Fibonacci numbers, golden ratio, Fibonacci sequence and Illuminati. Golden ratio - what is it? What are Fibonacci numbers? What do a DNA helix, a shell, a galaxy and the Egyptian pyramids have in common?
is a comprehensive manifestation of structural harmony. It is found in all spheres of the universe in nature, science, art, everything that a person can come into contact with. Once having become acquainted with the golden rule, humanity no longer betrayed it.
Surely you have often wondered why Nature is able to create such amazing harmonious structures that delight and delight the eye. Why artists, poets, composers, architects create amazing works of art from century to century. What is the secret and what laws underlie these harmonious creatures? No one can definitely answer this question, but in our book we will try to lift the veil and tell you about one of the secrets of the universe - the Golden Section or, as it is also called, the Golden or Divine Proportion. The Golden Ratio is called the number PHI (Phi) in honor of the great ancient Greek sculptor Phidias, who used this number in his sculptures.
For centuries, scientists have been using the unique mathematical properties of the PHI number, and this research continues to this day. This number found wide application in all areas of modern science, which we will also try to popularly talk about on the pages. There are also a number of What is this You will find out further...
Definition of the golden ratio
The simplest and most succinct definition of the golden ratio is that a small part relates to a larger part, just as a large part relates to the whole. Its approximate value is 1.6180339887. In a rounded percentage value, the proportions of the parts of the whole will correspond as 62% to 38%. This relationship operates in the forms of space and time.
The ancients saw the golden ratio as a reflection of cosmic order, and Johann called it one of the treasures of geometry. Modern science considers the golden ratio as an asymmetrical symmetry, calling it in a broad sense universal rule, reflecting the structure and order of our world order.
Fibonacci numbers in history
The ancient Egyptians had an idea about the golden proportions, they knew about them in Rus', but for the first time the golden ratio was scientifically explained by the monk Luca Pacioli in the book Divine Proportion, illustrations for which were supposedly made by Leonardo. Pacioli saw the divine trinity in the golden section: the small section personified the Son, the larger section the Father, and the whole the Holy Spirit.
The name of the Italian Leonardo is directly associated with the rule of the golden ratio. As a result of solving one of the problems, the scientist came up with a sequence of numbers, now known as the series: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. The ratio of neighboring numbers in a series in the limit tends to the Golden Ratio. I drew attention to the relationship of this sequence to the golden proportion: It is arranged in such a way that the two junior terms of this never-ending proportion add up to the third term, and any two last terms, if added, give the next term. Now the series is the arithmetic basis for calculating the proportions of the golden section in all its manifestations.
Golden ratio formula
Fashion designers and clothing designers make all calculations based on the proportions of the golden ratio. Man is universal form can mean: The shape of an object - the relative position of the boundaries (contours) of an object, object, as well as the relative position of points on a line to test the laws of the golden ratio. Of course, by nature, not all people have ideal proportions, which creates certain difficulties with the selection of clothes.
In Leonardo's diary there is a drawing of a naked man inscribed in a circle, in two superimposed positions. Based on the research of the Roman architect Vitruvius, Leonardo similarly tried to establish the proportions of the human body. Later, the French architect Le Corbusier, using Leonardo's Vitruvian Man, created his own scale of harmonic proportions, which influenced the aesthetics of 20th-century architecture.
Adolf Zeising, studying the proportionality of a person, did a colossal job. He measured about two thousand human bodies, as well as many ancient statues, and concluded that the golden ratio expresses the average statistical law. IN person living, intelligent social, subject of socio-historical activity and culture Almost all parts of the body are subordinate to it, but the main indicator gold something made of gold sections are divisions body In mathematics: Body (algebra) - a set with two operations (addition and multiplication) that has certain properties navel point.
As a result of measurements, the researcher found that the proportions of the male body 13:8 are closer to golden section a multi-valued term meaning: Section in drawing - unlike a section, the image of only a figure formed by dissecting a body by a plane (planes) without depicting the parts behind this than the proportions of the female body 8:5.
The art of spatial forms
The artist Vasily Surikov said that there is an immutable law in composition, when in a picture you cannot remove or add anything, you cannot even add an extra point, this is real. For a long time artists followed this law intuitively, but after Leonardo Di Ser Piero (Italian The process of creating a painting is no longer complete without solving geometric problems. For example, Albrecht Durer for the definition points can mean: Point - an abstract object in space that does not have any measurable characteristics other than coordinates The golden ratio was used by the proportional compass he invented.
Art critic F.V. Kovalev, having examined in detail the painting by Nikolai Ge Alexander Sergeevich Pushkin in the village of Mikhailovskoye, notes that every detail of the canvas, be it a fireplace, a bookcase, an armchair or the poet himself, is strictly inscribed in golden proportions.
Researchers of the golden ratio tirelessly study and measure architectural masterpieces, claiming that they became such because they were created according to the golden canons: their list includes the Great Pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, and the Parthenon.
And today, in any art of spatial forms, they try to follow the proportions of the golden section, since, according to art critics, they facilitate the perception of the work and form an aesthetic feeling in the viewer.
Word, sound and film
The forms of temporary art in their own way demonstrate to us the principle of the golden division. Literary scholars, for example, have noticed that the most popular number of lines in poems of the late period of Pushkin’s work corresponds to the series 5, 8, 13, 21, 34.
The rule of the golden section also applies in individual works of the Russian classic. So the climax Queen of Spades is a dramatic scene between Herman and the Countess, ending with the death of the latter. The story has 853 lines, and the climax occurs on line 535 (853:535 = 1.6), this is the point of the golden ratio.
Soviet musicologist E.K. Rosenov notes the amazing accuracy of the ratios of the golden section in the strict and free forms of the works of Johann Sebastian Bach, which corresponds to the thoughtful, concentrated, technically verified style of the master. This is also true of the outstanding works of other composers, where the most striking or unexpected musical solution usually occurs at the golden ratio point.
