Livio-GoldenRatio-ajk.txt (* txt -> HTML)

1,4396,244,mat,eng,20150227,20150403,5,Mario Livio:The Golden Ratio

20150227-20150403, 244 pages, 5* SalesInfo o eng

1. | Sisällysluettelo | Contents | Содержание | |

2. | Muistiinpanot | Highlights | Примечание | |

3. | Sanasto | Vocabulary | Словарь | |

4. | Yhteenvedot | Reviews | Резюме | |

5. | Huomautukset | Remarks | Замечания |

1 | 0001 | Prelude to a number |

11 | 0002 | The pitch and the Pentagram |

13 | 000201 | THREE IS A CROWD |

15 | 000202 | COUNTING MY NUMBERLESS FINGERS |

21 | 000203 | OUR NUMBERS, OUR GODS |

23 | 000204 | PYTHAGORAS AND THE PYTHAGOREANS |

35 | 000205 | TO THE RATIONAL BEING ONLY THE IRRATIONAL IS UNENDURABLE |

40 | 0003 | under a star-Y-POINTING A PYRAMID |

58 | 000301 | PLATO |

63 | 000302 | Aristotle: cosmic fifth essence (" quintessence"). |

66 | 000303 | THE VIRGIN'S PLACE |

69 | 000304 | EXTREME AND MEAN RATIO |

75 | 000305 | SURPRISES GALORE |

77 | 000306 | TOWARD THE DARK AGES |

84 | 000307 | ALL THE THOUGHTS OF A RABBIT ARE RABBITS |

91 | 000308 | AS THE SUNFLOWER TURNS ON HER GOD |

102 | 000309 | ALTHOUGH CHANGED, I RISE AGAIN THE SAME |

112 | 000310 | UNSUNG HERO OF THE RENAISSANCE? |

119 | 000311 | MYSTERIUM COSMOGRAPHICUM |

137 | 000312 | :THE ARTIST'S SECRET GEOMETRY? |

150 | 000313 | THE SENSES DELIGHT IN THINGS DULY PROPORTIONED |

157 | 000314 | GOLDEN MUSIC |

173 | 000315 | THE TILED ROAD TO QUASI-CRYSTALS |

180 | 000316 | FRACTALS |

189 | 000317 | A GOLDEN TOUR OF WALL STREET |

193 | 000318 | RABBITS AND COIN TOSSES |

202 | 000319 | THE UNREASONABLE POWER OF MATHEMATICS |

The most famous of these is the number pi (n), which is the ratio of the circumference of any circle to its diameter. The value o...

Less known than pi is another number, phi (cp), which is in many respects even more fascinating. Suppose I ask you, for exa...

"Golden Number," "Golden Ratio," and "Golden Section." A book published

The first clear definition of what has later become known as the Golden Ratio was given around 300 B.C. by the founder o...

In other words, if we look at Figure 2, line AB is certainly longer than the segment AC; at the same time, the segment AC ...

Albert Einstein (1879- 1955) valued so much. In Einstein's own words: "The fairest thing we can experience is the ...

The realization that there exist numbers, like the Golden Ratio, that go on forever without displaying any repetition or patt...

The precise date for the discovery of numbers that are neither whole nor fractions, known as irrational numbers, i...

After all, since the Golden Ratio has been denned as a geometrical proportion, perhaps we should not be too astonish...

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As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to r...

-ALBERT EINSTEIN (1879- 1955)

I see a certain order in the universe and math is one way of making it visible.

—MAY SARTON (1912- 1995)

No one knows for sure when humans started to count, that is, to measure multitude in a quantitative way.

In fact, we do not even know with certainty whether numbers like "one," "two," "three" (the cardinal numbers) p...

Originally it was assumed that counting developed specifically to address simple day- to-day needs, which clearly argued ...

that required the successive appearanc...

Clearly, an even bigger mental leap was required to move from the simple counting of objects to an actual understanding o...

Is it one wolf or a pack of wolves?—

Even in English, different names often are associated with the same numbers of different aggregations. We say "a yoke ...

