Number 13 At the energies of the Earth channel, the number 13, like a four, has a green color - the level of sound and information. This is the fourth digit of the new reality, its double sign. The number 13 adds up to the number 4, the fourth point of reality. In the Natural understanding, it is a flower awaiting pollination

Number 14 At the energies of the Earth channel, the number 14 manifests itself in the representatives of the new, not yet mastered by our civilization, the first intellectual level of the Sky-blue color. People born on the last day of the year come under the code number 14. These people are not

Number 11 At the energies of the Cosmic Channel, the number 11 personifies the energy of two worlds: the manifest and the unmanifest. Symbolically, this is the Sun reflected in the water, two Suns: in the sky and in the water, two units. It is a sign of play, a sign of creativity. The person of this sign is a mirror that

Number 12 On the energies of the Cosmic Channel, the number 12 personifies the harmony and completeness of space at a new level of reality, which includes three basic concepts of life: past, present and future. Number 12 contains one - the sign of the leader and two - the sign of the owner

Number 13 At the energies of the Cosmic Channel, the number 13 personifies the wind energy of all four cardinal points, mobility, sociability at a new level of development. Symbolically, the energy of number 13 looks like the same Rose of Winds as that of the number 4, but without space limitations.

Number 14 On the energies of the Cosmic Channel, the number 14 is the messenger of the Cosmos. The royal number 13 is not the last in the levels of development of our civilization. There is one more day of the year when missionaries come from the Cosmos itself, these people do not have a clear body code (Earth channel), they do not have

Step one. We calculate the number of birth, or the number of personality The number of birth reveals the natural characteristics of a person, it, as we have already said, remains unchanged for life. Unless we are talking about the numbers 11 and 22, which can "simplify" to 2 and 4

5th number. "Bor" Bor is often lucky at birth, and he inherits some capital, "factories" and "ships". Perhaps he will not squander the inheritance, and pass it on to his heirs. His personal preferences are vague - either he loves harmony and feels, or loves power and

Definition

Fibonacci numbers or Fibonacci Sequence is a numerical sequence that has a number of properties. For example, the sum of two adjacent numbers of the sequence gives the value of the next one (for example, 1 + 1 = 2; 2 + 3 = 5, etc.), which confirms the existence of the so-called Fibonacci ratios, i.e. constant ratios.

The Fibonacci sequence starts like this: 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 next one more and more tends to 0.618 as the ordinal number increases. The ratio of each number to the previous one tends to 1.618 (inverse to 0.618). The number 0.618 is called (PI).

2. When dividing each number by the next one, after one, the number 0.382 is obtained; on the contrary - respectively 2.618.

3. By choosing the ratios in this way, we obtain the basic set of Fibonacci coefficients:… 4.235, 2.618, 1.618, 0.618, 0.382, 0.236.

The connection between the Fibonacci sequence and the "golden ratio"

The Fibonacci sequence asymptotically (approaching more and more slowly) tends to some constant ratio. However, this ratio is rational, that is, it is a number with an infinite, unpredictable sequence of decimal digits in the fractional part. It is impossible to express it precisely.

If any member of the Fibonacci sequence is divided by the one that precedes it (for example, 13: 8), the result will be a value that fluctuates around the irrational value 1.61803398875 ... and, once in a while, then it does not reach it. But even having touched Eternity on it, it is impossible to know the ratio exactly, up to the last decimal digit. For the sake of hardness, we will translate it in the form of 1.618. Special names for this ratio began to be given even before Luca Pacioli (a mid-century mathematician) called it the Divine Proportion. Among its modern names there are such as Golden Ratio, Golden Mean and the ratio of rotating squares. Keplep called this relationship one of the "treasures of geometry". In algebra, its designation is generally accepted by the Greek letter phi

Let's imagine the golden ratio using a line segment as an example.

