1,6180339887 4989484820 4586834365 6381177203 0917980576 2862135448 6227052604 6281890244 9707207204 1893911374 8475408807 5386891752 1266338622 2353693179 3180060766 7263544333 8908659593 9582905638 3226613199 2829026788 0675208766 8925017116 9620703222 1043216269 5486262963 1361443814 9758701220 3408058879 5445474924 6185695364 8644492410 4432077134 4947049565 8467885098 7433944221 2544877066 4780915884 6074998871 2400765217 0575179788 3416625624 9407589069 7040002812 1042762177 1117778053 1531714101 1704666599 1466979873 1761356006 7087480710 1317952368 9427521948 4353056783 0022878569 9782977834 7845878228 9110976250 0302696156 1700250464 3382437764 8610283831 2683303724 2926752631 1653392473 1671112115 8818638513 3162038400 5222165791 2866752946 5490681131 7159934323 5973494985 0904094762 1322298101 7261070596 1164562990 9816290555 2085247903 5240602017 2799747175 3427775927 7862561943 2082750513 1218156285 5122248093 9471234145 1702237358 0577278616 0086883829 5230459264 7878017889 9219902707 7690389532 1968198615 1437803149 9741106926 0886742962 2675756052 3172777520 3536139362

Fibonacci numbers and golden section They constitute the basis of the surrounding world, building its shape and the optimal visual perception by a person with the help of which it can feel beauty and harmony.

The principle of determining the size of the golden section underlies the perfection of the whole world and its parts in its structure and functions, its manifestation can be seen in nature, art and technique. The teaching of the gold proportion was laid as a result of research by ancient scientists of the nature of numbers.

The evidence of the use of the ancient thinkers of the golden proportion is given in the book of Evklida "Beginning", written in 3rd. BC, who applied this rule to build the right 5-kalons. In Pythagoreans, this figure is considered sacred, since it is simultaneously symmetric and asymmetric. Pentagram symbolized life and health.

Fibonacci numbers

The famous book Liber Abaci Mathematics from Italy Leonardo Pisansky, who later became known as Fibonacci, saw the light in 1202. In it, the scientist first leads the pattern of numbers, in a number of which each number is the sum of 2 previous numbers. The sequence of Fibonacci numbers is as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc.

Also, the scientist led a number of patterns:

Any number from a series, divided by the subsequent, will be equal to a value that seeks to 0.618. Moreover, the first numbers of Fibonacci do not give such a number, but as it turns out from the beginning of the sequence, this ratio will be increasingly accurate.

If you divide the number from a number to the previous one, the result will rush to 1.618.

One number divided by the next one will show the value seeking to 0.382.

The use of communication and patterns of the golden section, the number of Fibonacci (0.618) can be found not only in mathematics, but also in nature, in history, in architecture and construction and in many other sciences.

For practical purposes, limited to an approximate value φ \u003d 1.618 or φ \u003d 1.62. In the percentage rounded value, the golden cross section is dividing any value in relation to 62% and 38%.

Historically, the division of the segment of the segment with into two parts (a smaller segment of the AU and a larger segment of the Sun) was called historically in a golden cross section (a smaller segment of the speaker and a larger segment) so that for the lengths of the segments it was right AC / BC \u003d BC / AV. Speaking with simple words, the golden section of the segment is dissected into two unequal parts so that a smaller part refers to a greater, as large to the whole segment. Later, this concept was distributed to arbitrary values.

The number φ is also called Golden number.

The golden cross section has many wonderful properties, but, in addition, many fictional properties are attributed to him.

Now details:

The definition of the CP is the division of the segment into two parts in such a relation, in which most relates to the smaller, as their sum (the entire segment) to the greater.

That is, if we take the entire segment C for 1, then the segment A will be 0.618, the segment B is 0.382. Thus, if you take the structure, for example, a temple built on the principle of the CP, then when it is height, we say 10 meters, the height of the drum with the dome will be equal to 3.82 cm, and the height of the structure of the structure will be 6, 18 cm. (It is clear that the numbers taken smooth for clarity)

And what about the connection between the zs and the numbers of Fibonacci?

Fibonacci sequence numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597…

The pattern of numbers is that each subsequent number is equal to the sum of the two previous numbers.
0 + 1 = 1;
1 + 1 = 2;
2 + 3 = 5;
3 + 5 = 8;
5 + 8 = 13;
8 + 13 \u003d 21, etc.,

and the relationship of adjacent numbers is approaching the ratio of the ZS.
So, 21: 34 \u003d 0.617, and 34: 55 \u003d 0.618.

That is, the basis of the CC is the number of Fibonacci sequences.

It is believed that the term "golden section" introduced Leonardo da Vinci, who said, "Let no one, without a mathematician, will not bother reading my work" and showed the proportions of the human body on his famous picture "Vitruvian man." "If we are a human figure - the most perfect creation of the universe - the belt to the belt and one then, then the distance from the belt to the feet, then this value will refer to the distance from the same belt to the macushkin, as the entire human growth to the length of the belt to the feet."

A number of Fibonacci numbers are clearly simulated (materialized) in the form of a helix.

And in nature Spiral ZS looks like this:

At the same time, the spiral is observed everywhere (in nature and not only):

Seeds in most plants are spiral
- Spider weave the web on the spiral
- Hurricane spiral twists
- A frightened flock of reindeer is running around the spiral.
- DNK molecule is twisted with a double helix. The DNA molecule is two vertically intertwined spirals 34 animals and a width of 21 angstroms. Numbers 21 and 34 follow each other in Fibonacci sequence.
- The embryo develops in the form of a spiral
- Spiral "Snails in the inner ear"
- Water goes into the drained spiral
- Spiral dynamics shows the development of the personality of man and its values \u200b\u200bon the helix.
- And of course, the galaxy itself has the form of a spiral

In this way, it can be argued that the nature itself is built on the principle of the golden section, because this proportion is harmoniously perceived by the human eye. It does not require "corrections" or additions to the resulting picture of the world.