Film director Sergei Eisenstein deliberately coordinated the script of his film Battleship Potemkin with the rule of the golden ratio, dividing the film into five parts. In the first three sections the action takes place on a ship, and in the last two in Odessa. The transition to scenes in the city is golden mean film.
Harmony of the Golden Ratio
Scientific and technological progress has a long history and went through several stages in its historical development (Babylonian and ancient Egyptian culture, the culture of Ancient China and Ancient India, ancient greek culture, Middle Ages, Renaissance, industrial Revolution 18th century, great scientific discoveries 19th century, scientific and technological revolution of the 20th century) and entered the 21st century, which opens new era in the history of mankind - the era of Harmony. It was during the ancient period that a number of outstanding mathematical discoveries were made that had a decisive influence on the development of material and spiritual culture, including the Babylonian 60-digit number system and the positional principle of representing numbers, trigonometry and Euclidean geometry, incommensurable segments, the Golden Section and Platonic solids, principles number theory and measurement theory. And, although each of these stages has its own specifics, at the same time it necessarily includes the content of the previous stages. This is the continuity in the development of science. Succession can take place in various forms. One of the essential forms of its expression is fundamental scientific ideas that permeate all stages of scientific and technological progress and influence various fields of science, art, philosophy and technology.
The category of such fundamental ideas includes the idea of Harmony, associated with the Golden Section. According to B.G. Kuznetsov, a researcher of the work of Albert Einstein, the great physicist firmly believed that science, physics in particular, has always had as its eternal fundamental goal “to find objective harmony in the labyrinth of observed facts.” The deep faith of the outstanding physicist in the existence of universal laws of harmony of the universe is evidenced by another widely famous saying Einstein: “The religiosity of a scientist consists of an enthusiastic admiration for the laws of harmony.”
In ancient Greek philosophy, Harmony opposed Chaos and meant the organization of the Universe, the Cosmos. The brilliant Russian philosopher Alexei Losev assesses the main achievements of the ancient Greeks in this area as follows:
“From the point of view of Plato, and indeed from the point of view of the entire ancient cosmology, the world is a kind of proportional whole, subject to the law of harmonic division - the Golden Section... Their (the ancient Greeks) system of cosmic proportions is often depicted in literature as a curious result of unbridled and wild imagination. This kind of explanation reveals the anti-scientific helplessness of those who declare it. However, this historical-aesthetic phenomenon can only be understood in connection with a holistic understanding of history, that is, using a dialectical-materialist idea of culture and looking for an answer in the peculiarities of ancient social existence.”
“The law of the golden division must be a dialectical necessity. This is an idea that, as far as I know, I am pursuing for the first time.”, Losev spoke with conviction more than half a century ago in connection with the analysis of the cultural heritage of the ancient Greeks.
And here is another statement regarding the Golden Ratio. It was made in the 17th century and belongs to the brilliant astronomer Johannes Kepler, the author of the three famous “Kepler’s Laws”. He expressed his admiration for the Golden Section in the following words:
“There are two treasures in geometry - the division of a segment in extreme and mean ratio. The first can be compared to the value of gold, the second can be called a precious stone."
Let us recall that the ancient problem of dividing a segment in extreme and mean ratio, which is mentioned in this statement, is the Golden Ratio!
Numbers in science
IN modern science there are many scientific groups professionally studying the Golden Ratio, numbers and their numerous applications in mathematics, physics, philosophy, botany, biology, medicine, computer science. Many artists, poets, and musicians use the “Golden Section Principle” in their work. Modern science has made a number of outstanding discoveries based on numbers and the Golden Ratio. The discovery of “quasi-crystals” made in 1982 by the Israeli scientist Dan Shechtman, based on the Golden Section and “pentagonal” symmetry, has revolutionary significance for modern physics. A breakthrough in modern ideas about the nature of the formation of biological objects was made in the early 90s by the Ukrainian scientist Oleg Bodnar, who created a new geometric theory of phyllotaxis. The Belarusian philosopher Eduard Soroko formulated the “Law of structural harmony of systems”, based on the Golden Section and playing important role in processes of self-organization. Thanks to the research of American scientists Elliott, Prechter and Fisher, numbers actively entered the field of business and became the basis for optimal strategies in business and trade. These discoveries confirm the hypothesis of the American scientist D. Winter, head of the “Planetary Heartbeats” group, according to which not only the energetic framework of the Earth, but also the structure of all living things are based on the properties of the dodecahedron and icosahedron - two “Platonic solids” associated with the Golden Ratio. And finally, perhaps most importantly, the DNA structure of the genetic code of life is a four-dimensional development (along the time axis) of a rotating dodecahedron! Thus, it turns out that the entire Universe - from the Metagalaxy to the living cell - is built according to one principle - the dodecahedron and icosahedron infinitely inscribed into each other, located in the proportion of the Golden Section!
Ukrainian professor and doctor of sciences Stakhov A.P. was able to create some . The essence of this generalization is extremely simple. If you specify a non-negative integer p = 0, 1, 2, 3, ... and divide the segment “AB” by point C in such a proportion that it is.
Have you ever heard that mathematics is called the “queen of all sciences”? Do you agree with this statement? As long as mathematics remains for you a set of boring problems in a textbook, you can hardly experience the beauty, versatility and even humor of this science.
But there are topics in mathematics that help make interesting observations about things and phenomena that are common to us. And even try to penetrate the veil of mystery of the creation of our Universe. There are interesting patterns in the world that can be described using mathematics.
Introducing Fibonacci numbers
Fibonacci numbers name the elements of a number sequence. In it, each next number in a series is obtained by summing the two previous numbers.