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

AC / CB = CB / AB or

You can think of it like this: A ----- C -------- B

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment refers to the larger part as much as the larger part itself refers to the smaller one; or in other words, a smaller segment relates to a larger one as a larger one to everything.

The segments of the golden ratio are expressed by the infinite irrational fraction 0.618 ..., if AB is taken as one, AC = 0.382 .. As we already know the numbers 0.618 and 0.382 are the coefficients of the Fibonacci sequence.

Fibonacci and Golden Ratios in nature and history

It is important to note that Fibonacci, as it were, reminded his sequence to humanity. She was known even to the ancient Greeks and Egyptians. Indeed, since then in nature, architecture, fine arts, mathematics, physics, astronomy, biology and many other fields, patterns have been found described by the Fibonacci coefficients. It's amazing how many constants can be calculated using the Fibonacci sequence, and how its members appear in a huge number of combinations. However, it would not be an exaggeration to say that this is not just a game with numbers, but the most important mathematical expression of natural phenomena ever discovered.

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

1. The shell is spirally wound. If you unfold it, you get a length slightly inferior to the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. The shape of the spiral-wound shell attracted the attention of Archimedes. The point is that the ratio of the measurements of the shell curls is constant and equal to 1.618. Archimedes studied the spiral of shells and derived the equation for the spiral. The spiral drawn from this equation is named after him. The increase in her step is always uniform. Currently, the Archimedes spiral is widely used in technology.

2. Plants and animals ... Even Goethe emphasized the tendency of nature to spiral. The helical and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians has shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch of sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore the law of the golden section manifests itself. The spider weaves the web in a spiral manner. A hurricane is spinning in a spiral. A frightened herd of reindeer scatters in a spiral. The DNA molecule is twisted in a double helix. Goethe called the spiral "the curve of life".

Among the roadside grasses, an unremarkable plant grows - chicory. Let's take a closer look at him. A process has formed from the main stem. The first sheet is located right there. The shoot makes a strong ejection into space, stops, releases a leaf, but is shorter than the first, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and ejects again. If the first emission is taken as 100 units, then the second is equal to 62 units, the third is 38, the fourth is 24, etc. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. The impulses of its growth gradually decreased in proportion to the golden section.

The lizard is viviparous. In a lizard, at first glance, proportions pleasant to our eyes are caught - the length of its tail is as much related to the length of the rest of the body as 62 to 38.

In both the plant and the animal world, the formative tendency of nature is persistently breaking through - symmetry with respect to the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth. Nature has carried out the division into symmetrical parts and golden proportions. In the parts, the repetition of the structure of the whole is manifested.

Pierre Curie at the beginning of this century formulated a number of profound ideas of symmetry. He argued that one cannot consider the symmetry of any body without considering the symmetry of the environment. The patterns of golden symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and space systems, in the genetic structures of living organisms. These patterns, as indicated above, are in the structure of individual organs of a person and the body as a whole, and are also manifested in biorhythms and the functioning of the brain and visual perception.

3. Space. It is known from the history of astronomy that I. Titius, a German astronomer of the 18th century, with the help of this series (Fibonacci) found the regularity and order in the distances between the planets of the solar system

However, one case that seemingly contradicted the law: there was no planet between Mars and Jupiter. Concentrated observation of this region of the sky led to the discovery of the asteroid belt. This happened after the death of Titius at the beginning of the 19th century.

The Fibonacci series is widely used: it is used to represent the architectonics of living beings, and man-made structures, and the structure of the Galaxies. These facts are evidence of the independence of the number series from the conditions of its manifestation, which is one of the signs of its universality.

4. Pyramids. Many have tried to unravel the secrets of the pyramid at Giza. Unlike other Egyptian pyramids, this is not a tomb, but rather an insoluble puzzle of number combinations. The remarkable ingenuity, skill, time and labor of the architects of the pyramid, which they used in the construction of the eternal symbol, indicate the extreme importance of the message that they wanted to convey to future generations. Their era was preliterate, pre-hieroglyphic, and symbols were the only means of recording discoveries. The key to the geometrical-mathematical secret of the pyramid in Giza, which had been a mystery to mankind for so long, was actually given to Herodotus by the temple priests, who informed him that the pyramid was built so that the area of ​​each of its faces was equal to the square of its height.