Film. The number of God. Irrefutable proof of God; The Number of God. The Incontrovertible Proof of God.

Gold proportions in the structure of the DNA molecule

All information about the physiological features of living beings is stored in the microscopic DNA molecule, the structure of which also contains the law of the golden proportion. The DNA molecule consists of two vertically twisted spirals. The length of each of these spirals is 34 angstroms, width 21 angstrom. (1 angstrom - one velomillion share of centimeter).

21 and 34 are numbers, following each other in the sequence of Fibonacci numbers, that is, the ratio of the length and width of the logarithmic spiral of the DNA molecule carries the formula of the Golden section 1: 1,618

Golden section in the structure of micromirov

Geometric shapes are not limited to a triangle, square, five or hexagon. If you connect these figures in a different way among themselves, we will get new three-dimensional geometric shapes. Examples of this are such figures as a cube or pyramid. However, in addition to them, there are also other three-dimensional figures with which we did not have to meet in everyday life, and whose names we hear may be for the first time. Among such three-dimensional figures, a tetrahedron can be called (the right four-sided figure), octahedron, dodecahedron, Ikosahedron, etc. Dodecahedron consists of 13 pentagons, Ikosahedron from 20-triangles. Mathematics note that these figures are mathematically very easily transformed, and their transformation occurs in accordance with the formula of the logarithmic spiral of the golden section.

In the microworld, three-dimensional logarithmic forms built on gold proportions are common everywhere. For example, many viruses have a three-dimensional geometric shape of the Ikosahedron. Perhaps the most famous of these viruses is the ADENO virus. The protein sheath of the adeno virus is formed from 252 units of protein cells located in a specific sequence. In each corner of the Ikosahedron, 12 units of protein cells are located in the form of a pentagonal prism and from these angles are shi-like structures.

For the first time, the golden cross section in the structure of viruses was found in the 1950s. Scientists from London Birkbek College A. Klug and D.Kaspar. 13 The first logarithmic form revealed the Polyo virus. The form of this virus turned out to be similar to the form of the Rhino 14 virus.

The question arises how the viruses form so complex three-dimensional forms, the device of which contains a golden cross section, which even our human mind construct quite difficult? The discoverer of these forms of viruses, Virologist A. Klug gives such a comment:

"Dr. Kaspar and I have shown that for the spherical shell of the virus, the most optimal form is the symmetry of the type of the shape of the ikoshedron. Such an order minimizes the number of binding elements ... Most of the geodesic hemispherical cubes of the wagers of fuller are built on a similar geometric principle. 14 Installation of such cubes requires an extremely accurate and detailed explanation scheme. Whereas the unconscious viruses themselves build a complex shell of elastic, flexible protein cellular units. "

Leonardo Fibonacci is one of the greatest mathematicians of the Middle Ages. In one and their works, the "Computing Book" Fibonacci described an Indo-Arabic calculation system and the advantages of its use before Roman.

Definition

Fibonacci numbers or fibonacci sequence is a numerical sequence with a number of properties. For example, the sum of two adjacent sequence numbers gives the value of following them (for example, 1 + 1 \u003d 2; 2 + 3 \u003d 5, etc.), which confirms the existence of so-called Fibonacci coefficients, i.e. permanent relations.

Fibonacci sequence begins as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ...

Fibonacci sequence properties

1. The ratio of each number to the subsequent is more and more striving to 0.618 to increase the sequence number. The relation of each number to the previous one seeks 1.618 (reverse to 0.618). The number 0.618 is called (FI).

2. When dividing each number to the following, after one, the number 0.382 is obtained; On the contrary - respectively 2.618.

3. Selecting the relationship in this way, we obtain the main set of Fibonachchic coefficients: ... 4.235, 2.618, 1.618, 0.618, 0.382, 0.236.

Communication of the Fibonacci sequence and "golden section"

The sequence of fibonaccm asymptotically (everything is slower and slower) is struting to some permanent ratio. However, this ratio is both, that is, it becomes a number with an infinite, unpredictable sequence of decimal digit in the diotype. Its impossible to express exactly.

If any member of the Fibonacci sequence is to subjection on the one with it (apply, 13: 8), the result will be the value that fluctuates around the and -ational value of 1.61803398875 ... and the passage of it is that there is no one that does not reach it. But even eclipping on this eternity, it is impossible to know the amount of accurately, until the last decimal digit. Padi's kpatness, we will try it in the form of 1.618. Special names of this relation began to be given even before Luka Pacioli (Papereko Mathematics) called him the Divine Topping. The conversion names of it are such as a golden cross section, gold down and the switching of vending quads. Keeplep called this ratio by one of the "juice of geometry". In the algebree, its designation of the Gpeech letter fi

Imagine a golden section on the example of a segment.

Consider a segment with the ends of A and B. Let the point C divides the segment AB so,

AC / CB \u003d CB / AB or

You can submit this as follows: a ----- C -------- b

The gold cross section is such a proportional division of the segment to unequal parts, in which the entire segment belongs to the most part, as most of the most relate to the smaller; Or in other words, a smaller cut is so relate to more as greater than everything.

