Irrational Number Phi 1 618. Numerology. Who And How Discovered ...

Number 13 On 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 Nature's understanding, this is a flower waiting for pollination.

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

Number 11 On the energies of the Cosmic Channel, the number 11 personifies the energy of two worlds: manifested and unmanifested. Symbolically, this is the Sun reflected in water, two Suns: in the sky and in water, two units. This 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. The number 12 contains one - the sign of the leader and two - the sign of the owner

Number 13 On 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 the number 13 looks like the same Wind Rose as the number 4, but without space restrictions.

Number 14 On the energies of the Cosmic Channel, the number 14 is the messenger of the Cosmos. Royal number 13 is not the last in the levels of development of our civilization. There is one more day in 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 characteristic 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 certain capitals, "factories" and "steamboats". Perhaps he will not squander the inheritance, and will pass it on to his heirs. His personal preferences are vague - whether he loves harmony and feels, or loves power and

So, please meet... PHI number = 1.618 * And it should not be confused with "pi", because, as mathematicians say:- the letter "H" makes it much cooler!Do you know that...

– The PHI number is the most important and significant number in the visual arts. The PHI number is considered by all to be the most beautiful number in the universe.

This number is derived from the Fibonacci sequence: - mathematical progression, known not only to those that the sum of two neighboring numbers in it is equal to the next number, but also because that the quotient of two neighboring numbers has a unique property - proximity to the number 1, 618, that is, to the number PHI!

Despite its almost mystical origin, the PHI number has played a unique role in its own way. The role of the brick in the foundation of building all life on earth. All plants, animals and even human beings are endowed with physical proportions, approximately equal to the root of the ratio of the number of PHI to 1.

This omnipresence of PHI in nature indicates the connection of all living beings. It used to be believed that the PHI number was predetermined by the Creator of the universe. Scientists of antiquity called the number = 1.618 "divine proportion."

Do you know that if you divide the number of females by the number of males in any hive in the world, then you always get the same number? PHI number.

If you look at the spiral-shaped sea shell nautilus (cephalopod), then the ratio of the diameter of each turn of the spiral to the next = 1.618.

Again PHI - Divine Proportion.

• Sunflower flower with mature seeds.
• Sunflower seeds are arranged in spirals, counterclockwise.
• The ratio of the diameter of each of the spirals to the diameter of the next = PHI.

If you look at the spiraling leaves on the cob of corn, arrangement of leaves on plant stems, segmentation parts of insect bodies, then all of them in their structure obediently follow the law of "divine proportion".

What does this have to do with art? The famous drawing by Leonardo da Vinci depicting a naked man in a circle. "Vitruvian Man" (named after Marcus Vitruvius, a brilliant Roman architect, who praised "divine proportion" in his Ten Books on Architecture).

No one better than da Vinci understood the divine structure of the human body, its structure. Da Vinci was the first to show that the human body is made up of "building blocks" the ratio of the proportions of which is always equal to our cherished number.

Don't believe? Then, when you go to the shower, do not forget to take a centimeter with you. Everyone is so arranged. Both boys and girls. Check it out yourself.

Measure the distance from the top of your head to the floor. Then divide by your height. And see what the number will be. Measure from shoulder to fingertips then divide it by the distance from the elbow to the same fingertips. The distance from the top of the thigh divided by the distance from the knee to the floor and again PHI. Phalanges of the fingers. Phalanges of the toes. And again PHI... PHI...

As you can see, behind the apparent chaos of the world lies order. And the ancients, who discovered the number PHI, were sure that they had found that building stone, which the Lord God used to create the world. Many of us glorify Nature, as the pagans did, It's just that they don't fully understand why.

Man simply plays by the rules of Nature, and therefore art is nothing but as an attempt by man to imitate the beauty created by the Creator of the universe.

Considering the works of Michelangelo,

Albrecht Durer,

Leonardo da Vinci

And many other artists (J.-L. David. Cupid and Psyche. 1817)

Then we will see that each of them strictly followed the "divine proportions" in the construction of their compositions.

This magic number is found in architecture, in the proportions of the Greek Parthenon,

Egyptian Pyramids,

Even the UN buildings in New York.

PHI manifested itself in the strictly organized structures of Mozart's sonatas, in Beethoven's Fifth Symphony, as well as in the works of Bartók, Debussy and Schubert.

The PHI number was used in Stradivari's calculations when creating his unique violin.

Five-pointed star - this symbol is one of the most powerful images. It is known as the pentagram, or pentacle, as the ancients called it.