Triangle area

356 x 440/2 = 78320

Square area

280 x 280 = 78400

The length of the edge of the base of the pyramid at Giza is 783.3 feet (238.7 m), the height of the pyramid is 484.4 feet (147.6 m). The length of the base rib divided by the height leads to the ratio Ф = 1.618. A height of 484.4 feet corresponds to 5813 inches (5-8-13) - these are numbers from the Fibonacci sequence. These interesting observations suggest that the design of the pyramid is based on the proportion Φ = 1.618. Some modern scholars are inclined to interpret that the ancient Egyptians built it with the sole purpose of transmitting knowledge that they wanted to preserve for future generations. Intensive studies of the pyramid at Giza have shown how extensive knowledge in mathematics and astrology was at that time. In all internal and external proportions of the pyramid, the number 1.618 plays a central role.

Pyramids in Mexico. Not only are the Egyptian pyramids built in accordance with the perfect proportions of the golden ratio, the same phenomenon is found in the Mexican pyramids. The idea arises that both Egyptian and Mexican pyramids were erected at approximately the same time by people of common descent.

About the Fibonacci sequence of the Order of the Illuminati.

This is, in fact, stored in the once secret records of the Illuminati society, founded in 1776 by Professor Adam Weishaupt, a sequence of Fibonacci numbers written 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
34421183201802881412747
44316884721845939278143
54740999990722332030592
62976611238327983316988
25393126200650370288447
82866694044730794710476
12558658375298623625099
98232335971550723383833
24408152577819336426263
04330265895817080045127
88731159355877472172564
94700051636672577153920
98409503274511215368730
09121996295227659131637
09396860727134269262315
47533043799331658110736
96431421719794340563915
51210810813626268885697
48068060116918941750272
29874158699179145349946
24441940121978586013736
60828690722365147713912
68742096651378756205918
54328888341742920901563
13328319357562208971376
56309785015631549824564
45865424792935722828750
60848145335135218172958
79329911710032476222052
19464510536245051298843
08713444395072442673514
62861799183233645983696
37632722575691597239543
83052086647474238151107
92734948369523964792689
93698324917999502789500
06045966131346336302494
99514808053290179029751
82515875049007435187983
51183603272277260171740
45355716588555782972910
61958193517105548257930
70910057635869901929721
79951687311755631444856
48100220014254540554292
73458837116020994794572
08237804368718944805636
89182580244499631878342
02749101533579107273362
53289069334741238022220
11626277119308544850295
41913200400999865566651
77566409536561978978183
80451030356510131589458
90287186108690589394713
68014845700183664956472
03294334374298946427412
55143590584348409195487
01523614031739139036164
40198455051049121169792
00120199960506994966403
03508636929039410070194
50532016234872763232732
44943963048089055425137
97233147518520709102506
36859816795304818100739
42453170023880475983432
34504142584314063612721
09602282423378228090279
76596077710849391517488
73168777135223900911711
73509186006546200990249
75852779254278165970383
49505801062615533369109
37846597710529750223173
07412177834418941184596
58610298018778742744563
86696612772450384586052
64151030408982577775447
41153320764075881677514
97553804711629667771005
87664615954967769270549
62393985709255070274069
97814084312496536307186
65337180605874224259816
53070525738345415770542
92162998114917508611311
76577317209561565647869
54744892713206080635457
79462414531066983742113
79816896382353330447788
31693397287289181036640
83269856988254438516675
86228993069643468489751
48408790396476042036102
06021717394470263487633
65439319522907738361673
89811781242483655781050
34169451563626043003665
74310847665487778012857
79236454185224472361713
74229255841593135612866
37167032807217155339264
63257306730639108541088
68085742838588280602303
34140855039097353872613
45119629264159952127893
11354431460152730902553
82710432596622674390374
55636122861390783194335
70590038148700898661315
39819585744233044197085
66967222931427307413848
82788975588860799738704
47020316683485694199096
54802982493198176579268
29855629723010682777235
16274078380743187782731
82119196952800516087915
72128826337968231272562
87000150018292975772999
35790949196407634428615
75713544427898383040454
70271019458004258202120
23445806303450336581472
18549203679989972935353
91968121331951653797453
99111494244451830338588
41290401817818821376006
65928494136775431745160
54093871103687152116404
05821934471204482775960
54169486453987832626954
80139150190389959313067
03186616706637196402569
28671388714663118919268
56826919952764579977182
78759460961617218868109
45465157886912241060981
41972686192554787899263
15359472922825080542516
90681401078179602188533
07623055638163164019224
54503257656739259976517
53080142716071430871886
28598360374650571342046
70083432754230277047793
31118366690323288530687
38799071359007403049074
59889513647687608678443
23824821893061757031956
38032308197193635672741
96438726258706154330729
63703812751517040600505
75948827238563451563905
26577104264594760405569
50959840888903762079956
63880178618559159441117