The segments of the gold proportion are expressed by an endless irrational fraction of 0.618 ... if AB is taken per unit, AC \u003d 0.382 .. Kak We already know the number 0.618 and 0.382 are the coefficients of Fibonacci sequence.

The proportions of Fibonacci and the Golden section in nature and history

It is important to note that Fibonacci seemed to reminded his sequence to humanity. She was also known to the ancient Greeks and Egyptians. And indeed, since then in nature, architecture, visual art, mathematics, physics, astronomy, biology and many other areas, the patterns described by Fibonacci coefficients were found. It is just surprising how many permanent can be calculated using Fibonacci sequence, and how its members appear in a huge amount of combinations. However, it will not be an exaggeration to say that this is not just a game with numbers, and the most important mathematical expression of natural phenomena from everyone ever open.

The following examples show some interesting applications of this mathematical sequence.

1. The Pakin is spinled on the helix. If it is deployed, it turns out the length, a little inferior to the length of the snake. A small decade-inthimeter shell has a 35 cm in a length of 35 cm. The shape of a spiral-curled shell attracted the attention of Archimedes. The fact is that the relation of measuring the curls of the shell is constantly equal to 1.618. Archimeda studied the spiral of the shell and removed the spiral equation. Pillar, drawn along this equation, is called his name. The increase in its step is always evenly. Currently, the Archimeph Spiral is widely used in the technique.

2. Plants and animals . Gethete also emphasized the trend of nature to spiral. The screw and spiral arrangement of the leaves on the branches of the trees was noticed for a long time. Pillar saw in the location of sunflower seeds, in pine cones, pineapples, cactus, etc. The seeing work of botany and mathematicians shed light on these amazing phenomena of nature. It turned out that in the location of the leaves on the branch of the seeds of sunflower, the pine cones show itself a number of fibonacci, and therefore, the law of the golden section is manifested. Spider rods spiral spiral. A hurricane is twisted. A frightened flock of reindeer is running around the spiral. DNK molecule is twisted with a double helix. Goethe called the spiral of the "curve of life."

Caring roadside herbs is growing no noticeable plant - chicory. I look at it carefully. From the main stem, the process was formed. Immediately the first sheet is located. The process makes a strong release into space, stops, produces a sheet, but already shorter than the first, again makes a release into space, but already less power, releases a leaflet of even smaller size and emissions again. If the first emission is taken for 100 units, the second is 62 units, the third - 38, the fourth - 24, etc. The length of the petals is also subordinate to the golden proportion. In growth, the conquest of space, the plant retained certain proportions. The impulses of its growth gradually decreased in the proportion of the Golden Section.

Lizard whip. In a lizard at first glance, pleasant for our eye proportion - the length of her tail is as follows to the length of the rest of the body, like 62 to 38.

Both in the plant, and in the animal world persistently breaks through the formative tendency of nature - symmetry relative to the direction of growth and movement. Here, the golden cross section is manifested in the proportions of parts perpendicular to the direction of growth. Nature made division into symmetric parts and gold proportions. In parts manifests the repetition of the structure of the whole.

Pierre Kuri at the beginning of our century formulated a number of deep ideas of symmetry. He argued that it was impossible to consider the symmetry of any body without taking into account the symmetry of the environment. The patterns of gold symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and space systems, in the gene structures of living organisms. These patterns, as indicated above, are in the structure of individual human and body bodies as a whole, and also manifest themselves in biorhythms and the functioning of the brain and visual perception.

3. Cosmos. From the history of astronomy, it is known that I. Titius, a German astronomer of the XVIII century, with the help of this series (Fibonacci) found regularity and order within the distance between the planets of the solar system

However, one case, which, it would seem to contradict the law: There was no planet between Mars and Jupiter. Conducted observation of this section of the sky led to the opening of the belt of asteroids. It happened after the death of Tizius at the beginning of the XIX century.

Pyad Fibonacci is widely used: it is helpful to architectonics and living beings, and man-made structures, and the structure of galaxies. These facts are evidence of the independence of the numerical series from the conditions of its manifestation, which is one of the signs of its versatility.

4. Pyramids. Many tried to solve the secrets of the pyramid in Giza. In contrast to other Egyptian pyramids, this is not a tomb, but as an unresolved puzzle from numerical combinations. Wonderful inventivity, skill, time and labor of the pyramids, used by these eternal symbol, indicate the extreme importance of the message they wanted to convey to future generations. Their era was complementary, dupieroglyphic and symbols were the only means of recording discoveries. The prior to the geometro-mathematical secret of the pyramid in Giza, so long for humanity for humanity, the temple priests were transferred to Herodotus, who told him that the pyramid was constructed so that the area of \u200b\u200beach of her faces was equal to the square of her height.

Square Tinger

356 x 440/2 \u003d 78320

Square Kvadpat

280 x 280 \u003d 78400

The length of the pyramid base ribs in Giza is 783.3 feet (238.7 m), the height of the pyramid -484.4 feet (147.6 m). The length of the base ribs, divided into height, leads to the ratio F \u003d 1.618. The height 484.4 feet corresponds to 5813 inches (5-8-13) - these are numbers from Fibonacci sequence. These interesting observations suggest that the design of the pyramid is based on the proportion of F \u003d 1.618. Some modern scientists are inclined to interpret that the ancient Egyptians built it with the sole purpose - to convey the knowledge they wanted to maintain for the upcoming generations. Intensive studies of the pyramid in Giza showed how extensive were in those times of knowledge in mathematics and astrology. In all internal and external proportions of the pyramid, the number 1.618 plays a central role.