Example sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…
You can write it like this:
F 0 = 0, F 1 = 1, F n = F n-1 + F n-2, n ≥ 2
You can start a series of Fibonacci numbers with negative values n. Moreover, the sequence in this case is two-sided (i.e., it covers negative and positive numbers) and tends to infinity in both directions.
An example of such a sequence: -55, -34, -21, -13, -8, 5, 3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
The formula in this case looks like this:
F n = F n+1 - F n+2 or else you can do this: F -n = (-1) n+1 Fn.
What we now know as “Fibonacci numbers” was known to ancient Indian mathematicians long before they began to be used in Europe. And this name is generally one continuous historical anecdote. Let's start with the fact that Fibonacci himself never called himself Fibonacci during his lifetime - this name began to be applied to Leonardo of Pisa only several centuries after his death. But let's talk about everything in order.
Leonardo of Pisa, aka Fibonacci
The son of a merchant who became a mathematician, and subsequently received recognition from posterity as the first major mathematician of Europe during the Middle Ages. Not least thanks to the Fibonacci numbers (which, let us remember, were not called that yet). Which he described at the beginning of the 13th century in his work “Liber abaci” (“Book of Abacus”, 1202).
I travel with my father to the East, Leonardo studied mathematics with Arab teachers (and in those days they were in this field, and in many other sciences, one of the best specialists). He read the works of mathematicians of Antiquity and Ancient India in Arabic translations.
Having thoroughly comprehended everything he had read and using his own inquisitive mind, Fibonacci wrote several scientific treatises on mathematics, including the above-mentioned “Book of Abacus.” In addition to this I created:
- "Practica geometriae" ("Practice of Geometry", 1220);
- "Flos" ("Flower", 1225 - a study on cubic equations);
- "Liber quadratorum" ("Book of Squares", 1225 - problems on indefinite quadratic equations).
He was a big fan of mathematical tournaments, so in his treatises he paid a lot of attention to the analysis of various mathematical problems.
There is very little biographical information left about Leonardo's life. As for the name Fibonacci, under which he entered the history of mathematics, it was assigned to him only in the 19th century.
Fibonacci and his problems
After Fibonacci there remained a large number of problems that were very popular among mathematicians in subsequent centuries. We will look at the rabbit problem, which is solved using Fibonacci numbers.
Rabbits are not only valuable fur
Fibonacci set the following conditions: there is a pair of newborn rabbits (male and female) of such an interesting breed that they regularly (starting from the second month) produce offspring - always one new pair of rabbits. Also, as you might guess, a male and a female.
These conditional rabbits are placed in a confined space and breed with enthusiasm. It is also stipulated that not a single rabbit dies from some mysterious rabbit disease.
We need to calculate how many rabbits we will get in a year.
- At the beginning of 1 month we have 1 pair of rabbits. At the end of the month they mate.
- The second month - we already have 2 pairs of rabbits (a pair has parents + 1 pair is their offspring).
- Third month: The first pair gives birth to a new pair, the second pair mates. Total - 3 pairs of rabbits.
- Fourth month: The first pair gives birth to a new pair, the second pair does not waste time and also gives birth to a new pair, the third pair is still only mating. Total - 5 pairs of rabbits.
Number of rabbits in n th month = number of pairs of rabbits from the previous month + number of newborn pairs (there are the same number of pairs of rabbits as there were pairs of rabbits 2 months before now). And all this is described by the formula that we have already given above: Fn = Fn-1 + Fn-2.
Thus, we obtain a recurrent (explanation about recursion– below) number sequence. In which each next number is equal to the sum of the previous two:
- 1 + 1 = 2
- 2 + 1 = 3
- 3 + 2 = 5
- 5 + 3 = 8
- 8 + 5 = 13
- 13 + 8 = 21
- 21 + 13 = 34
- 34 + 21 = 55
- 55 + 34 = 89
- 89 + 55 = 144
- 144 + 89 = 233
- 233+ 144 = 377 <…>
You can continue the sequence for a long time: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987<…>. But since we have set a specific period - a year, we are interested in the result obtained on the 12th “move”. Those. 13th member of the sequence: 377.
The answer to the problem: 377 rabbits will be obtained if all stated conditions are met.
One of the properties of the Fibonacci number sequence is very interesting. If you take two consecutive pairs from a series and divide the larger number by the smaller number, the result will gradually approach golden ratio(you can read more about it later in the article).
In mathematical terms, "the limit of relations a n+1 To a n equal to the golden ratio".
More number theory problems
- Find a number that can be divided by 7. Also, if you divide it by 2, 3, 4, 5, 6, the remainder will be one.
- Find square number. It is known about it that if you add 5 to it or subtract 5, you again get a square number.
We suggest you search for answers to these problems yourself. You can leave us your options in the comments to this article. And then we will tell you whether your calculations were correct.
Explanation of recursion
Recursion– definition, description, image of an object or process that contains this object or process itself. That is, in essence, an object or process is a part of itself.
Recursion is widely used in mathematics and computer science, and even in art and popular culture.
Fibonacci numbers are determined using a recurrence relation. For number n>2 n- e number is equal (n – 1) + (n – 2).
Explanation of the golden ratio
Golden ratio- dividing a whole (for example, a segment) into parts that are related according to the following principle: the larger part is related to the smaller one in the same way as the entire value (for example, the sum of two segments) is to the larger part.
The first mention of the golden ratio can be found in Euclid in his treatise “Elements” (about 300 BC). In the context of constructing a regular rectangle.
The term familiar to us was introduced into circulation in 1835 by the German mathematician Martin Ohm.
If we describe the golden ratio approximately, it represents a proportional division into two unequal parts: approximately 62% and 38%. IN numerically The golden ratio represents a number 1,6180339887 .