In the records of the members of this secret society, this set of numbers plays a very important role. But which one? What were the Illuminati hiding behind these numbers?

The fact is that according to the surviving data, the Illuminati possessed extensive knowledge not only in the field of occult sciences, but also mathematics, astronomy, astrology, chemistry and alchemy, medicine and psychology. They also had access to some ancient sources of knowledge.

Many researchers believe that behind these numbers may be hidden a universal code of life, a recipe for a philosophical stone, etc.

Golden ratio and Fibonacci numbers in photography

Created on 08.24.2012 09:49

This article is devoted to the basic rules and concepts related both directly to the shooting process and to the subsequent processing of the resulting image in graphic editors. It will be about the rules of the "Golden Section", geometric proportions, which, if used correctly and competently, make it possible to create amazing and harmonious works.

The golden ratio is truly the first thing a beginner photographer should know about! It is sometimes called the rule of thirds. The aesthetic value of this rule was known in ancient times. Consciously use the rule of thirds - the great da Vinci, after him other artists began to use this rule, and after them photographers and cameramen, architects and designers. Let's start with the math.

Mathematical interpretation

Mathematically, the "Golden Ratio" is defined as follows - the ratio of the whole to the greater part should be equal to the ratio of the greater part to the lesser. If we divide a segment of a straight line into two unequal parts so that its length (a + b) relates to the greater part (a) in the same way as this greater part to the smaller part (b), we get the result, which is called the "Golden Section". This number equals 1.618 or 0.618. Parts of the whole segment (a + b), taken as 1, are expressed in relative values: a = 0.62 ..., b = 0.38 or in percentages of 62% and 38%.

These numbers are called "gold".

An example of using the “Golden Section” rule in photography can be the location of the main components of the frame at special points - “visual centers”. Often four points are used, located at a distance of 3/8 and 5/8 from the corresponding edges of the plane.

Fig. 2 Practical use of the "Golden Section" rule when composing a shot.

Of course, at the time of shooting, we are not able to calculate and visually postpone the necessary proportions in our mind. Therefore, at the time of shooting, a simplified version of the “Golden Section” or “Thirds” rule is used. It consists in the following: we mentally divide the frame into three parts horizontally and vertically and, at the intersection points of the imaginary lines, place the key details of the scene being shot. The simplest grid of "Thirds" looks like this: (Fig. 3).

Thus, a frame formed according to the rule of the golden section may look, for example, like this: (Figure 4, 5)

Of course, we can combine subject placement depending on the intent of the photographer and the subject. In fig. Figures 6-9 show different uses of the rule.