Many studies show that the largest number we are able to capture at a glance, without counting, is about four or five.

Finally, five men were sent: as before, all entered the tower, and one remained while the other four came out and wen...

Thus, while normal plurals end in "im" (for items considered masculine) or "ot" (for feminine items), the plural form for eyes...

For practical reasons, no symbolic system that has a uniquely different name or different representing object for every ...

We are so familiar in everyday life with base 10 that it is almost difficult to imagine that other bases could have been chosen.

This important rule of position, the place- value system, was first invented by the Babylonians (who used 60 as their base,...

In case you wonder, by the way, where "eleven" and "twelve" in English came from, "eleven" derives from "an" (one)...

In some Malay-Polynesian languages, the word for "hand," "lima," is actually the same as the word for "five."

For the Inuit (Eskimo) people, for example, the number "twenty" is expressed by a phrase with the meaning "a man is com...

L'Opital de Quinze-Vingts (The Hospital of Fifteen Twenties), because it was originally designed to contain 300 beds for blind ...

If we write the number 555 in a purely sexagesimal system, what we mean is 5 x (60) 2 + 5 x (60) + 5. or 18.305 in our... 60: It is the first number that is divisible by 1, 2, 3, 4, 5, and 6.

Base 12, for which we find many vestiges even

today— for example, in the British system of weights and measures— may have had its origins in the number of joints in the ...

base 18, then 4 x 5 = 20 will indeed be written as 12, because 20 is 1 unit of 18 and 2 units of

What lends plausibility to this explanation is of course the fact that Charles Dodgson (" Lewis Carroll" was his pen name) lecture...

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Pythagoras was born around 570 B.C. in the island of Samos in the Aegean Sea (off Asia Minor), and he emigrated sometim...

Pythagoras and the Pythagoreans are best known for their presumed role in the development of mathematics and for th... numbers p2 - q2; 2pq; p2 - q2. For example, suppose q is 1 and p is 4. The...

Porphyry says about Pythagoras: "He himself could hear the harmony of the Universe, and understood the music of...

The number 1, for example, was considered the generator of all other numbers and thus not regarded as a nu...

The number 2 was the first female number and also the number of

opinion and of division.

The fact that 1+2 + 3+ 4 = 10 generated a close association between 10 and 4. At the same time, this relation m...

Five represented the union of the first female number, 2, with the first male number, 3, and as such it was the num...

The nineteenth-century mathematician Leopold Kronecker (1823- 1891) expressed his opinion on this matter by ...

For those during the Renaissance who had trouble memorizing how to add or subtract fractions, some writers of mathematical...

mathematics. I Hinhlinht

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean...

pupil of the ancient Greeks, when it comes to mathematics, science, philosophy, art, and literature.

In the span of only four hundred years, for example, from Thales of Miletus (at ca. 600 B.C.) to "the Great Geometer" Apolloni...

Plato (428/ 427 B.C.- 348/ 347 B.C.), one of the most influential minds of ancient Greece and of western civilization in ge...

Plato: rather than observing the stars in their motions, he advocates to "leave the heavens alone" and concentrate on the...

The distinction here is between the beauty of the cosmos itself and the beauty of the theory that explains the universe.

optimal number of citizens in a state is 5,040 because: (a) it is the product of 12, 20, and 21; (b) the twelfth part of it ca...

61 1081: 20150303@ sum nf the ruhes nf the sides nf the

The result of the Michelson-Morley experiment set Einstein on the road to the theory of relativity.

a surprising turn of events, in 1998 two groups of astronomers discovered that not only is our universe expanding (a fact air... total shock,

gravitational force exerted by all the matter in the universe should cause the cosmic expansion to decelerate.

"dark energy"

that manifests itself as a repulsive force.

Luc Besson's 1997 science fiction movie The Fifth Element, the "fifth element" of the title was taken to be the life force i...

Plato, the complex phenomena that we observe in the universe are not what really matters; the truly fundamental things ar...

The original fascination of the Pythagoreans with polyhedra may have originated from observations of pyrite cr...

Pyrite, commonly known as fool's gold, often has crystals with a dodecahedral shape.