And for many centuries and in many cultures, this symbol was considered both divine and magical. Because when you draw a pentagram, the lines are automatically divided into segments, corresponding to the "divine proportion". The ratio of line segments in a five-pointed star is always equal to the number of PHI, which makes this symbol the highest expression of "divine proportion". It is for this reason that the five-pointed star has always been a symbol of beauty and perfection. and was associated with the goddess and the sacred feminine.

It is proved that Leonardo was a consistent admirer of ancient religions, associated with the feminine. The Last Supper has become one of the most amazing examples of worship Leonardo da Vinci Golden Section.

The Renaissance is associated with the names of such "titans", like Leonardo da Vinci, Michelangelo, Raphael, Nicolaus Copernicus, Albert Durer, Luca Pacioli. And the first place in this list is rightfully occupied by Leonardo da Vinci, the greatest artist, engineer and scientist of the Renaissance.

There is a lot of authoritative evidence that it was Leonardo da Vinci who was one of the first to introduce the term "Golden Section" itself. The term "golden section" (aurea sectio) comes from Claudius Ptolemy, who gave this name to the number 0.618. This term was fixed and became popular thanks to Leonardo da Vinci, who often used it.

For Leonardo da Vinci himself, art and science were inextricably linked. Giving the palm to painting in the "dispute of the arts", Leonardo da Vinci understood it as a universal language (similar to mathematics in the field of sciences), which embodies by means of proportion and perspective all the manifold manifestations of the rational principle reigning in nature. According to the artistic canons of Leonardo, the golden ratio corresponds not only dividing the body into two unequal parts by the waist line, at which the ratio of the greater part to the lesser is equal to the ratio of the whole to the greater part (this ratio is approximately 1.618).

The ratio of the height of the face (to the roots of the hair) to the vertical distance between the arches of the eyebrows and the lower part of the chin; distance between the bottom of the nose and the bottom of the chin to the distance between the corners of the lips and the bottom of the chin This is also the golden ratio.

The most striking evidence of the enormous role of Leonardo da Vinci in the development of the theory of the Golden Section is its influence on the work of the outstanding Italian Renaissance mathematician Luca Pacioli who called himself Luca di Borgo San Sepolcro.

The latter was already a famous mathematician, author of the book "Sum on Arithmetic, Geometry, Proportions and Proportions", when he met Leonardo da Vinci. Leonardo da Vinci became the third great man (after Piero della Francesco and Leon Battista Alberti), met on the life path of Luca Pacioli.

It is believed that it was under the influence of Leonardo da Vinci that Luca Pacioli begins to write his "the second great book", called by him "On the Divine Proportion". This book was published in 1509. Leonardo made illustrations for this book. About the authorship of Leonardo, the testimony of Pacioli himself has been preserved: “... those were made by the most worthy painter, perspectiveist, architect, musician and all the perfections gifted by Leonardo da Vinci, Florentine, in the city of Milan ... ".

Vitruvius also described other anthropometric patterns. Actually, "Vitruvian Man" in the literature of subsequent centuries was called such images, showing the proportions of the human body and their relationship with architecture.

1. C. Caesariano. Edition of Vitruvius, 3rd vol. Como, 1521

2. Ibid. Unlike its square counterpart, this one has an erection

3. J. Martin. Architecture, or the art of building. Paris, 1547. Engraving by J. Goujon

4. F. Giocondo. Manuscript of Vitruvius with Giocondo corrections, with illustrations and a table of contents for reading and understanding. 3rd volume. Venice, 1511

5. P. Cataneo. The first four books on architecture. Venice, 1554. The figure is inscribed in the cruciform plan of the church

6. V. Scamozzi. The idea of ​​universal architecture. Part I, book 1. London, 1676. Central fragment of engraving

Nowadays, the Vitruvian Man in the version of Da Vinci is no longer perceived like a geometric diagram of the human body. He has become nothing less than into a symbol of man, humanity and the universe.

And we don't mind...

Even true opinions are worth little until someone connects them with the link of causal reasoning.

D. Brown's book "The Da Vinci Code" helped me to start developing this material. As a code, the hero of the book uses several numbers from the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, ... I found additional material on this topic and. As a result, many of my lesson developments have been replenished.

For example, the first lesson of mathematics in the fifth grade on the topic: "Denotation of natural numbers." Speaking about the infinite sequence of natural numbers, I noted the presence of other series, for example, the Fibonacci series and the series of "triangular numbers": 1, 3, 6, 10, ...