When using the “Golden Section” rule, one should not forget about the horizon line.

The correct setting of the horizon should correspond, depending on the composition, to one of the lines of the horizontal thirds, upper or lower. Figure 10 shows the positioning of the horizon along the bottom thirds line.

One can talk endlessly about the "golden ratio". Below I want to give different variants of grids, created according to the rule of "Golden Ratio", for different compositional options. In order to understand these principles, you need to experimentally try to match the grids with your photos on your own. Basic grids look like this (Figure 11-17):

Equilibrium rule.

Compositionally, the frame must be built so that the objects on it are balanced. What does it mean? This means that pictures will look harmonious in which either symmetry is observed (Fig. 18 - in this case, the balancing elements are the pillars on the right and left), or the main subject of the exposure is compensated for by an additional or secondary one (Fig. 19 - the crane on the left balances the composition on the right ).

As V. Tsoi sang: “We need a place to step forward”!

Any picture, even built according to the rule of the "Golden Section", can be misunderstood and misunderstood, just because the direction of movement (gaze, action) of the subject is not taken into account. In Fig. 20, the girl has absolutely no room to continue her movement (she leaves the frame), although the frame is built in the proportions of the “Golden Section”. In Figure 21, she has such a space. I repeat, this rule applies not only to the movement (people, animal machines), but also the gaze (portrait), the dynamics of the rotation of the body, face, or plot action.

Text: D.I. Zhamkov

Fibonacci numbers are elements of a numerical sequence in which each subsequent number is equal to the sum of the two previous numbers. The name is named after the Italian mathematician of medieval Europe Leonardo of Pisa, nicknamed Fibonacci, which means "a good son was born."

Fibonacci numbers are also called the golden ratio. Without going into mathematics, we can only say one thing - images that agree with the golden ratio and Fibonacci numbers are especially favorable for the human eye.

Many photographers and designers stick to 1: 1.618 aspect ratios for better composition.

This sequence was well known in India, where it was used in the metric sciences. Later, many researchers began to notice this sequence in nature and space.

The next two videos and the images that follow will help you better understand how this works in practice.

Below are photos that were taken using the Fibonacci proportions.

However, this is not all that can be done with the golden ratio. If we divide the unit by 0.618, then we get 1.618, if we square it, then we get 2.618, if we square it, we get the number 4.236. These are Fibonacci expansion ratios. The only thing missing is the number 3.236, which was proposed by John Murphy.

What do experts think of sequencing

Someone might say that these numbers are already familiar because they are used in technical analysis programs to determine the magnitude of retracements and expansions. In addition, these same series play an important role in Eliot's wave theory. They are its numerical basis.

Our expert Nikolay Verified portfolio manager of the Vostok investment company.

• - Nikolay, do you think the appearance of the Fibonacci numbers and its derivatives on the charts of various instruments is accidental? And can we say: "Fibonacci series practical application" takes place?
• - I have a bad attitude towards mysticism. And even more so on the stock exchange charts. Everything has its reasons. in the book "Fibonacci Levels" he beautifully told where the golden ratio appears, so he was not surprised that it appeared on the stock exchange quotes charts. But in vain! In many of the examples he gave, pi often appears. But for some reason it is not in the price ratios.
• - So you do not believe in the effectiveness of the Eliot wave principle?
• - No, that's not the point. The wave principle is one thing. The numerical ratio is different. And the reasons for their appearance on price charts are the third
• - What, in your opinion, are the reasons for the appearance of the golden ratio on stock charts?
• - The correct answer to this question may be able to earn the Nobel Prize in economics. While we can guess about the true reasons. They are clearly not in harmony with nature. There are many exchange pricing models. They do not explain the indicated phenomenon. But not understanding the nature of a phenomenon should not deny the phenomenon as such.
• - And if ever this law is opened, will it be able to destroy the exchange process?
• - As the same theory of waves shows, the law of changes in stock prices is pure psychology. It seems to me that knowledge of this law will not change anything and will not be able to destroy the exchange.