The golden ratio finds practical application in fine arts(paintings by Leonardo da Vinci and other Renaissance painters), architecture, cinema (“Battleship Potemkin” by S. Esenstein) and other areas. For a long time it was believed that the golden ratio is the most aesthetic proportion. This opinion is still popular today. Although, according to research results, visually most people do not perceive this proportion as the most a good option and is considered too elongated (disproportionate).
- Length of the segment With = 1, A = 0,618, b = 0,382.
- Attitude With To A = 1, 618.
- Attitude With To b = 2,618
Now let's get back to Fibonacci numbers. Let's take two consecutive terms from its sequence. Divide the larger number by the smaller number and get approximately 1.618. And now we use the same larger number and the next member of the series (i.e., an even larger number) - their ratio is early 0.618.
Here's an example: 144, 233, 377.
233/144 = 1.618 and 233/377 = 0.618
By the way, if you try to do the same experiment with numbers from the beginning of the sequence (for example, 2, 3, 5), nothing will work. Almost. The golden ratio rule is hardly followed for the beginning of the sequence. But as you move along the series and the numbers increase, it works great.
And in order to calculate the entire series of Fibonacci numbers, it is enough to know three terms of the sequence, coming one after another. You can see this for yourself!
Golden Rectangle and Fibonacci Spiral
Another interesting parallel between the Fibonacci numbers and the golden ratio is the so-called “golden rectangle”: its sides are in proportion 1.618 to 1. But we already know what the number 1.618 is, right?
For example, let's take two consecutive terms of the Fibonacci series - 8 and 13 - and construct a rectangle with the following parameters: width = 8, length = 13.
And then we will divide the large rectangle into smaller ones. Required condition: The lengths of the sides of the rectangles must correspond to the Fibonacci numbers. Those. The side length of the larger rectangle must be equal to the sum of the sides of the two smaller rectangles.
The way it is done in this figure (for convenience, the figures are signed in Latin letters).
By the way, you can build rectangles in reverse order. Those. start building with squares with side 1. To which, guided by the principle stated above, figures with sides are completed, equal numbers Fibonacci. Theoretically, this can be continued indefinitely - after all, the Fibonacci series is formally infinite.
If we connect the corners of the rectangles obtained in the figure with a smooth line, we get a logarithmic spiral. Or rather, its special case is the Fibonacci spiral. It is characterized, in particular, by the fact that it has no boundaries and does not change shape.
A similar spiral is often found in nature. Clam shells are one of the most striking examples. Moreover, some galaxies that can be seen from Earth have a spiral shape. If you pay attention to weather forecasts on TV, you may have noticed that cyclones have a similar spiral shape when photographed from satellites.
It is curious that the DNA helix also obeys the rule of the golden section - the corresponding pattern can be seen in the intervals of its bends.
Such amazing “coincidences” cannot but excite minds and give rise to talk about some single algorithm to which all phenomena in the life of the Universe obey. Now do you understand why this article is called this way? And what kind of amazing worlds can mathematics open for you?
Fibonacci numbers in nature
The connection between Fibonacci numbers and the golden ratio suggests interesting patterns. So curious that it is tempting to try to find similar to numbers Fibonacci sequences in nature and even during historical events. And nature really gives rise to such assumptions. But can everything in our life be explained and described using mathematics?
Examples of living things that can be described using the Fibonacci sequence:
- the arrangement of leaves (and branches) in plants - the distances between them are correlated with Fibonacci numbers (phyllotaxis);
- arrangement of sunflower seeds (the seeds are arranged in two rows of spirals twisted in different directions: one row clockwise, the other counterclockwise);
- arrangement of pine cone scales;
- flower petals;
- pineapple cells;
- ratio of the lengths of the phalanges of the fingers on the human hand (approximately), etc.
Combinatorics problems
Fibonacci numbers are widely used in solving combinatorics problems.
Combinatorics is a branch of mathematics that studies the selection of a certain number of elements from a designated set, enumeration, etc.
Let's look at examples of combinatorics problems designed for high school level (source - http://www.problems.ru/).
Task #1:
Lesha climbs a staircase of 10 steps. At one time he jumps up either one step or two steps. In how many ways can Lesha climb the stairs?
The number of ways in which Lesha can climb the stairs from n steps, let's denote and n. It follows that a 1 = 1, a 2= 2 (after all, Lesha jumps either one or two steps).
It is also agreed that Lesha jumps up the stairs from n> 2 steps. Let's say he jumped two steps the first time. This means, according to the conditions of the problem, he needs to jump another n – 2 steps. Then the number of ways to complete the climb is described as a n–2. And if we assume that the first time Lesha jumped only one step, then we describe the number of ways to finish the climb as a n–1.
From here we get the following equality: a n = a n–1 + a n–2(looks familiar, doesn't it?).
Since we know a 1 And a 2 and remember that according to the conditions of the problem there are 10 steps, calculate all in order and n: a 3 = 3, a 4 = 5, a 5 = 8, a 6 = 13, a 7 = 21, a 8 = 34, a 9 = 55, a 10 = 89.
Answer: 89 ways.
Task #2:
You need to find the number of words 10 letters long that consist only of the letters “a” and “b” and must not contain two letters “b” in a row.
Let's denote by a n number of words length n letters that consist only of the letters “a” and “b” and do not contain two letters “b” in a row. Means, a 1= 2, a 2= 3.
In sequence a 1, a 2, <…>, a n we will express each of its next members through the previous ones. Therefore, the number of words of length is n letters that also do not contain a double letter “b” and begin with the letter “a” are a n–1. And if the word is long n letters begin with the letter “b”, it is logical that the next letter in such a word is “a” (after all, there cannot be two “b” according to the conditions of the problem). Therefore, the number of words of length is n in this case we denote the letters as a n–2. In both the first and second cases, any word (length of n – 1 And n – 2 letters respectively) without double “b”.