In the eighth grade, when studying irrational numbers, along with the number "pi", I give the number "phi" (Ф = 1.618 ...). (D. Brown calls this number "pfi", which, according to the author, is even cooler than "pi"). I ask students to think of two numbers, and then form a series according to the "principle" of the Fibonacci series. Each calculates its sequence up to the tenth term. For example, 7 and 13. Let's build a sequence: 7, 13, 20, 33, 53, 86, 139, 225, 364, 589, ... Even when dividing the ninth term by the eighth, the Fibonacci number appears.

Life story.

The Italian merchant Leonardo of Pisa (1180-1240), better known by the nickname Fibonacci, was an important medieval mathematician. The role of his books in the development of mathematics and the dissemination of mathematical knowledge in Europe can hardly be overestimated.

The life and scientific career of Leonardo is closely connected with the development of European culture and science.

The Renaissance was still far away, but history gave Italy a short period of time that could well be called a rehearsal for the impending Renaissance. This rehearsal was led by Frederick II, Holy Roman Emperor. Brought up in the traditions of southern Italy, Frederick II was internally deeply far from European Christian chivalry. Frederick II did not recognize knightly tournaments at all. Instead, he cultivated mathematical competitions, in which opponents exchanged not blows, but problems.

At such tournaments, the talent of Leonardo Fibonacci shone. This was facilitated by a good education, which was given to his son by the merchant Bonacci, who took him with him to the East and assigned Arab teachers to him. The meeting between Fibonacci and Frederick II took place in 1225 and was an event of great importance for the city of Pisa. The emperor rode at the head of a long procession of trumpeters, courtiers, knights, officials, and a wandering menagerie of animals. Some of the problems that the Emperor posed to the famous mathematician are detailed in the Book of the Abacus. Fibonacci, apparently, solved the problems posed by the Emperor, and forever became a welcome guest at the Royal Court. When Fibonacci revised the Book of the Abacus in 1228, he dedicated the revised edition to Frederick II. In total, he wrote three significant mathematical works: the Book of the Abacus, published in 1202 and reprinted in 1228, Practical Geometry, published in 1220, and the Book of Quadratures. These books, surpassing in their level Arabic and medieval European writings, taught mathematics almost until the time of Descartes. As stated in documents from 1240, the admiring citizens of Pisa said that he was "a sensible and erudite man", and not so long ago, Joseph Guise, the editor-in-chief of the Encyclopædia Britannica, declared that future scientists at all times "will pay their debt to Leonardo of Pisa, as one of the world's greatest intellectual pioneers."

Rabbit problem.

Of greatest interest to us is the essay "The Book of the Abacus". This book is a voluminous work containing almost all the arithmetic and algebraic information of that time and played a significant role in the development of mathematics in Western Europe over the next few centuries. In particular, it was from this book that Europeans got acquainted with Hindu (Arabic) numerals.

The material is explained by examples of tasks that make up a significant part of this path.

In this manuscript, Fibonacci placed the following problem:

"Someone placed a pair of rabbits in a certain place, fenced on all sides by a wall, in order to find out how many pairs of rabbits would be born during the year, if the nature of the rabbits is such that in a month a pair of rabbits gives birth to another pair, and rabbits give birth from the second months after his birth.

It is clear that if we consider the first pair of rabbits as newborns, then in the second month we will still have one pair; on the 3rd month - 1+1=2; on the 4th - 2 + 1 = 3 pairs (because of the two available pairs, only one pair gives offspring); on the 5th month - 3 + 2 = 5 pairs (only 2 couples born on the 3rd month will give offspring on the 5th month); on the 6th month - 5 + 3 = 8 pairs (because only those pairs that were born on the 4th month will give offspring), etc.

Thus, if we denote the number of pairs of rabbits available in the nth month as Fk, then F1=1, F2=1, F3=2, F4=3, F5=5, F6=8, F7=13, F8=21 etc., and the formation of these numbers is regulated by the general law: Fn=Fn-1+Fn-2 for all n>2, because the number of pairs of rabbits in the nth month is equal to the number Fn-1 of pairs of rabbits in the previous month plus the number newly born pairs, which coincides with the number of Fn-2 pairs of rabbits born in the (n-2)th month (because only these pairs of rabbits give offspring).

The numbers Fn that form the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... are called "Fibonacci numbers", and the sequence itself is called the Fibonacci sequence.

Special names for this ratio began to be given even before Luca Pacioli (a medieval mathematician) called it the Divine Proportion. Kepler called this relation one of the treasures of geometry. In algebra, its designation is generally accepted by the Greek letter "phi" (Ф=1.618033989…).