Pyramids in Mexico. He is only Egyptian pinamides are postponed in accordance with the counseling of the golden section, the same phenomenon is also unpaved in Mexican pipamides. There is a thought that both Egyptian and Mexican pipamids were erected in one of the people with common origin.

On the Fibonacci sequence of the Orden of the Illuminati.

This is essentially stored in the once secret records of the Illuminati society, founded in 1776 by Professor Adam Weisgaupt, the sequence of Fibonacci numbers recorded in a row:
58683436563811772030917
98057628621354486227052
60462818902449707207204
18939113748475408807538
68917521266338622235369
31793180060766726354433
38908659593958290563832
26613199282902678806752
08766892501711696207032
22104321626954862629631
36144381497587012203408
05887954454749246185695
36486444924104432077134
49470495658467885098743
39442212544877066478091
58846074998871240076521
70575179788341662562494
07589069704000281210427
62177111777805315317141
01170466659914669798731
76135600670874807101317
95236894275219484353056
78300228785699782977834
78458782289110976250030
26961561700250464338243
77648610283831268330372
42926752631165339247316
71112115881863851331620
38400522216579128667529
46549068113171599343235
97349498509040947621322
29810172610705961164562
99098162905552085247903
52406020172799747175342
77759277862561943208275
05131218156285512224809
39471234145170223735805
77278616008688382952304
59264787801788992199027
07769038953219681986151
43780314997411069260886
74296226757560523172777
52035361393621076738937
64556060605921658946675
95519004005559089502295
30942312482355212212415
44400647034056573479766
39723949499465845788730
39623090375033993856210
24236902513868041457799
56981224457471780341731
26453220416397232134044
44948730231541767689375
21030687378803441700939
54409627955898678723209
51242689355730970450959
56844017555198819218020
64052905518934947592600
73485228210108819464454
42223188913192946896220
02301443770269923007803
08526118075451928877050
21096842493627135925187
60777884665836150238913
49333312231053392321362
43192637289106705033992
82265263556209029798642
47275977256550861548754
35748264718141451270006
02389016207773224499435
30889990950168032811219
43204819643876758633147
98571911397815397807476
15077221175082694586393
20456520989698555678141
06968372884058746103378
10544439094368358358138
11311689938555769754841
49144534150912954070050
19477548616307542264172
93946803673198058618339
18328599130396072014455
95044977921207612478564
59161608370594987860069
70189409886400764436170
93341727091914336501371
57660114803814306262380
51432117348151005590134
56101180079050638142152
70930858809287570345050
78081454588199063361298
27981411745339273120809
28972792221329806429468
78242748740174505540677
87570832373109759151177
62978443284747908176518
09778726841611763250386
12112914368343767023503
71116330725869883258710
33632223810980901211019
89917684149175123313401
52733843837234500934786
04979294599158220125810
45982309255287212413704
36149102054718554961180
87642657651106054588147
56044317847985845397312
86301625448761148520217
06440411166076695059775
78325703951108782308271
06478939021115691039276
83845386333321565829659
77310343603232254574363
72041244064088826737584
33953679593123221343732
09957498894699565647360
07295999839128810319742
63125179714143201231127
95518947781726914158911
77991956481255800184550
65632952859859100090862
18029775637892599916499
46428193022293552346674
75932695165421402109136
30181947227078901220872
87361707348649998156255
47281137347987165695274
89008144384053274837813
78246691744422963491470
81570073525457070897726
75469343822619546861533
12095335792380146092735
10210119190218360675097
30895752895774681422954
33943854931553396303807
29169175846101460995055
06480367930414723657203
98600735507609023173125
01613204843583648177048
48181099160244252327167
21901893345963786087875
28701739359303013359011
23710239171265904702634
94028307668767436386513
27106280323174069317334
48234356453185058135310
85497333507599667787124
49058363675413289086240
63245639535721252426117
02780286560432349428373
01725574405837278267996
03173936401328762770124
36798311446436947670531
27249241047167001382478
31286565064934341803900
41017805339505877245866
55755229391582397084177
29833728231152569260929
95942240000560626678674
35792397245408481765197
34362652689448885527202
74778747335983536727761
40759171205132693448375
29916499809360246178442
67572776790019191907038
05220461232482391326104
32719168451230602362789
35454324617699757536890
41763650254785138246314
65833638337602357789926
72988632161858395903639
98183845827644912459809
37043055559613797343261
34830494949686810895356
96348281781288625364608
42033946538194419457142
66682371839491832370908
57485026656803989744066
21053603064002608171126
65995419936873160945722
88810920778822772036366
84481532561728411769097
92666655223846883113718
52991921631905201568631
22282071559987646842355
20592853717578076560503
67731309751912239738872
24682580571597445740484
29878073522159842667662
57807706201943040054255
01583125030175340941171
91019298903844725033298
80245014367968441694795
95453045910313811621870
45679978663661746059570
00344597011352518134600
65655352034788811741499
41274826415213556776394
03907103870881823380680
33500380468001748082205
91096844202644640218770
53401003180288166441530
91393948156403192822785
48241451050318882518997
00748622879421558957428
20216657062188090578088
05032467699129728721038
70736974064356674589202
58656573978560859566534
10703599783204463363464
85489497663885351045527
29824229069984885369682
80464597457626514343590
50938321243743333870516
65714900590710567024887
98580437181512610044038
14880407252440616429022
47822715272411208506578
88387124936351068063651
66743222327767755797399
27037623191470473239551
20607055039920884426037
08790843334261838413597
07816482955371432196118
95037977146300075559753
79570355227144931913217
25564401283091805045008
99218705121186069335731
53895935079030073672702
33141653204234015537414
42687154055116479611433
23024854404094069114561
39873026039518281680344
82525432673857590056043
20245372719291248645813
33441698529939135747869
89579864394980230471169
67157362283912018127312
91658995275991922031837
23568272793856373312654
79985912463275030060592
56745497943508811929505
68549325935531872914180
11364121874707526281068
69830135760524719445593
21955359610452830314883
91176930119658583431442
48948985655842508341094
29502771975833522442912
57364938075417113739243
76014350682987849327129
97512286881960498357751
58771780410697131966753
47719479226365190163397
71284739079336111191408
99830560336106098717178
30554354035608952929081
84641437139294378135604
82038947912574507707557
51030024207266290018090
42293424942590606661413
32287226980690145994511
99547801639915141261252
57282806643312616574693
88195106442167387180001
10042184830258091654338
37492364118388856468514
31500637319042951481469
42431460895254707203740
55669130692209908048194
52975110650464281054177
55259095187131888359147
65996041317960209415308
58553323877253803272763
29773721431279682167162
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In the records of the members of this secret society, this set of figures occupies a very important role. But what? What hid the illuminations behind these numbers?