We were able to justify why a n = a n–1 + a n–2.
Let us now calculate a 3= a 2+ a 1= 3 + 2 = 5, a 4= a 3+ a 2= 5 + 3 = 8, <…>, a 10= a 9+ a 8= 144. And we get the familiar Fibonacci sequence.
Answer: 144.
Task #3:
Imagine that there is a tape divided into cells. It goes to the right and lasts indefinitely. Place a grasshopper on the first square of the tape. Whatever cell of the tape he is on, he can only move to the right: either one cell, or two. How many ways are there in which a grasshopper can jump from the beginning of the tape to n-th cells?
Let us denote the number of ways to move a grasshopper along the belt to n-th cells like a n. In this case a 1 = a 2= 1. Also in n+1 The grasshopper can enter the -th cell either from n-th cell, or by jumping over it. From here a n + 1 = a n – 1 + a n. Where a n = Fn – 1.
Answer: Fn – 1.
You can create similar problems yourself and try to solve them in math lessons with your classmates.
Fibonacci numbers in popular culture
Of course, such an unusual phenomenon as Fibonacci numbers cannot but attract attention. There is still something attractive and even mysterious in this strictly verified pattern. It is not surprising that the Fibonacci sequence somehow “lit up” in many works of modern popular culture a variety of genres.
We will tell you about some of them. And you try to search for yourself again. If you find it, share it with us in the comments – we’re curious too!
- Fibonacci numbers are mentioned in Dan Brown's bestseller The Da Vinci Code: the Fibonacci sequence serves as the code used by the book's main characters to open a safe.
- In the 2009 American film Mr. Nobody, in one episode the address of a house is part of the Fibonacci sequence - 12358. In addition, in another episode main character must call a phone number, which is essentially the same, but slightly distorted (extra digit after the 5) sequence: 123-581-1321.
- In the 2012 series “Connection”, the main character, a boy suffering from autism, is able to discern patterns in events occurring in the world. Including through Fibonacci numbers. And manage these events also through numbers.
- Java game developers for mobile phones Doom RPG placed a secret door in one of the levels. The code that opens it is the Fibonacci sequence.
- In 2012, the Russian rock band Splin released the concept album “Optical Deception.” The eighth track is called “Fibonacci”. The verses of the group leader Alexander Vasiliev play on the sequence of Fibonacci numbers. For each of the nine consecutive terms there is a corresponding number of lines (0, 1, 1, 2, 3, 5, 8, 13, 21):
0 The train set off
1 One joint snapped
1 One sleeve trembled
2 That's it, get the stuff
That's it, get the stuff
3 Request for boiling water
The train goes to the river
The train goes through the taiga<…>.
- A limerick (a short poem of a specific form - usually five lines, with a specific rhyme scheme, humorous in content, in which the first and last lines are repeated or partially duplicate each other) by James Lyndon also uses a reference to the Fibonacci sequence as a humorous motif:
The dense food of Fibonacci's wives
It was only for their benefit, nothing else.
The wives weighed, according to rumor,
Each one is like the previous two.
Let's sum it up
We hope that we were able to tell you a lot of interesting and useful things today. For example, you can now look for the Fibonacci spiral in the nature around you. Maybe you will be the one who will be able to unravel “the secret of life, the Universe and in general.”
Use the formula for Fibonacci numbers when solving combinatorics problems. You can rely on the examples described in this article.
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Golden ratio and Fibonacci sequence numbers. June 14th, 2011
Some time ago, I promised to comment on Tolkachev’s statement that St. Petersburg is built according to the principle of the Golden Section, and Moscow is built according to the principle of symmetry, and that this is why the differences in the perception of these two cities are so noticeable, and this is why a St. Petersburger, coming to Moscow, “gets a headache” ”, and a Muscovite “gets a headache” when he comes to St. Petersburg. It takes some time to tune in to the city (like when flying to the states - it takes time to tune in).
The fact is that our eye looks - feeling the space with the help of certain eye movements - saccades (in translation - the clap of a sail). The eye makes a “clap” and sends a signal to the brain “adhesion to the surface has occurred. Everything is fine. Information such and such." And over the course of life, the eye gets used to a certain rhythm of these saccades. And when this rhythm changes radically (from a city landscape to a forest, from the Golden Section to symmetry), then some brain work is required to reconfigure.
Now the details:
The definition of GS is the division of a segment into two parts in such a ratio in which the larger part is related to the smaller one, as their sum (the entire segment) is to the larger one.
That is, if we take the entire segment c as 1, then segment a will be equal to 0.618, segment b - 0.382. Thus, if we take a building, for example, a temple built according to the 3S principle, then with its height, say, 10 meters, the height of the drum with the dome will be 3.82 cm, and the height of the base of the structure will be 6.18 cm (it is clear that the numbers I took them flat for clarity)
What is the connection between ZS and Fibonacci numbers?
The Fibonacci sequence numbers are:
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 = 21, etc.,
and the ratio of adjacent numbers approaches the ratio of ZS.
So, 21: 34 = 0.617, and 34: 55 = 0.618.
That is, the GS is based on the numbers of the Fibonacci sequence.
This video once again clearly demonstrates this connection between GS and Fibonacci numbers
Where else are the 3S principle and Fibonacci sequence numbers found?
Plant leaves are described by the Fibonacci sequence. Sunflower grains, pine cones, flower petals, and pineapple cells are also arranged according to the Fibonacci sequence.
bird egg
The lengths of the phalanges of human fingers are approximately the same as the Fibonacci numbers. The golden ratio is visible in the proportions of the face.
Emil Rosenov studied GS in the music of the Baroque and Classical eras using the examples of works by Bach, Mozart, and Beethoven.