Golden Rectangle (a rectangle in which the ratio of length to width is the Golden

Athena Parthenos (Athena the Virgin).

authors, such as Miloutine Borissavlievitch in The Golden Number and the Scientific Aesthetics of Architecture (1958), whil...

Museum) in Alexandria. This institution included a library, which, following an immense gathering effort, was reputed ...

Abraham Lincoln wanted to understand the true meaning of the word "proof in the legal profession, he started to stud...

encounter with Euclid's Elements (at age eleven!) as "one of the great events of my life, as dazzling as first love."

In the Elements, Euclid attempted to encompass most of the mathematical knowledge of his time. Books I to VI de synonymous with Euclid's name (Euclidean

Elements was translated into Arabic three times.

Yusuf ibn Matar, at the request of Caliph Harun ar-Rashid (ruled 786- 809), who is familiar to us through the stories in The...

Western Europe through Latin translations of the Arabic versions. English Benedictine monk Adelard of Bath (ca. 1070- 1145)...

into Latin around 1120. LULQUUII UU1 While Euclid himself may not have been the greatest mathematician who ever lived, he was certainly the greatest tea...

The textbook he wrote remained in use practically unaltered for more than two thousand years,

The Golden Ratio appears in the Elements in several places.

The Golden Ratio has the unique properties that we produce its square by simply adding the number 1 and its red...

Mathematics and the Golden Ratio in particular provide a rich treasury of such surprises.

Because of the "divine" properties attributed to the Golden Ratio, mathematician Clifford A. Pickover sugg...

The great Alexandrian library was destroyed by a series of attacks, first by the Romans and then by Christians and ...

In fact, the whole enterprise of science was essentially transferred in its entirety to India and the Arab world.

significant event of this period was the introduction of the so-called Hindu-Arabic numerals and of decimal notation.

Had it not been for the intellectual surge in Islam during the eighth century, most of the ancient mathematics would have b... caliph decided to have all the ancient Greek works translated.

word "algorithm," used for any special method for solving a mathematical problem using a collection of exact procedural st...

Book on What Is Needed from the Science of Arithmetic for Scribes and Businessmen, and according to Abu’l-W^fa, it "compris...

George Bernard Shaw once expressed his views on progress by the statement: "The reasonable man adapts himself to the w...

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The nickname Fibonacci (from the Latin filius Bonacci, son of the Bonacci family, or "son of good nature")

Arithmetic operations with Roman numerals are not fun. For example, to obtain the sum of 3,786 and 3,843, you would ne...

typical abacus had four wires, with beads on the bottom wire representing units, those on the one above it tens, those ...

Now consider the following entirely different problem. A child is trying to climb a staircase. The maximum number of st...

We find that the numbers of possibilities, 1, 2, 3, 5, 8,..., form a Fibonacci sequence.

golden ratio formula

The suggestion is therefore that phyllotaxis simply represents a state of minimal energy for a system of mutually repelling buds.

I hope that the next time you eat a pineapple, send a red rose to a loved one, or

admire van Gogh's sunflower paintings, you will remember that the growth pattern of these plants embodies this wonderful n...

they are "only a fascinatingly prevalent tendency."

They appear in phenomena covering a range in sizes from the microscopic to that of giant galaxies.

Bernoulli

famous for their bitter interfamilial rivalries as they were for their numerous mathematical achievements.

Nature loves logarithmic spirals.

logarithmic spiral remained essentially unchanged for millions of years.

Describing the violent flow of water Leonardo wrote: "The sudden waters rush into the pond that contains them, striki...

"the eye of God").

Peregrine falcons are some of the fastest

birds on Earth, plummeting toward their targets at speeds of up to two hundred miles per hour.

Because of the spiral's equiangular property, this path allows them to keep their target in view while maximizing sp...

Milky way" which philosopher Immanuel Kant (1724- 1804) speculated about long before they were actually observed (Fig...

English poet, painter, and mystic William Blake (1757- 1827), when he wrote: To see a World in a Grain of Sand, And a H...