The fact is that according to the preserved data, the Illuminati had extensive knowledge not only in the area of \u200b\u200boccult sciences, but also mathematics, astronomy, astrology, chemistry and alchemy, medicine and psychology. Also, they were also available some ancient sources of knowledge.

Many researchers believe that the universal code of life may hide behind these figures, the recipe for the philosovsky stone, etc. ...

This harmony is striking with its scale ...

Hello, friends!

Have you heard anything about Divine Harmony or Golden section? Did you think about why something seems to us perfect and beautiful, and something repels?

If not, you successfully hit this article, because in it we will discuss the golden cross section, find out what it is, as it looks in nature and in man. Let's talk about his principles, find out what a number of fibonacci and much more, including the concept of a golden rectangle and a gold spiral.

Yes, in the article many images, formulas, as-in no way, the golden cross section is also mathematics. But everything is described quite simple, clearly. And also at the end of the article, you will know why everyone loves the cats \u003d)

What is a golden section?

If it is simple, then the golden cross section is a certain rule of proportion, which creates harmony?. That is, if we do not violate the rules of these proportions, then we obtain a very harmonious composition.

The most capacious definition of the golden section states that a smaller part relates to a greater, as large for all.

But, besides this, the golden cross section is mathematics: it has a specific formula and a specific number. Many mathematics, in general, consider it the formula of divine harmony, and is called "asymmetric symmetry".

Until our contemporaries, the golden section came from the times of ancient Greece, however, there is an opinion that the Greeks themselves have already spied the golden cross section among the Egyptians. Because many works of art of ancient Egypt are clearly built on the canons of this proportion.

It is believed that the concept of the golden section of Pythagores introduced the first. The Euclides have come to this day (with the help of a golden section built the right pentagons, which is why such a pentagon is called "gold"), and the number of golden section is named after the ancient Greek architect Fidia. That is, this is our number "FI" (denoted by the Greek letter φ), and equally it 1.6180339887498948482 ... Naturally, this value is rounded: φ \u003d 1,618 or φ \u003d 1.62, and in the percentage ratio, the gold cross section looks like 62% and 38%.

What is the uniqueness of this proportion (and if she, believe me)? Let's first try to figure out the example of the segment. So, we take a segment and divide it on unequal parts in such a way that his smaller part belongs to more, as big to everything. I understand not very clear what I will try to illustrate clearly on the example of segments:

So, we take a segment and divide it into two others, so that a smaller cut A, relate to a larger segment B, as well as the segment B refers to a whole, that is, the whole line (A + B). Mathematically it looks like this:

This rule works infinitely, you can divide the seglings for how long. And see how simple it is. The main thing is to understand it once.

But now consider a more complex example that comes across very often, since the golden cross section is still represented in the form of a golden rectangle (the aspect ratio of which is φ \u003d 1.62). This is a very interesting rectangle: if you "cut off" from it, then we will get a golden rectangle again. And so infinite many times. See:

But mathematics would not be mathematics if there was no formulas in it. So, friends, now will be a little "hurt". The golden proportion solution hidden under the spoiler, a lot of formulas, but without them I do not want to leave an article.

Fibonacci row and golden cross section

We continue to create and observe the magic of mathematics and the golden section. In the Middle Ages there was such a friend - Fibonacci (or fibonaci, everywhere write differently). He loved mathematics and tasks, he had an interesting task with the reproduction of rabbits \u003d) But not the essence. He opened a numeric sequence, the number in it is also called "Fibonacci Numbers".

The sequence itself looks like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ... and further indefinitely.

If in words, the Fibonacci sequence is such a sequence of numbers, where each subsequent number is equal to the sum of the previous two.

Where is the golden section? Now you will see.

Spiral Fibonacci

To see and feel the entire relationship between the numerical row of Fibonacci and the Golden section, you need to look at the formula again.

In other words, from the 9th member of the Fibonacci sequence, we begin to get the values \u200b\u200bof the golden section. And if we visualize the whole picture, then we will see how Fibonacci sequence creates rectangles getting closer and closer to the golden rectangle. Here is such a connection.

Now let's talk about Spiral Fibonacci, it is also called the "golden spiral".

The gold spiral is a logarithmic spiral, the growth coefficient of which is φ4, where φ is a golden cross section.

In general, from the point of view of mathematics, the golden cross section is the perfect proportion. But on this, its miracles are just beginning. The principles of the golden section are subject to almost the whole world, nature created this proportion. Even esoterica, and those see numerical power in it. But it is definitely not about this article to talk about it, so you can not miss anything, you can subscribe to the site updates.