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The overlap of the foundations of the principles of mathematics in a variety of theories seems incredible. Maybe it’s a fantasy or a fit for the final result. Wait and see. Much of what was previously considered unusual or not possible: space exploration, for example, has become commonplace and does not surprise anyone. Also, the wave theory, which may be incomprehensible, will become more accessible and understandable over time. What was previously unnecessary, in the hands of an experienced analyst, will become a powerful tool for predicting future behavior.

Fibonacci numbers in nature.

Watch

Now, let's talk about how you can refute the fact that the Fibonacci digital series is involved in any patterns in nature.

Take any other two numbers and build a sequence with the same logic as the Fibonacci numbers. That is, the next term in the sequence is equal to the sum of the two previous ones. For example, let's take two numbers: 6 and 51. Now let's build a sequence, which we end with two numbers 1860 and 3009. Note that when dividing these numbers, we get a number close to the golden ratio.

At the same time, the numbers that were obtained by dividing other pairs decreased from the first to the last, which allows us to assert that if this series is continued infinitely, then we will receive a number equal to the golden ratio.

Thus, Fibonacci numbers do not stand out by themselves. There are other sequences of numbers, of which there are infinitely many, which give, as a result of the same operations, the golden number phi.

Fibonacci was not esoteric. He didn’t want to put any mysticism into numbers, he was just solving an ordinary problem about rabbits. And he wrote a sequence of numbers that followed from his problem, in the first, second and other months, how many rabbits there will be after breeding. Within a year, he received that very sequence. And I didn't make a relationship. There was no golden proportion, the Divine attitude was out of the question. All this was invented after him during the Renaissance.

Before mathematics, the merits of Fibonacci are enormous. He adopted the system of numbers from the Arabs and proved its validity. It was a hard and long struggle. From the Roman numeral system: heavy and inconvenient for counting. She disappeared after the French Revolution. Fibonacci has nothing to do with the golden ratio.

There are infinitely many spirals, the most popular are: natural logarithm spiral, Archimedes spiral, hyperbolic spiral.

Now let's take a look at the Fibonacci spiral. This piecewise composite aggregate consists of several quarters of a circle. And it is not a spiral as such.

Output

No matter how long we look for confirmation or refutation of the applicability of the Fibonacci series on the stock exchange, such a practice exists.

Huge masses of people act according to the Fibonacci ruler, which is found in many user terminals. Therefore, whether we like it or not: Fibonacci numbers have an impact on, and we can take advantage of this influence.

Be sure to read the article -.

GOU Gymnasium №1505

"Moscow city pedagogical gymnasium-laboratory"

abstract

Fibonacci numbers and the Golden ratio

Azov Nikita

Supervisor: Shalimova M.N.

Introduction ………………………………………………….……………2

Chapter 1

History of Fibonacci numbers. ……………………………… .. …… ..5

Chapter 2

Fibonacci numbers as a reverse progression ……… ...… ... …………………………………… ..… ..... 12

CHAPTER 3

Fibonacci numbers and the Golden ratio ………………………

Conclusion …………………………………………………...…...16

Bibliography ………………………………………………………………….……..20

Introduction.

The relevance of research. In my opinion, these days little attention is paid to mathematical theorems and facts known from the history of the development of science. Using the Fibonacci numbers as an example, I would like to show how global they can and are widely applicable not only in mathematics, but also in everyday life.

The aim of my work is to study the history, properties, applications and connections of Fibonacci numbers with the golden ratio.

Chapter 1. Fibonacci numbers and their history.

The Abacus Book can be divided into five parts according to content. The first five chapters of the book are devoted to the arithmetic of integers based on decimal numbering. Chapter 6-7 describes operations with ordinary fractions. Chapter 8-10 describes techniques for solving problems using proportions. Chapter 11 deals with mixing problems, chapter 12 deals with the so-called Fibonacci numbers. The following sections describe some more techniques with numbers and tasks on various topics.