Newton's laws of motion show that as a result of this dependence on the distance, the orbits of the planets around the Su...

All of this is but a small tribute to the man who used rabbits to discover a world- embracing mathematical concept. As im...

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Tho mioct’ fnr mir nrinin ic ci'.raof frnit-'c juice which maintains satisfaction in the minds of the philosophers. —LUCA ...

interesting contributions to mathematics. Not surprisingly perhaps, the mathematical investigations of all three painters were ...

Who was this highly controversial mathematician Luca Pacioli? Was he the greatest mathematical plagiarist of all ti...

in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet...

"Vitruvian man,"’ drawn beautifully by Leonardo

Johannes Kepler is best remembered as an outstanding astronomer responsible (among other things) for the three law...

We can only be astonished how, with this background and with the violent ups and downs of his tumultuous life, Kepler wa...

Realizing that fate had forced upon him the career of a mathematician, Kepler became determined to fulfill what he regarded a...

Why were there precisely six planets? and What was it that determined that the planetary orbits would be spaced as the...

He sent copies of the book to various astronomers for comments, including a copy to one of the foremost figures of...

of the great Galileo Galilei (1564- 1642), who informed Kepler that he too believed in Copernicus' model but lamented the ...

discovery of the planets Uranus (next after Saturn in terms of increasing distance from the Sun) in 1781 and Neptune (next af...

"Seldom in history has so wrong a book been so seminal in directing the future course of science." Kepler took the Pyt...

falsifiable by observations that could be made subsequently. These are precisely the ingredients required by the "scientific

In 1610, Galileo discovered with his telescope four new celestial bodies in the Solar System. Had these proven to be ...

Similarly, modern theories known as string theories use basic entities (strings) which are extremely tiny (more than a billion b...

"divine proportion" played a crucial role in architecture.

Kepler's First Law states that the orbits of the known planets around the Sun are not exact circles but rather ellipses, with th...

Kepler's Second Law establishes that the planet moves fastest when it is closest to the Sun (the point known as perihelion...

It took the genius of Isaac Newton (1642- 1727) to deduce that the force holding the planets in their orbits is gravity.

Kepler's heroic efforts in the calculations of Mars' orbit (many hundreds of sheets of arithmetic and their interpretation; dub...

Kepler's apartment itself, however, is not marked in any special way and is not open to the public, being occupied by one of...

In other words, Kepler discovered that the ratio of consecutive Fibonacci numbers converges to the Golden Ratio.

Kepler refers to the Golden Ratio as "the divine proportion as the geometers of today call it."

Only strings plucked at lengths with ratios corresponding to simple numbers produced consonant tones. A ratio of 2: 3 sound...

As Kepler was convinced that "before the origin of things, geometry was coeternal with the Divine Mind," much of the Har...

In general, the word "tiling" is used to describe a pattern or structure that comprises of one or more shapes of "til...

The Harmony of the World. This particular tiling pattern is composed of four shapes, all related to the Golden Ratio: pentago...

The force is stronger the closer the planet is to the Sun, so inner planets must move faster to avoid falling toward the Sun.

used to measure the heavens, Now the Earth’s shadows I measure My mind was in the heavens, Now the shadow of my b...

Painting isn't an aesthetic operation; it’s a form of magic designed as a mediator between this strange hostile world and ...

The Renaissance produced a significant change in direction in the history of the Golden Ratio. No longer was this conce...

The name "Cubism" was coined by art critic Louis Vauxcelles (who, by the way, had also been responsible for "Expressionism...

One of the strongest advocates for the application of the Golden Ratio to art and architecture was the famous Swiss-Fren...

(Charles-Edouard Jeanneret, 1887- 1965).

Jeanneret did not take the name "Le Corbusier" (co-opted from ancestors on his mother's side called Lecorbesier) until h...

Matila Ghyka's influential book Aesthetics of Proportions in Nature and in the Arts, and his Golden Number, Pythagorean Rites a...

The Modulor was supposed to provide "a harmonic measure to the human scale, universally applicable to architecture and...

fact no more than a rephrasing of Protagoras' famous saying from the fifth- century B.C. "Man is the measure of all t...

would give harmonious proportions to everything, from the sizes of cabinets a...