Golden section in nature, man, art

Before we start, I would like to clarify a number of inaccuracies. First, the determination of the golden section in this context is not entirely true. The fact is that the very concept of "cross section" is the term geometric, denoting always plane, but not the sequence of Fibonacci numbers.

And, secondly, a numerical row and the ratio of one to another, of course, turned into a stencil, which can be imposed on everything that seems suspicious, and very rejoice when there are coincidences, but still, common sense should not lose.

However, "everything was mixed in our kingdom" and one became synonymous with another. So in general, the meaning of this was not lost. And now to the case.

You will be surprised, but the golden cross section, or rather the proportions as close as possible to it, you can see almost everywhere, even in the mirror. Do not believe? Let's start and start.

You know when I learned to draw, we were explained how easier to build a person's face, his body, and so on. Everyone must count on something else.

Everything, absolutely everything is proportional to: bones, our fingers, palm, distances on the face, the distance of the elongated hands towards the body and so on. But even this is not all, the inner structure of our organism, even it is equal or almost equal to the golden section. Here are the distances and proportions:

from shoulders to the top to the size of the head \u003d 1: 1.618

from the navel to the top to the segment from the shoulders to the top \u003d 1: 1.618

from the navel to the knees and from the knee to the feet \u003d 1: 1.618

from the chin to the extreme point of the upper lip and from it to the nose \u003d 1: 1.618

Isn't it amazing!? Harmony in pure form, both inside and outside. And that is why, on some subconscious, any level, some people do not seem beautiful to us, even if they have a strong taut body, velvet skin, beautiful hair, eyes, and so on and everything else. But, it's all the same, the slightest violations of the proportions of the body, and the appearance is slightly "cuts eyes."

In short, the more beautifully the person seems to us, the closer to its proportion to the perfect. And this, by the way, not only to the human body can be attributed.

Golden section in nature and its phenomena

The classic example of the golden section in nature is the mollusk sink Nautilus Pompilius and ammonite. But this is not all, there are many more examples:

in the curls of the human ear we can see the Golden Spiral;

her (or approximate to it) in spirals, for which galaxies are tightered;

and in the DNA molecule;

for a number of Fibonacci, the sunflower center is arranged, cones, middle colors, pineapple and many other fruits grow.

Friends, examples so much that I will just leave a video clip here (it is slightly lower) so as not to overload the article. Because, if you dig this topic, then you can go deep into such a debris: still the ancient Greeks argued that the universe and, in general, all the space, is planned on the principle of the golden cross section.

You will be surprised, but these rules can be found even in sound. See:

The highest point of sound, causing pain and discomfort in our ears, is 130 decibels.

We divide the proportion of 130 to the number of golden section φ \u003d 1.62 and we get 80 decibels - the sound of the human scream.

We continue to proportionately and we get, let's say, the normal volume of human speech: 80 / φ \u003d 50 decibels.

Well, and the last sound that we obtain thanks to the formula is a pleasant sound of whisper \u003d 2.618.

According to this principle, it is possible to determine the optimal-comfortable, minimum and maximum number of temperature, pressure, humidity. I did not check, and I do not know how much this theory is true, but you will agree, it sounds impressive.

Absolutely in all live and not alive, you can read the highest beauty and harmony.

The main thing is not to get involved in this, because if we want to see something in something, we will see, even if this is not there. So I, for example, drew attention to the design of the PS4 and saw a gold cross section there \u003d) however, this console is so cool that it is not surprised if the designer, and the truth, something is wiser there.

Golden section in art

Also a very large and extensive topic, which is worth considering separately. Here we only have a few base moments. The most remarkable thing is that many works of art and architectural masterpieces of antiquity (and not only) are made according to the principles of the golden cross section.

Egyptian and Mayan Pyramids, Notre Dame De Paris, Greek Parthenon and so on.

In the musical works of Mozart, Chopin, Schubert, Bach and others.

In painting (there it is clearly seen): All the most famous paintings of famous artists are made taking into account the rules of the golden section.

These principles can be found in the verses of Pushkin, and in the bust of the beauties of Nefertiti.

Even now, the rules of the golden proportion are used, for example, in the photo. Well, and of course, in all other arts, including cinema and design.

Golden cats Fibonacci

And finally, about the quotes! Did you think about why all the cats love so much? They also flooded the Internet! Cats everywhere and it is wonderful \u003d)

But the thing is that cats are perfect! Do not believe? Now I will prove it mathematically!

See? The secret is revealed! Cats are ideal from the point of view of mathematics, nature and universe \u003d)

* I'm kidding, of course. No, cats, really, ideal) But nobody measured them mathematically, probably.

On this, in general, all, friends! We will see in the following articles. Good luck to you!

P. S. Images taken from Medium.com.

Fibonacci sequence in mathematics and in nature

Fibonacci sequence, known to all by the film "Da Da Vinci" film - a number of numbers described in the form of a riddle of the Italian mathematician Leonardo Pisansky, more famous for the nickname Fibonacci, in the XIII century. In short, the essence of the riddles:

Someone placed a couple of rabbits in a certain closed space to find out how many pairs of rabbits will be born during the year, if the nature of rabbits is such that each month the pair of rabbits is on the other pair, and the ability to produce offspring them appear to achieve Two-month age.

As a result, it turns out such a sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 Where, through the comma, the number of rabbit pairs in each of the twelve months is shown.

This sequence can be continued indefinitely. Its essence is that each next number is the sum of the two previous ones.