It is known that Sergei Eisenstein artificially constructed the film “Battleship Potemkin” according to the rules of the Legislature. He broke the tape into five parts. In the first three, the action takes place on the ship. In the last two - in Odessa, where the uprising is unfolding. This transition to the city occurs exactly at the golden ratio point. And each part has its own fracture, which occurs according to the law of the golden ratio. In a frame, scene, episode there is a certain leap in the development of the theme: plot, mood. Eisenstein believed that since such a transition is close to the golden ratio point, it is perceived as the most logical and natural.
Many decorative elements, as well as fonts, were created using ZS. For example, the font of A. Durer (in the picture there is the letter “A”)
It is believed that the term “Golden Ratio” was introduced by Leonardo Da Vinci, who said, “let no one who is not a mathematician dare to read my works” and showed the proportions of the human body in his famous drawing “Vitruvian Man”. “If we tie a human figure - the most perfect creation of the Universe - with a belt and then measure the distance from the belt to the feet, then this value will relate to the distance from the same belt to the top of the head, just as the entire height of a person relates to the length from the waist to the feet.”
The famous portrait of Mona Lisa or Gioconda (1503) was created according to the principle of golden triangles.
Strictly speaking, the star or pentacle itself is a construction of the Earth.
The Fibonacci number series is visually modeled (materialized) in the form of a spiral
And in nature, the GS spiral looks like this:
At the same time, the spiral is observed everywhere(in nature and not only):
- Seeds in most plants are arranged in a spiral
- The spider weaves a web in a spiral
- A hurricane is spinning like a spiral
- A frightened herd of reindeer scatters in a spiral.
- The DNA molecule is twisted in a double helix. The DNA molecule is made up of two vertically intertwined helices, 34 angstroms long and 21 angstroms wide. The numbers 21 and 34 follow each other in the Fibonacci sequence.
- The embryo develops in a spiral shape
- Cochlear spiral in the inner ear
- The water goes down the drain in a spiral
- Spiral dynamics shows the development of a person’s personality and his values in a spiral.
- And of course, the Galaxy itself has the shape of a spiral
Thus, it can be argued that nature itself is built according to the principle of the Golden Section, which is why this proportion is more harmoniously perceived by the human eye. It does not require “correction” or addition to the resulting picture of the world.
Now about the Golden Ratio in architecture
The Cheops pyramid represents the proportions of the Earth. (I like the photo - with the Sphinx covered in sand).
According to Le Corbusier, in the relief from the temple of Pharaoh Seti I at Abydos and in the relief depicting Pharaoh Ramses, the proportions of the figures correspond to the golden ratio. In the facade ancient greek temple The Parthenon also features golden proportions.
Notredame de Paris Cathedral in Paris, France.
One of the outstanding buildings made according to the GS principle is the Smolny Cathedral in St. Petersburg. There are two paths leading to the cathedral along the edges, and if you approach the cathedral along them, it seems to rise in the air.
In Moscow there are also buildings made using ZS. For example, St. Basil's Cathedral
However, development using the principles of symmetry prevails.
For example, the Kremlin and the Spasskaya Tower.
The height of the Kremlin walls also nowhere reflects the principle of the Civil Code regarding the height of towers, for example. Or take the Russia Hotel, or the Cosmos Hotel.
At the same time, buildings built according to the GS principle represent a larger percentage in St. Petersburg, and these are street buildings. Liteiny Avenue.
So the Golden Ratio uses a ratio of 1.68 and the symmetry is 50/50.
That is, symmetrical buildings are built on the principle of equality of sides.
Another important characteristic of the ES is its dynamism and tendency to unfold, due to the sequence of Fibonacci numbers. Whereas symmetry, on the contrary, represents stability, stability and immobility.
In addition, the additional ZS introduces into the plan of St. Petersburg an abundance of water spaces, splashed throughout the city and dictating the city’s subordination to their bends. And Peter’s diagram itself resembles a spiral or an embryo at the same time.
The Pope, however, expressed a different version of why Muscovites and St. Petersburg residents have “headaches” when visiting the capitals. Dad relates this to the energies of cities:
St. Petersburg – has masculine and accordingly masculine energies,
Well, Moscow - accordingly - is feminine and has feminine energies.
So, for residents of the capitals, who are attuned to their specific balance of feminine and masculine in their bodies, it is difficult to readjust when visiting a neighboring city, and someone may have some difficulties with the perception of one or another energy and therefore the neighboring city may not be at all be in love!
This version is also confirmed by the fact that all Russian empresses ruled in St. Petersburg, while Moscow saw only male tsars!
Resources used.
Leonardo Fibonacci is one of greatest mathematicians Middle Ages. In one of his works, “The Book of Calculations,” Fibonacci described the Indo-Arabic system of calculation and the advantages of its use over the Roman one.Definition
Fibonacci numbers or Fibonacci Sequence is a number sequence that has a number of properties. For example, the sum of two adjacent numbers in a 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 coefficients, i.e. constant ratios.
The Fibonacci sequence starts like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...
Properties of the Fibonacci sequence
1. The ratio of each number to the next one tends more and more to 0.618 as the serial number increases. The ratio of each number to the previous one tends to 1.618 (the reverse of 0.618). The number 0.618 is called (FI).
2. When dividing each number by the one following it, the number after one is 0.382; on the contrary - respectively 2.618.
3. Selecting the ratios in this way, we obtain the main set of Fibonacci ratios: ... 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 slower and slower) tends to some constant relationship. However, this ratio is irrational, that is, it represents 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 its predecessor (for example, 13:8), the result will be a value that fluctuates around the irrational value 1.61803398875... and sometimes exceeds it, sometimes does not reach it. But even after spending Eternity on this, it is impossible to find out the ratio exactly, down to the last decimal digit. For the sake of brevity, we will present it in the form of 1.618. Special names began to be given to this ratio even before Luca Pacioli (a medieval mathematician) called it the Divine proportion. Among its modern names are the Golden Ratio, the Golden Average and the ratio of rotating squares. Kepler called this relationship one of the “treasures of geometry.” In algebra, it is generally accepted to be denoted by the Greek letter phi
Let's imagine the golden ratio using the example of a segment.