Le Corbusier was very proud of the fact that he had the opportunity to present the Modulor even to Albert Einstein, in ...

letter from Einstein, in which the great man said this of the Modulor: "It is a scale of proportions which makes the bad diff...

I shall describe, the term "beautiful" was actually shunned. Rather, words like "pleasing" or "attractive" have been use...

inspiration for the research came to him when he "saw the vision of a unified world of thought, spirit and matter, linked tog...

German playing cards tended to be somewhat more elongated than the Golden Rectangle, while French playing ...

156 2787: 20150307@ Figure 84

Did you pick the Golden Rectangle as your first choice? (It is the fifth from the left in the fourth row.)

in the ancient Greek curriculum and up to medieval times, music was considered a part of mathematics, and musicians con...

Johann Sebastian Bach (1685- 1750) had a fascination for the kinds of games that can be played with musical notes and n...

Mnrf nf fhrtrA nvnrtf+r Ir rill nlrn 4"oil imi i fhnf than mere mindless computation. —RO...

On one hand, the Pythagorean motto "all Is number" has materialized

spectacularly, in the role that the Golden Ratio plays in natural phenomena ranging from phyllotaxis to the shape of galaxies...

One of the most startling properties of any Pen-rose kite-dart tiling design is that the number of kites is about 1.618 times th...

1976, mathematician Robert Ammann discovered a pair of "cubes" (Figure 106), one "squashed" and one "stretched," k...

Golden Sequence. Start with the number 1, and then replace 1 by 10. From then on, replace each 1 by 10 and each 0 by...

Note that the numbers of Is in the sequence of lines 1, 1, 2, 3, 5, 8... form a Fibonacci sequence, and so do the num...

we get the sequence: 2122121... (the first overlap with the second, the third is on...

name "fractal" (from the Latin fractus, meaning "broken, fragmented") was coined by the famous Polish-French-Am...

Fractal curves like the path of a bolt of lightning, on the other hand, wiggle so aggressively that they fell somewhere b...

"How long is the coast of Britain?" Mandelbrot's surprising answer is that the length of the coastline actually depends...

The point is that every time you decrease the size of your ruler, you get a larger value for the length, because you always disc...

idea is to examine how many small objects make up a larger object in any number of dimensions.

Once you get used to the concept, you realize that the world around us is füll of fractals.

breakdown of a complete market cycle, according to Elliott, might look as follows. A generally upward trend consisting of fiv...

Benoit Mandelbrot published a book entitled Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, whic...

The defining property of the Fibonacci sequence— that each new number is the sum of the previous two numbers— wa...

Fnr examnle. the series nf tosses HTTHHTHTTH will produce the sequence 1, 1, 2, -1, 3, 2, 5, -3, 2, -5, 7, 2. On ...

essentially 100 percent probability, the one hundredth number in any of the sequences generated in this way was al...

The importance of Viswanath’s work lies not only in the discovery of a new mathematical constant, a significant fea...

The power emitted by the Sun’s surface determined (and continues to determine) the temperature on Earth’s surface and...

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I should attempt to treat human vice and folly geometrically... the passions of hatred, anger, envy, and so on, considered in th...

Euclid defined the Golden Ratio because he was interested in using this simple proportion for the construction of the p...

Somehow the Golden Ratio always makes an unexpected appearance at the juxtaposition of the simple and the com...

"Pure and Applied Mathematics in the People's Republic of China." By "pure," mathematicians usually refer to the typ...

MATHEMATICS SHOULD SURPRISE

number 1 appeared as the first digit in about 32 percent of the numbers, 2 appeared in about 17 percent, 3 in 14 ...

Astronomer and mathematician Simon Newcomb (1835- 1909) first discovered this "first-digit phenomenon" in 1881.

formula is now known as Benford’s law.

(recall that the nth Fibonacci number is close to

So, is there an infinite number of Fibonacci primes (as there is an infinite number of primes, in general)? No one actually kno...