The following are the ratios of the second term to the first, the third to the second, the fourth to the third, and so on:

1:1 = 1.0000, which is less than phi by 0.6180

2:1 = 2.0000, which is 0.3820 more phi

3:2 = 1.5000, which is less than phi by 0.1180

5:3 = 1.6667, which is 0.0486 more phi

8:5 = 1.6000, which is less than phi by 0.0180

As we move along the Fibonacci summation sequence, each new term will divide the next with more and more approximation to the unattainable "phi". Fluctuations of ratios around the value of 1.618 by a larger or smaller value, we will find in the Elliott Wave Theory, where they are described by the Rule of Alternation. It should be noted that it is precisely the approximation to the number "phi" that occurs in nature, while mathematics operates with a "pure" value. It was introduced by Leonardo da Vinci and called the "golden section" (golden proportion). Among its modern names there are such as "golden mean" and "rotating squares ratio". The golden ratio is the division of the segment AC into two parts in such a way that its greater part AB relates to the smaller part BC in the same way that the entire segment AC relates to AB, that is: AB: BC = AC: AB = F (exact irrational number " fi").

When dividing any member of the Fibonacci sequence by the next one, the value inverse to 1.618 is obtained (1: 1.618=0.618). This is also a very unusual, even remarkable phenomenon. Since the original ratio is an infinite fraction, this ratio must also have no end.

When dividing each number by the next one after it, we get the number 0.382.

Selecting ratios in this way, we obtain the main set of Fibonacci coefficients: 4.235, 2.618, 1.618, 0.618, 0.382, 0.236. All of them play a special role in nature and in particular in technical analysis.

It is simply amazing how many constants can be calculated using the Fibonacci sequence, and how its terms appear in a huge number of combinations. However, it would not be an exaggeration to say that this is not just a number game, but the most important mathematical expression of natural phenomena ever discovered.

These numbers are undoubtedly part of a mystical natural harmony that feels good, looks good, and even sounds good. Music, for example, is based on an 8-note octave. On a piano this is represented by 8 white keys and 5 black keys for a total of 13.

A more visual representation can be obtained by studying spirals in nature and works of art. Sacred geometry explores two types of spirals: the golden section spiral and the Fibonacci spiral. Comparison of these spirals allows us to draw the following conclusion. The golden ratio spiral is perfect: it has no beginning and no end, it continues indefinitely. Unlike it, the Fibonacci spiral has a beginning. All natural spirals are Fibonacci spirals, and works of art use both spirals, sometimes at the same time.

Maths.

The pentagram (pentacle, five-pointed star) is one of the frequently used symbols. The pentagram is a symbol of a perfect person standing on two legs with outstretched arms. We can say that a person is a living pentagram. This is true both physically and spiritually - a person possesses five virtues and manifests them: love, wisdom, truth, justice and kindness. These are the virtues of Christ, which can be represented by a pentagram. These five virtues, necessary for human development, are directly related to the human body: kindness is associated with the feet, justice with the hands, love with the mouth, wisdom with the ears, eyes with the truth.

Truth belongs to the spirit, love to the soul, wisdom to the intellect, kindness to the heart, justice to the water. There is also a correspondence between the human body and the five elements (earth, water, air, fire and ether): will corresponds to earth, heart to water, intellect to air, soul to fire, spirit to ether. Thus, by his will, intellect, heart, soul, spirit, man is connected with the five elements working in the cosmos, and he can consciously work in harmony with it. This is the meaning of another symbol - a double pentagram, a person (microcosm) lives and acts inside the universe (microcosm).

The inverted pentagram pours energy into the earth and is therefore a symbol of materialistic tendencies, while the normal pentagram directs energy upward, thus being spiritual. On one point everyone agrees: the pentagram certainly represents the "spiritual form" of the human figure.

Note CF:FH=CH:CF=AC:CH=1.618. The actual proportions of this symbol are based on a sacred proportion called the golden ratio: this is the position of a point on any line drawn when it divides the line so that the smaller part is in the same ratio to the larger part as the larger part to the whole. In addition, the regular pentagon in the center suggests that the proportions are preserved for infinitesimal pentagons. This "divine proportion" is manifested in each individual ray of the pentagram and helps to explain the awe with which mathematicians have looked at this symbol at all times. Moreover, if the side of the pentagon is equal to one, then the diagonal is equal to 1.618.

Many have tried to unravel the secrets of the Giza pyramid. Unlike other Egyptian pyramids, this is not a tomb, but rather an unsolvable puzzle of numerical 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 pre-literate, pre-hieroglyphic, and symbols were the only means of recording discoveries.