Consider a segment with ends A and B. Let point C divide the segment AB so that,
AC/CB = CB/AB or
You can imagine it something like this: A----------C--------B
The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole.
Segments of the golden proportion are expressed as an 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 proportions and the golden ratio in nature and history
It is important to note that Fibonacci seemed to remind humanity of his sequence. It was known to the ancient Greeks and Egyptians. And indeed, since then, patterns described by Fibonacci ratios have been found in nature, architecture, fine arts, mathematics, physics, astronomy, biology and many other fields. It's amazing how many constants can be calculated using the Fibonacci sequence, and how its terms appear in a huge number of combinations. However, it is no 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 sink is twisted in a spiral. If you unfold it, you get a length slightly shorter than the length of the snake. The small ten-centimeter shell has a spiral 35 cm long. The shape of the spirally curled shell attracted the attention of Archimedes. The fact is that the ratio of the dimensions of the shell curls is constant and equal to 1.618. Archimedes studied the spiral of shells and derived the equation of the spiral. The spiral drawn according to this equation is called by his name. The increase in her step is always uniform. Currently, the Archimedes spiral is widely used in technology.
2. Plants and animals . Goethe also emphasized the tendency of nature towards spirality. The helical and spiral arrangement of leaves on tree branches was noticed a long time ago. The spiral was seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians shed light on these amazing phenomena nature. It turned out that in the arrangement of leaves on a branch of sunflower seeds and pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden ratio manifests itself. The spider weaves its web in a spiral pattern. A hurricane is spinning like 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 herbs grows an unremarkable plant - chicory. Let's take a closer look at it. A shoot has formed from the main stem. The first leaf was located right there. The shoot makes a strong ejection into space, stops, releases a leaf, but this time it is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again. If the first emission is taken as 100 units, then the second is equal to 62 units, the third - 38, the fourth - 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.
The lizard is viviparous. At first glance, the lizard has proportions that are pleasant to our eyes - the length of its tail is related to the length of the rest of the body as 62 to 38.
In both the plant and animal worlds, the formative tendency of nature persistently breaks through - symmetry regarding 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 division into symmetrical parts and golden proportions. The parts reveal a repetition of the structure of the whole.
Pierre Curie at the beginning of this century formulated a number of profound ideas about symmetry. He argued that one cannot consider the symmetry of any body without taking into account the symmetry environment. The laws of golden symmetry are manifested in the energy transitions of elementary particles, in the structure of some chemical compounds, in planetary and cosmic systems, in the gene structures of living organisms. These patterns, as indicated above, exist in the structure of individual human organs and the body as a whole, and also manifest themselves in the biorhythms and functioning of the brain and visual perception.
3. Space. From the history of astronomy it is known that I. Titius, a German astronomer of the 18th century, with the help of this series (Fibonacci) found a pattern and order in the distances between the planets of the solar system
However, one case that seemed to contradict the law: there was no planet between Mars and Jupiter. Focused observation of this part of the sky led to the discovery of the asteroid belt. This happened after the death of Titius in early XIX V.
The Fibonacci series is widely used: it is used to represent the architectonics of living beings, man-made structures, and the structure of 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 unsolvable puzzle of number combinations. The remarkable ingenuity, skill, time and labor that the pyramid's architects employed in constructing the eternal symbol indicate the extreme importance of the message they wished to convey to future generations. Their era was preliterate, prehieroglyphic, and symbols were the only means of recording discoveries. The key to the geometric-mathematical secret of the Pyramid of 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.
Area of a triangle
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 edge divided by the height leads to the ratio Ф=1.618. The height of 484.4 feet corresponds to 5813 inches (5-8-13) - these are the numbers from the Fibonacci sequence. These interesting observations suggest that the design of the pyramid is based on the proportion Ф=1.618. Some modern scholars are inclined to interpret that the ancient Egyptians built it for the sole purpose of passing on knowledge that they wanted to preserve for future generations. Intensive studies of the pyramid at Giza showed how extensive the knowledge of 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 were the Egyptian pyramids built in accordance with the perfect proportions of the golden ratio, the same phenomenon was found in the Mexican pyramids. The idea arises that both the Egyptian and Mexican pyramids were erected at approximately the same time by people of common origin.
About the Fibonacci sequence of the Illuminati order.
This is essentially, kept 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
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43192637289106705033992
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47275977256550861548754
35748264718141451270006
02389016207773224499435
30889990950168032811219
43204819643876758633147
98571911397815397807476
15077221175082694586393
20456520989698555678141
06968372884058746103378
10544439094368358358138
11311689938555769754841
49144534150912954070050
19477548616307542264172
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18328599130396072014455
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70189409886400764436170
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51432117348151005590134
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70930858809287570345050
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09778726841611763250386
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71116330725869883258710
33632223810980901211019
89917684149175123313401
52733843837234500934786
04979294599158220125810
45982309255287212413704
36149102054718554961180
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56044317847985845397312
86301625448761148520217
06440411166076695059775
78325703951108782308271
06478939021115691039276
83845386333321565829659
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33953679593123221343732
09957498894699565647360
07295999839128810319742
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95518947781726914158911
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10210119190218360675097
30895752895774681422954
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06480367930414723657203
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21901893345963786087875
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02780286560432349428373
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34362652689448885527202
74778747335983536727761
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32719168451230602362789
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62393985709255070274069
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31693397287289181036640
83269856988254438516675
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In the very records of the members of this secret society, this set of numbers plays a very important role. But which one? What was the Illuminati hiding behind these numbers?