This sequence has a number of mathematical features that need to be touched. This sequence is asymptotically (approaching everything slower and slower) tends to some permanent ratio. However, this ratio is irrationally, that is, is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. Its impossible to express exactly.

So the ratio of any member of the sequence to the previous one hesitates near the number 1,618 Through the PAs is superior, without reaching it. Attitude to the next similarly approaching 0,618 what is inversely proportional 1,618 . If we share the elements of the sequence through one, then we get the number 2,618 and 0,382 which are also inversely proportional. These are the so-called Fibonacci coefficients.

What is all this? So we are approaching one of the most mysterious phenomena of nature. Fibonacci essentially did not open anything new, he just reminded the world about such a phenomenon as Golden cross sectionwhich is not inferior to the importance of Pythagora theorem

We distinguish all the items around us in the form. Some of us like more, some less, some pushing the look at all. Sometimes interest can be dictated by the life situation, and sometimes the beauty of the observed object. Symmetric and proportional form, contributes to the best visual perception and causes a feeling of beauty and harmony. The holistic image always consists of parts of different sizes that are in a certain relationship with each other and integer.

Golden cross section - Higher manifestation of the perfection of the whole and its parts in science, art and nature.

If on a simple example, then the gold cross section is the division of a segment into two parts in such a ratio, in which most of which relates to a smaller, as their sum (the entire segment) to the greater.

If we take the whole segment c. per 1 , then cut a. will be equal 0,618 , section b. - 0,382 , only so the Gold Condition Conduct (0.618 / 0.382 \u003d 1,618 ; 1/0,618=1,618 ). Attitude c. to a. equally 1,618 , but from to b2,618.. These are all the same, already familiar to us, Fibonacci coefficients.

Of course there is a golden rectangle, a golden triangle and even a gold cuboid. The proportions of the human body in many ratios are close to the golden cross section.

Picture: marcus-frings.de.

But the most interesting begins when we combine the knowledge gained. The figure shows the link between the Fibonacci sequence and the golden section. We start with two squares of the first size. Top add the second size square. Pain next square with a party equal to the sum of the parties of the two previous, third size. By analogy, the square of the fifth size appears. And so on until you get bored, the main thing is that the length of the side of each next square is equal to the sum of the lengths of the parties of the previous two. We see a series of rectangles, the lengths of the parties, which are Fibonacci numbers, and, as not strange, they are called Fibonacci rectangles.

If we do smooth lines through the angles of our squares, we will get anything other than the Archimedian spiral, the increase in the step of which is always evenly.

Nothing reminds?

Photo: ethanhein On Flickr.

And not only in the shell of the mollusk you can find Spirals of Archimedes, and in many colors and plants, they are simply not such obvious.

Scarlet multi-sample:

Photo: brewbooks. On Flickr.

Photo: beart.org.uk.

Photo: esdrascalderan. On Flickr.

Photo: mandj98. On Flickr.

And here is the time to remember the golden section! Neither one of the most beautiful and harmonious creations of nature are depicted in these photos? And this is not all. Looking around, you can find similar patterns in many forms.

Of course, the statement that all these phenomenons are built on Fibonacci sequences sounds too loud, but the trend on the face. And besides, the sequence itself is far from perfect, as well as everything in this world.

There is an assumption that the Fibonacci sequence is an attempt to adapt to a more fundamental and perfect gold logarithmic sequence, which is almost the same, just starts from nowhere and goes nowhere. The nature must necessarily need some kind of principle, from which you can push off, it cannot create something from nothing. The relations of the first members of Fibonacci sequence are far from the golden section. But the farther we are moving on it, the more these deviations are smoothed. To determine any sequence, it is enough to know three of its member, walking on each other. But not for the gold sequence, it is enough for it, it is geometric and arithmetic progress at the same time. You might think it seems to be the basis for all other sequences.

Each member of the golden logarithmic sequence is a degree of gold proportion ( z.). Part of the row looks like this: ... z -5; z -4; z -3; z -2; z -1; z 0; z 1; z 2; z 3; z 4; z 5 ... If we round the value of the golden proportion to three characters, then we get z \u003d 1,618., then the row looks like this: ... 0,090 0,146; 0,236; 0,382; 0,618; 1; 1,618; 2,618; 4,236; 6,854; 11,090 ... Each next member can be obtained not only by multiplying the previous on 1,618 , but also the addition of the two previous ones. Thus, exponential growth in the sequence is ensured by simply addition of two adjacent elements. This is a number without start and end, and it is at him that it is trying to be a similar Fibonacci sequence. Having a well-defined start, she strives for the ideal, never reaching it. That is life.

Nevertheless, in connection with all seen and read, quite natural questions arise:
Where did these numbers come from? Who is this architect of the Universe, who tried to make it perfect? Was it all the way he wanted? And if so, why did it come down? Mutations? Free choice? What will be next? Spiral twists or spinned?

Finding an answer to one question, you will get the next. I solve it, you will get two new ones. You'll figure them out, three more will appear. Deciding and to have them, get five unresolved. Then eight, then thirteen, 21, 34, 55 ...

Sacred geometry. Energy Codes of Harmony Prokopenko Iolaant

The number "FI" \u003d 1,618

The number "FI" \u003d 1,618

To connect two parts with the third perfect manner, a proportion that would brought them into a single integer. At the same time, one part of the whole should be treated like this, as a whole to most.

The number of fi is considered the most beautiful number in the world, the basis of the foundations of the whole living. One of the sacred places of ancient Egypt hides in his title this is the dizes. This number has many titles, it is known to mankind more than 2500 years.