We have to wonder, for example, how is it possible that planets in their orbits around the Sun were found to follow a curve (...

Why are the laws of physics themselves expressible as mathematical equations in the first place?

"Platonic view," is that mathematics is universal and timeless, and its existence is an objective fact, independent of us hu...

just as Michelangelo thought that his sculptures existed inside the marble and he merely uncovered them.

Douglas R. Hofstadter phrased this succinctly in his fantastic book Godel, Escher, Bach: An Eternal Golden Braid: "...

Clifford A. Pickover wrote in his lively book The Loom of God: "I do not know if God is a mathematician, but

mathematics is the loom upon which God weaves the fabric of the universe....

Mathematical objects have no objective reality— they are imaginary.

Marilyn vos Savant, the "world record holder" in IQ— an incredible 228.

Mathematics is simply assumed to be the symbolic counterpart of the universe.

Wolfram proposes simple computer programs instead. He suggests that nature's main secret is the use of simple...

Newton's first paper was on optics, and he continued to work on this subject for most of his life. In 1704 he published the first...

idea of particles of light.

photon— the particle of light— was introduced.

another theory of light— a wave theory—

The modern quantum theory of light has unified the classical notions of waves and particles in the concept of probabilities....

Mathematics itself could have originated from a subjective human perception of how nature works. Geometry may simpl...

naturalists Charles Darwin (1809- 1882) and Alfred Russel Wallace (1823- 1913) independently had the inspiration to int...

"Human logic [emphasis added was forced on us by the physical world and is therefore consistent with it. Mathematical idea of particles of light.

photon— the particle of light— was introduced.

another theory of light— a wave theory—

The modern quantum theory of light has unified the classical notions of waves and particles in the concept of probabilities....

Mathematics itself could have originated from a subjective human perception of how nature works. Geometry may simpl...

naturalists Charles Darwin (1809- 1882) and Alfred Russel Wallace (1823- 1913) independently had the inspiration to int...

"Human logic [emphasis added] was forced on us by the physical world and is therefore consistent with it. Mathemati...

The Golden Ratio is a product of humanly invented geometry. Humans had no idea, however, into what magical fairyland thi...

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It is only shallow people who do not judge by appearances. The mystery of the world is the visible, not the invisible. —OSCAR Wilde

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#eng The Most Astonishing Number by Mario Livio