Scientists have discovered that the three pyramids at Giza are arranged in a spiral. In the 1980s, it was found that both the golden spiral and the Fibonacci spiral were present there.

The key to the geometrical-mathematical secret of the Giza pyramid, so long a mystery to mankind, 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 face 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 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 numbers from the Fibonacci sequence.

These interesting observations suggest that the construction of the pyramid is based on the proportion Ф=1.618. Modern scholars lean toward the interpretation that the ancient Egyptians built it for the sole purpose of passing on the knowledge they wanted to preserve for future generations. Intensive studies of the pyramid at Giza showed 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.

Not only the Egyptian pyramids were 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 Egyptian and Mexican pyramids were built at approximately the same time by people of common origin.

Biology.

In the 19th century, scientists noticed that the flowers and seeds of sunflowers, chamomile, scales in pineapple fruits, coniferous cones, etc. are "packed" in double spirals, curling towards each other. At the same time, the numbers of "right" and "left" spirals always refer to each other as neighboring Fibonacci numbers (13:8, 21:13, 34:21, 55:34). Numerous examples of double helixes found throughout nature always follow this rule.

Even Goethe emphasized the tendency of nature to spirality. The spiral 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 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 spins its web in a spiral pattern. A hurricane is spiraling. A frightened herd of reindeer scatter in a spiral. The DNA molecule is twisted into a double helix. Goethe called the spiral "the curve of life."

Any good book will show the nautilus shell as an example. Moreover, in many publications it is said that this is a golden ratio spiral, but this is not true - this is a Fibonacci spiral. You can see the perfection of the arms of the spiral, but if you look at the beginning, it doesn't look so perfect. Its two innermost bends are actually equal. The second and third bends are a little closer to phi. Then, finally, this elegant smooth spiral is obtained. Remember the relationship of the second term to the first, the third to the second, the fourth to the third, and so on. It will be clear that the mollusk follows the mathematics of the Fibonacci series exactly.

Fibonacci numbers show up in the morphology of various organisms. For example, starfish. Their number of rays corresponds to a series of Fibonacci numbers and is equal to 5, 8, 13, 21, 34, 55. The well-known mosquito has three pairs of legs, the abdomen is divided into eight segments, and there are five antennae on the head. The mosquito larva is divided into 12 segments. The number of vertebrae in many domestic animals is 55. The proportion of "phi" is also manifested in the human body.

Drunvalo Melchizedek in The Ancient Secret of the Flower of Life writes: "Da Vinci calculated that if you draw a square around the body, then draw a diagonal from the feet to the tips of the outstretched fingers, and then draw a parallel horizontal line (the second of these parallel lines) from the navel to the side of the square, then this horizontal line will intersect the diagonal exactly in phi proportion, as well as the vertical line from the head to the feet.If we consider that the navel is at that perfect point, and not slightly higher for women or slightly lower for men, then this means that the human body is divided in phi proportion from the top of the head to the feet... If these lines were the only ones where there is a phi proportion in the human body, it would probably only be an interesting fact. In fact, the phi proportion is found in thousands of places throughout the body , and this is not just a coincidence.Here are some clear places in the human body where the proportion of phi is found.The length of each phalanx of the finger is in the proportion of phi to the next phalanx ... The same proportion is noted for all fingers and toes. If you correlate the length of the forearm with the length of the palm, then you get the proportion of phi, just as the length of the shoulder refers to the length of the forearm. Or take the length of the leg to the length of the foot and the length of the thigh to the length of the leg. The proportion of phi is found throughout the skeletal system. It is usually marked in places where something bends or changes direction. It is also found in the ratio of the sizes of some parts of the body to others. When you study it, you are always surprised.”

Conclusion.

Although he was the greatest mathematician of the Middle Ages, the only monuments to Fibonacci are a statue opposite the Leaning Tower of Pisa across the Arno River and two streets that bear his name, one in Pisa and the other in Florence.

If you put your open palm vertically in front of you, pointing your thumb to your face, and, starting with the little finger, successively clench your fingers into a fist, you get a movement that is a Fibonacci spiral.

Literature

1. Ensenzberger Hans Magnus Spirit of number. Math Adventures. - Per. from English. - Kharkov: Book Club "Family Leisure Club", 2004. - 272 p.

2. Encyclopedia of symbols / comp. V.M. Roshal. - Moscow: AST; St. Petersburg; Owl, 2006. - 1007 p.