letters contain most of the alphabets in the world, 33 major languages on the planet, and 33 important religion in the world, 33 of the term indicate the tempo of music, 32 colors in the palette the artist - it is (more or less reasonable) maximum color, ...
32 versions of the arrangement of atoms around the lattice site (see crystallography), 29 clusters of galaxies in the largest of the superclusters, 39 satellites of Jupiter (the maximum in the solar system); ...]
Pseudo-function in frequent worlds can also be expressed by (1.9), but under very different settings:
a = 0,2710; b = 0,9028 and abs (OP) = 9% in the interval [24, 10 ^ 20] (ie, within a small segment);
a = 0,3197; b = 0,8588 and abs (OD) = 5% in the interval [10 ^ 20, 10 ^ 61] (that is, until the end of the Great segment).
So, with the number N of its maximum possible type of Tmax (clearly, he realized frequent worlds) - is increasing, and there is a limit on the value of Tmax (for a given number N). In the first approximation to the large segment can be used quite simple (and purely empirical) pseudo-function:
Tmax = exp [(lnN) ^ 0,6692]. (1.10)
Fig. 1.1 upper limit types (Tmax) shows a dotted line. According to all my estimates (above), at the end of great length, the maximum possible type is Tmax = 7 ∙ 10 ^ 11, and this number from now on we shall call i-trillion (trillion Isaeva). Let's not very modest, but it is immediately clear about exactly what question (7 ∙ 10 ^ 11 - it's almost a trillion, or nearly 10 ^ 12), and it is immediately clear that the author attaches great importance to found the most important parameters for large segments: a giant natural numbers (whose record includes 61 figures!, because it is the number N of the order of 10 ^ 61) will be almost as much as a trillion factors. If every subgroup in the form of one millimeter, then the divisors trillion will stretch on one ... a million miles!
Reflektsiya 1.4. [At the end of the Great segment (in our modern age) Tmax of the world about 8 times greater than the Tmax of the worlds rare (another "magic" number 7). Yet we find very important in the virtual relationship cosmology: Tmax / Ts = 5 ∙ 10 ^ 9, where Ts = 140,39 - arithmetic mean type of all integers in large stretches. Moreover, note that in the real cosmology one of the main parameters of our universe is the ratio of the number of photons to the number of baryons: 10 ^ 9 to 3,33 ∙ 10 ^ 9.
I have found numerous "reflection" i-trillion (Tmax = 7 ∙ 10 ^ 11) in the real physical world - are collected in the last (so far, relatively speaking - in the 100-meter) section of my book in the chapter "A Trillion Isayev"]
Range of the worlds - a picture of the distribution of all worlds (T), Total (K) of positive integers, they contained (in this particular segment). Thus, the spectrum of the worlds my desktop segment [1; 520000] can be represented in graphical form (Fig. 1.3) or in Table 1.1. Fig. 1.3 clearly shows that with increasing of T the maximum possible number (Kmax) numbers in these worlds are exponentially decreasing. A Table. 1.1 clearly shows that seven ("magic" number 7) the most "populated" part (rare) worlds contain 85.44% (88.77%) of all natural numbers of the working segment (incidentally, the "cap" in Table 1.1. In terms of rare Worlds contains a bug: the "cap" are confused in some places the letters T and N).
Worlds-phantoms
It is easy to see that, say, on the interval [1, 360] do not appear worlds with the numbers (of type T) 11, 13, 17, 19, 21, 22, 23 - those worlds we call worlds-phantoms. Since it is clear that increasing the right margin of the segment (N> 360) worlds Phantom sooner or later there will be (each - in their "time"), ceasing to be a phantom (phantoms will be other worlds). Because the numbers of all worlds (types T for all natural numbers N) also form an infinite sequence of natural numbers (starting from one).
On an arbitrary interval [1; N] number of world-phantoms Kf (as in frequent, and in rare worlds) rushes to the maximum possible value of type Tmax (in their respective worlds). In this case, of course, will always satisfy the condition: CF <Tmax.
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