For the first time, the mention of this number is found in the work of the ancient Greek mathematics Euclida "Beginning" (about 300 years BC). There this number is used to build a regular pentagon, which is based on the ideal "Platonic Body" - dodecahedron, the symbol of the perfect universe.

The number of fi - the trazane number and is expressed by an infinite decimal fraction. Leonardo Pisa, Contemporary Leonardo da Vinci, more famous as Fibonacci, called this the number "divine proportion". Later, the "FI" constant was founded "Golden section". The term "golden section" was introduced in 1835 by Martin Ohm.

The proportion of "FI" in the statue of the dorifera spear

Fibonacci row (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc.) in ancient times was considered a unique key to the laws of the universe. You can find a private between two next numbers and approach the "fi", but it is impossible to achieve it.

The permanent constant "FI" was used in constructing a peyramid of Heops, as well as to create bas-reliefs, household goods and jewelry from Tutankhamon's tomb. The proportion of the "golden section" is used everywhere to this day in the works of artists, sculptors, architects and even choreographers and musicians.

The French architect Le Corbusier found the meaning of the "fi" constant in the relief from the temple in Abidos, the relief of Pharaoh Ramses, the facade of Greek Parfenon. Gold proportions are also hidden in the Circle of the Ancient Roman city of Pompei. The proportion of "FI" is also present in the architecture of the human body. (For details, see the section "Golden Section".)

From the book the number of life. Fate code. Read this book if you were born 3rd, 12th, 21st or 30th Author Hardy Titania

From the book the number of life. Fate code. Read this book if you were born on the 4th, 13th, on the 22nd or 31st Author Hardy Titania

Number of day if your birthday is a two-digit number, fold its numbers to make a clear number. Birth rate - 22rd: 2 + 2 \u003d 4.Dime - 13th Number: 1 + 3 \u003d

From the book the number of life. Fate code. Read this book if you were born on the 5th, 14th or 23rd Author Hardy Titania

Number of day if your birthday is a two-digit number, fold its numbers to get an unambiguous number. Examples Birthday - February 14: 1 + 4 \u003d 5.The birth - August 23: 2 + 3 \u003d

From the book Mystery named Author Zgur Maria Pavlovna

Number of name and number of birth (fate) With the help of numbers, you can define a cipher of your name, to correlate it with the number that denotes the code of birth, look into the secret of your character and fate and learn the compatibility of "like a loved one" with the people around you in business, family,

From the book plot of the Siberian healer. Release 09. Author Stepanova Natalia Ivanovna

The number of three numbers three is an amazing, unusually strong number because it marks the Holy Trinity (Father, Son and the Holy Spirit). This is the number of holiness, the number of true faith, strong and unshakable. This is what the troika allocates from all other numbers. How much the effect of the troika on

From the book Yoga and sexual practices author Douglas Nik.

From the book sacred geometry. Energy codes of harmony Author Prokopenko Iolaanta

The number "FI" \u003d 1.618 for connecting two parts with the third perfect manner is necessary proportion that would bore them into a single integer. At the same time, one part of the whole should be treated like this, as a whole to most. Plato The number of fi is considered the most beautiful number in

From the book a numerical code of birth and its influence on fate. How to calculate luck Author Mikheeva Irina Firsovna

The number 12 on the energies of the earth channel number 12 has like a triple (12 \u003d 1 + 2 \u003d 3), yellow color, but this is the third digit of a new reality, its double sign. Troika is a sprout of his variety, a triangle, a sign of immutability and unshakable . Psychological plan is a sign of hardness and

From the book how to call a child to be happy Author Stephanie Sister

The number 13 on the energies of the earth channel 13, like the Four, has a green color - the sound level and information. This is the fourth digit of a new reality, its double sign. The amount of 13 gives in the amount of Figure 4, the fourth point of reality. In natural understanding is a flower waiting for pollination

From the book Eternal Horoscope by the author Kuchin Vladimir

The number 14 on the energies of the earth channel number 14 is manifested in representatives of the new, not yet mastered by our civilization of the first intellectual level of heavenly blue color. The codenal numeric sign 14 people born on the last day of the year come. These people are ne.

From the book of the author

Number 11 on the energies of the space channel Number 11 manifests the energy of two worlds: manifested and uncompressed. And this is the sun, reflected in the water, two Suns: in the sky and in water, two units. This is a game sign, a sign of creativity. Man of this sign - a mirror that

From the book of the author

Number 12 on the energies of the space channel Number 12 personifies the harmony and the end of the space at the new level of reality, which includes three basic concepts of life: the past, present and future. 6 contains a unit - a sign of the leader and a two - the owner sign

From the book of the author

Number 13 on the energies of the space channel Number 13 manifests the wind energy of all four sides of the light, mobility, society at a new level of development. Symmetically energy of the number 13 looks like the same wind rose as in the number 4, but without limiting space.

From the book of the author

Number 14 on the energies of the space channel Number 14 is a messenger of space. Royal number 13 is not the last in the levels of development of our civilization. There is another day in a year, when missionaries come from the very cosmos, these people do not have a clear body code (earthly channel), they do not have

From the book of the author

Step one. Calculate the number of birth, or the number of personality the number of birth reveals the natural characteristics of the person, it, as we have already spoken, remains unchanged for life. If only we are talking about numbers 11 and 22, which can be "simplified" to 2 and 4

From the book of the author

5th. Bor Bor is often lucky at birth, and he inherits some capital, "factories" and "steamboats". Perhaps he does not bother the inheritance, and will give it to his heirs. His personal preferences are undefined - whether he loves harmony and feels, or loves power and