The Golden Ratio
The Story of Phi, the World's Most Astonishing NumberBook - 2003 | First edition.
The Golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as the greatest treasure of geometry; such Renaissance thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Goethe, Cezanne, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order, beauty, and eternal mystery will always coexist.
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In his book Fads and Fallacies in the Name of Science, Gardner writes: If one looks up the facts about the Washington Monument in the World Almanac, he will find considerable fiveness. Its height is 555 feet and 5 inches. The base is 55 feet square, and the windows are set at 500 feet from the base. If the base is multiplied by 60 (or five times the number of months in a year) it gives 3,300, which is the exact weight of the capstone in pounds. Also, the word “Washington” has exactly ten letters (two times five). And if the weight of the capstone is multiplied by the base, the result is 181,500—a fairly close approximation to the speed of light in miles per second.
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 reality. —ALBERT EINSTEIN (1879–1955)
Arithmetic operations with Roman numerals are not fun. For example, to obtain the sum of 3,786 and 3,843, you would need to add MMMDCCLXXXVI to MMMDCCCXLIII; if you think that is cumbersome, try multiplying those numbers.
For example, if your first two numbers were 2 and 5, you would have obtained the series 2, 5, 7, 12, 19, 31, 50, 81, 131…. Now use the calculator to divide your twentieth number by your nineteenth number. Does the result look familiar? It is, of course, φ.
The golden mean is quite absurd;
It’s not your ordinary surd.
If you invert it (this is fun!),
You’ll get itself, reduced by one;
But if increased by unity,
This yields its square, take it from me.
Richard Buckminster Fuller (1895–1983) once put it: “When I am working on a problem, I never think about beauty. I think only of how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong.
the Fibonacci sequence; 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,…, and this time let us look at the ratios of successive numbers (calculated here to the sixth decimal place):
1/1 = 1.000000
2/1 = 2.000000
3/2 = 1.500000
5/3 = 1.666666
8/5 = 1.600000
13/8 = …
610/377 = 1.6188037
987/610 = 1.618033
As we go farther and farther down the Fibonacci sequence, the ratio of two successive Fibonacci numbers oscillates about (being alternately greater or smaller) but comes closer and closer to the Golden Ratio. If we denote the nth Fibonacci number by Fn, and the next one by Fn + 1, then we discovered that the ratio Fn + 1/Fn approaches φ as n becomes larger.
George Burns (in his book How to Live to Be 100 or More) once put it: “How do you live to be 100 or more? There are certain things you have to do. The most important one is you have to be sure to make it to 99.”
“God created the natural numbers, all else is the work of man.”
The strength of the familiar electromagnetic force between two electrons, for example, is expressed in physics in terms of a constant known as the fine structure constant. The value of this constant, almost exactly 1/137, has puzzled many generations of physicists. A joke made about the famous English physicist Paul Dirac (1902–1984), one of the founders of quantum mechanics, says that upon arrival to heaven he was allowed to ask God one question. His question was: “Why 1/137?”
The sum of any ten (Fibonacci) consecutive numbers is always equal to 11 times the seventh number.
To see a World in a Grain of Sand, And a Heaven in a Wild Flower, Hold Infinity in the Palm of your hand, And Eternity in an hour.
The name “equiangular” reflects another unique property of the logarithmic spiral. If you draw a straight line from the pole to any point on the curve, it cuts the curve at precisely the same angle (Figure 42). Falcons use this property when attacking their prey… in the November 2000 issue of the Journal of Experimental Biology, show that falcons keep their head straight and follow a logarithmic spiral. Because of the spiral’s equiangular property, this path allows them to keep their target in view while maximizing speeds.
Archimedean spiral in the side of a roll of paper towels or a rope coiled on the floor. In this type of spiral, the distance between successive coils remains always the same.
Accepting the general principle of the Copernican heliocentric system, he set out to search for answers to the following two major questions: Why were there precisely six planets? and What was it that determined that the planetary orbits would be spaced as they are? These “why” and “what” questions were entirely new in the astronomical vocabulary. Unlike the astronomers before him, who satisfied themselves with simply recording the observed positions of the planets, Kepler was seeking a theory that would explain it all.
An idealized scientific method begins with the collection of facts, a model is then proposed, and the model’s predictions are tested through experiments or further observations. This process is sometimes summed up by the sequence: induction, deduction, verification
Similarly, modern theories known as string theories use basic entities (strings) which are extremely tiny (more than a billion billion times smaller than the atomic nucleus) to deduce the properties of all the elementary particles. Like a violin string, the strings can vibrate and produce a variety of “tones,” and all the known elementary particles simply represent these different tones.
This particular property of Fibonacci numbers gives rise to a puzzling paradox … Consider the square of eight units on the side (area of 8^2 = 64) . Now dissect it into four parts as indicated. The four pieces can be reassembled to form a rectangle of sides 13 and 5 with an area of 65!
Contemporary painter and draftsman David Hockney argues in his book Secret Knowledge (2001), for example, that starting with around 1430, artists began secretly using cameralike devices, including lenses, concave mirrors, and the camera obscura, to help them create realistic-looking paintings.
In an experiment conducted in 1966 by H. R. Schiffman of Rutgers University, subjects were asked to “draw the most aesthetically pleasing rectangle” that they could on a sheet of paper. After completion, they were instructed to orient the figure either horizontally or vertically (with respect to the long side) in the most pleasing position. While Schiffman found an overwhelming preference for a horizontal orientation, consistent with the shape of the visual field, the average ratio of length to width was about 1.9—far from both the Golden Ratio and the visual field’s “average rectangle.”
However, a thorough examination of all the data basically convinced Putz that Mozart did not use the Golden Ratio in his sonatas, nor is it clear why the simple matter of measures would give a pleasing effect.
Goethe expressed elsewhere his opinion on mathematics thus: “The mathematicians are a sort of Frenchmen: when you talk to them, they immediately translate it into their own language, and right away it is something entirely different.”
Each wife of Fibonacci, Eating nothing that wasn’t starchy, Weighed as much as the two before her, His fifth was some signora! Similarly, two lines from a poem by Katherine O’Brien read: Fibonacci couldn’t sleep— Counted rabbits instead of sheep.
Mephistopheles: Let me admit; a tiny obstacle Forbids my walking out of here: It is the druid’s foot upon your threshold.
Faust: The pentagram distresses you? But tell me, then, you son of hell. If this impedes you, how did you come in?
Mephistopheles: Observe! The lines are poorly drawn; That one, the angle pointing outward, Is, you see, a little open.
Build thee more stately mansions, O my soul, As the swift seasons roll! Leave thy low-vaulted past! Let each new temple, nobler than the last, Shut thee from heaven with a dome more vast, Till thou at length art free, Leaving thine outgrown shell by life’s unresting sea.
A fly and a flea in a flue
Were imprisoned, so what could they do?
Said the fly, “Let us flee!”
“Let us fly!” said the flea,
So they fled through a flaw in the flue.
All the attempts to disclose the (real or false) Golden Ratio in various works of art, pieces of music, or poetry rely on the assumption that a canon for ideal beauty exists and can be turned to practical account. History has shown, however, that the artists who have produced works of lasting value are precisely those who have broken away from such academic precepts. In spite of the Golden Ratio’s importance for many areas of mathematics,
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