The main problem explaining the appearance of a series of Fibonacci numbers is the problem of rabbits. The question of the problem sounds like this: "How many pairs of rabbits are born from one pair in one year?" An explanation is given to the problem that a couple of rabbits give birth to another pair in a month, and by nature, rabbits begin to give birth to offspring in the second month after their birth. The author gives us a solution to the problem. It turns out that in the first month the first couple will give birth to another one. In the second, the first couple will give birth to another one - there will be three couples. In the 3rd month, two couples will give birth - originally given and born in the first month. It turns out 5 pairs. And so on, using the same logic in reasoning, we get that in the fourth month there will be 8 pairs, in the fifth 13, in the sixth 21, in the seventh 34, in the eighth 55, in the ninth 89, in the tenth 144, in the eleventh 233, at the twelfth 377.

We can designate the number of rabbits in any of the twelve months as u n. We get a series of numbers:

In a series of these numbers, each term is equal to the sum of the two previous ones. It turns out that any term in the equation can be determined by the equation:

Consider an important special case for this equation when u 1 and u 2 = 1. We will receive a sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 ... We received the same sequence of numbers in the problem about rabbits. These numbers are named Fibonacci numbers after the author.

These numbers as well as equation (2) have many properties, which will be considered in my work.

Chapter 2. Relationship between a series of Fibonacci numbers and progressions. The main properties of the series.

In order to deduce the basic properties of the series, let us take as an example the first five numbers: 1, 1, 2, 3, 5, 8. We see that each new number is equal to the sum of the two previous ones. From here we can derive the formula for obtaining any number of the series, as well as the formula for the sum of any number of numbers from the series.

We see that the formulas are fundamentally different from the formulas inherent in arithmetic and geometric progressions. And also we can say that only the first two numbers from the row can refer to any progressions.

Arithmetic and geometric progressions have only the two previously mentioned formulas, and in order to calculate, for example, the sum of even, odd or the sum of squares of numbers, each time you have to solve a problem for a separate series. But since the series of Fibonacci numbers is unchanged (it has no steps, denominators and various first members of the progression), this means that for it you can derive a formula for obtaining the sums of the individual elements of the series. For example, here is a formula for obtaining the sum of the numbers in a series with even numbers:

There is a similar formula for odd-numbered numbers from a series:

There is also a formula for obtaining the sum of numbers from a series squared:

Fibonacci numbers have another unique property that is not typical for arithmetic and geometric progressions. The ratio of a number of numbers (the previous to the next) constantly tends to the value 0.618, a similar situation occurs when F n is divided by F n +2 (the ratio tends to 0.382), when F n is divided by F n +3 (the ratio tends to 0.236), and so Further. As a result, we got a set of relationships. The set of their values ​​and values ​​inverse to them are called Fibonacci coefficients. And the inverse value 0.618 - 1.618 is a number

Chapter 3. Golden ratio and Fibonacci numbers.

The golden ratio (golden ratio, division in extreme and average ratio) is the division of a continuous value into two parts in such a ratio in which the smaller part relates to the larger one as the larger one to the whole value.

Let's try to explain this using the example of an infinite straight line. Let us take the whole line c as a unit. Let's divide it into two parts a and b, which divide the line into segments equal with respect to 1, like 0.618 and 0.382, respectively. And these numbers are one of the coefficients of a series of Fibonacci numbers. We find that the ratio of large parts of this line to smaller ones asymptotically approaches the number

There are two main figures that reflect the principle of the golden ratio.

The golden ratio was already known to the ancient Greeks. Archimedes is considered the discoverer of the Archimedes spiral. Its meaning is that each new curl increases by a certain number, and the ratio of these curls is equal to the number

The second figure is the golden triangle. This is an isosceles triangle in which the ratio of the sides to the base is