The fact is that, according to surviving data, the Illuminati had 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 there may be a universal code of life, a recipe for the philosopher’s stone, etc....
Fibonacci sequence in mathematics and in nature
Fibonacci sequence, known to everyone from the film "The Da Vinci Code" - a series of numbers described in the form of a riddle by the Italian mathematician Leonardo of Pisa, better known by the nickname Fibonacci, in the 13th century. Briefly the essence of the riddle:
Someone placed a pair of rabbits in a certain enclosed space to find out how many pairs of rabbits would be born during the year, if the nature of rabbits is such that every month a pair of rabbits gives birth to another pair, and they become capable of producing offspring when they reach two months of age.
The result is the following sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 , where the number of pairs of rabbits in each of the twelve months is shown, separated by commas.
This sequence can be continued indefinitely. Its essence is that each next number is the sum of the previous two.
This sequence has a number of mathematical features that definitely need to be touched upon. This sequence asymptotically (approaching slower and slower) tends to some constant ratio. However, this ratio is irrational, 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.
Thus, the ratio of any member of the sequence to the one preceding it fluctuates around the number 1,618 , sometimes exceeding it, sometimes not achieving it. The ratio to the following similarly approaches the number 0,618 , which is inversely proportional 1,618 . If we divide the elements of the sequence through one, we get numbers 2,618 And 0,382 , which are also inversely proportional. These are the so-called Fibonacci ratios.
What is all this for? This is how we approach one of the most mysterious natural phenomena. Fibonacci essentially did not discover anything new, he simply reminded the world of such a phenomenon as Golden Ratio, which is not inferior in importance to the Pythagorean theorem
We distinguish all the objects around us by their shape. We like some more, some less, some are completely off-putting. Sometimes interest can be dictated by the life situation, and sometimes by the beauty of the observed object. The symmetrical and proportional shape promotes the best visual perception and evokes a feeling of beauty and harmony. A complete image always consists of parts different sizes, which are in a certain relationship with each other and the whole.
Golden ratio- the highest manifestation of the perfection of the whole and its parts in science, art and nature.
If on simple example, then the Golden Ratio is the division of a segment into two parts in such a ratio in which the larger part is related to the smaller one, as their sum (the entire segment) is to the larger one.
If we take the entire segment c behind 1 , then the segment a will be equal 0,618 , line segment b - 0,382 , only in this way will the condition of the Golden Section be met (0.618/0.382= 1,618 ; 1/0,618=1,618 ). Attitude c To a equals 1,618 , A With To b2.618. These are all the same Fibonacci ratios already familiar to us.
Of course there is a golden rectangle, a golden triangle and even a golden cuboid. The proportions of the human body are in many respects close to the Golden Section.
Image: marcus-frings.de
But the fun begins when we combine the knowledge we have gained. The figure clearly shows the relationship between the Fibonacci sequence and the Golden Ratio. We start with two squares of the first size. Add a second size square on top. Draw a square next to it with a side equal to the sum of the sides of the previous two, third size. By analogy, a square of size five appears. And so on until you get tired, the main thing is that the length of the side of each next square is equal to the sum of the lengths of the sides of the two previous ones. We see a series of rectangles whose side lengths are Fibonacci numbers, and, oddly enough, they are called Fibonacci rectangles.
If we draw smooth lines through the corners of our squares, we will get nothing more than an Archimedes spiral, the increment of which is always uniform.
Doesn't remind you of anything?
Photo: ethanhein on Flickr
And not only in the shell of a mollusk you can find Archimedes’ spirals, but in many flowers and plants, they’re just not so obvious.
Aloe multifolia:
Photo: brewbooks on Flickr
Photo: beart.org.uk
Photo: esdrascalderan on Flickr
Photo: manj98 on Flickr
And now it’s time to remember the Golden Section! Are some of the most beautiful and harmonious creations of nature depicted in these photographs? And that's not all. If you look closely, you can find similar patterns in many forms.
Of course, the statement that all these phenomena are based on the Fibonacci sequence sounds too loud, but the trend is obvious. And besides, the sequence itself is far from perfect, like everything in this world.
There is an assumption that the Fibonacci sequence is an attempt by nature to adapt to a more fundamental and perfect golden ratio logarithmic sequence, which is almost the same, only it starts from nowhere and goes to nowhere. Nature definitely needs some kind of whole beginning from which it can start; it cannot create something out of nothing. The ratios of the first terms of the Fibonacci sequence are far from the Golden Ratio. But the further we move along it, the more these deviations are smoothed out. To define any sequence, it is enough to know its three terms, following each other. But not for the golden sequence, two are enough for it, it is geometric and arithmetic progression simultaneously. One might think that it is the basis for all other sequences.
Each term of the golden logarithmic sequence is a power of the Golden Ratio ( z). Part of the series looks something 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 Golden Ratio to three decimal places, we get z=1.618, then the series 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 term can be obtained not only by multiplying the previous one by 1,618 , but also by adding the two previous ones. Thus, exponential growth in a sequence is achieved by simply adding two adjacent elements. It's a series without beginning or end, and that's what the Fibonacci sequence tries to be like. Having quite definite beginning, she strives for the ideal without ever achieving it. That is life.
And yet, in connection with everything we have seen and read, quite logical questions arise:
Where did these numbers come from? Who is this architect of the universe who tried to make it ideal? Was everything ever the way he wanted? And if so, why did it go wrong? Mutations? Free choice? What will be next? Is the spiral curling or unwinding?
Having found the answer to one question, you will get the next one. If you solve it, you'll get two new ones. Once you deal with them, three more will appear. Having solved them too, you will have five unsolved ones. Then eight, then thirteen, 21, 34, 55...