http://www.amazon.com/kindleforandroid/

Pagetop

1,4396,244,mat,eng,20150227,20150403,5,Mario Livio:The Golden Ratio

20150227-20150403, 244 pages, 5* SalesInfo o eng

## eng The Most Astonishing Number by Mario Liviohttp://www.amazon.com/kindleforandroid/ Indeed this is also a most astonishing book telling the story of a number. In bibliographical terms this book is mathematics, or slightly more narrowly history of mathematics. Nota bene! The number is not the well-known Pi (3,14..), related with circle, but equally well-known, not as a number but as ratio of long and short edge of an ideal four corner surface. Everybody knows and has a conception of what is a Golden Ratio without ever thinking it as a number. At least I bumped to the number reading this book, first time in my over 75 years of life. The magic number is PHI (1,618), the ratio of the long side to short side of any Golden cut surface. What is so special about that innocent-looking number? Read this book and you will be astonished. As if the whole Universe would be planned on the basis of this magic number: all 'natural' dimensions fron snow-flakes to the form of galaxies, masterpieces of painting, sculpture and music have this Golden ratio as the basic measure of their inner proportions. We seek it instinctively everywhere and are disappointed, if we do not find it. As an example I am extremely irritated of the brute deviation from this ideal of the format of paper journals; in addition to being unpleasant looking, they also are clumsily flabby for holding in hands. Despite of presenting the innocent looking simple numbeer Phi (with alternative formulas behind it) this book plunges right away to the deepest mysteries of mathematics referring to dozens, hundreds of authorities. And yet, you will have no difficulty reading and understanding the text. High level mathematics is usually thought as pages full of formulas and dissertations of 20 pages, which very few persons understand. This book is not that way, although it goes far beyond mathematics requiring technical knowledge and skills. Just that is the fascination of the book. Embracing structures from flowers to houses and galaxies you get a fantastic feeling of better understanding what you see. Also another very rare feature is included. The host of personalities contributing to this discovery of hidden secrets of our world view are presented on everyday grass root level. Such well-known as Pythagoras, Newton, Gauss, Kepler, Einstein along with many less known but very important geniuses. Believe or not you have the feeling of meeting and chatting with them persoanlly. A real magician this Mario Livio, five stars without any hesitation. Grateful to my friend Viljo, class mate beyond 60 years, who introduced this author to me. Reading already a second Amazon book by Livio, about The Impossible Equation. |
Review in English | ## fin Ihmeellinen suhdelukuhttp://www.amazon.com/kindleforandroid/ Todellakin tämä on myös kaikkein hämmästyttävä kirja kertoo tarinan useita. Vuonna bibliografiset kannalta tämä kirja on matematiikkaa, tai hieman suppeammin historian matematiikka. Huom! Numero ei ole tunnettu Pi (3,14 ..), liittyy ympyrä, mutta yhtä hyvin tiedossa, eikä sillä useat vaan suhde pitkä ja lyhyt reuna ihanteellinen neljä nurkka pinta. Kaikki tietävät ja on käsitys siitä, mitä on Golden Ratio koskaan ajatellut sitä lukuna. Ainakin Törmäsin lukumäärään luet tätä kirjaa, ensimmäinen kerta yli 75 vuotta elämää. Maaginen numero on PHI (1618), suhde pitkä sivu lyhyeen sivuun minkään Golden leikkauspinnan. Mikä on niin erikoista, että viattoman näköinen numero? Lue tämä kirja ja sinun on hämmästynyt. Ikään kuin koko maailmankaikkeus suunnitellaan perusteella tämä taika numero: kaikki "luonnollinen" mitat rajoista lumi-hiutaleita muotoon galakseja, mestariteoksia maalaus, kuvanveisto ja musiikki on tämä Golden suhde kuin perus mittana niiden sisäinen mittasuhteet . Haemme sen vaistomaisesti kaikkialla ja ovat pettyneitä, jos emme löydä sitä. Esimerkkinä olen äärimmäisen ärtynyt ja raa'alla poikkeama tästä ihanteellisen muodon paperin lehtiä; sen lisäksi, että epämiellyttävä näköinen, ne ovat myös kömpelösti vetelä varten tilalla käsissä. Huolimatta esitellä viattoman näköisen yksinkertainen numbeer Phi (vaihtoehtoisia kaavoja takana) tämä kirja sukeltaa heti syvimpiin mysteereihin matematiikan viitaten kymmeniä, satoja viranomaiset. Ja vielä, et ei ole vaikea lukea ja ymmärtää tekstiä. Korkean tason matematiikkaa on yleensä ajateltu kuin sivut täynnä kaavoja ja väitöskirjoista 20 sivua, joka hyvin harvat henkilöt ymmärtää. Tämä kirja ei ole näin, vaikka se menee paljon pidemmälle matematiikka edellyttää teknistä osaamista. Vain, että on kiehtovaa kirjan. Embracing rakenteita kukkia taloja ja galaksit saat fantastinen tunne ymmärtää paremmin, mitä näet. Myös toinen hyvin harvinainen ominaisuus on mukana. Isäntä persoonallisuuksia edistää tätä löytö piilotetut salaisuudet maailmamme mieltä esitellään jokapäiväiseen ruohonjuuritasolla. Tällaiset tunnetut Pythagoraana, Newton, Gauss, Kepler, Einstein yhdessä monien vähemmän tunnettuja, mutta erittäin tärkeä neroja. Uskokaa tai et on tunne kokous ja jutteleminen heidän persoanlly. Todellinen taikuri tämän Mario Livio, viisi tähteä epäröimättä. Kiitollinen ystäväni Viljo, luokan mate yli 60 vuotta joka esitteli tämä kirjailija minulle. Lukeminen jo toinen Amazon kirja Livio, noin Impossible yhtälö. |

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