Part 2 (2/2)

In case you do not remember precisely how to solve quadratic equations, Appendix 5 presents a brief reminder. The two solutions of the equation for the Golden Ratio are:

The positive solution )...gives the value of the Golden Ratio. We now see clearly that is irrational, being simply half the sum of 1 and the square root of 5. Even before we go any further, we can get a feeling that this number has some interesting properties by using a simple scientific pocket calculator. Enter the number 1.6180339887 and hit the [x )...gives the value of the Golden Ratio. We now see clearly that is irrational, being simply half the sum of 1 and the square root of 5. Even before we go any further, we can get a feeling that this number has some interesting properties by using a simple scientific pocket calculator. Enter the number 1.6180339887 and hit the [x2] b.u.t.ton. Do you see something surprising? Now enter the number again, and this time hit the [1/x] b.u.t.ton. Intriguing, isn't it? While the square of the number 1.6180339887... gives 2.6180339887..., its reciprocal (”one over”) gives 0.6180339887..., all having precisely the same digits after the decimal point! The Golden Ratio has the unique properties that we produce its square by simply adding the number 1 and its reciprocal by subtracting the number 1. Incidentally, the negative solution of the equation is equal precisely to the negative of 1/ . is equal precisely to the negative of 1/ .

Paul S. Bruckman of Concord, California, published in 1977 in the journal The Fibonacci Quarterly The Fibonacci Quarterly an amusing poem called ”Constantly Mean.” Referring to the Golden Ratio as the ”Golden Mean,” the first verse from that poem reads: an amusing poem called ”Constantly Mean.” Referring to the Golden Ratio as the ”Golden Mean,” the first verse from that poem reads: 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.

The fact that we now have an algebraic expression for the Golden Ratio allows us, in principle, to calculate it to a high precision. This is precisely what M. Berg did in 1966, when he used 20 minutes on an IBM 1401 mainframe computer to calculate to the 4,599th decimal place. (The result was published in the decimal place. (The result was published in the Fibonacci Quarterly.) Fibonacci Quarterly.) The same can be achieved today on almost any personal computer in less than two seconds. In fact, the Golden Ratio was computed to 10 million decimal places in December 1996, and it took about thirty minutes. For the true number enthusiasts, here is to the 2,000 The same can be achieved today on almost any personal computer in less than two seconds. In fact, the Golden Ratio was computed to 10 million decimal places in December 1996, and it took about thirty minutes. For the true number enthusiasts, here is to the 2,000th decimal place: decimal place:

Intriguing as they are, you may think that the properties of I have described so far hardly justify adjectives like ”Golden” or ”Divine,” and you would be right. But this has been just a first glimpse of the wonders to come.

SURPRISES GALORE.

Everyone is familiar with the feeling we experience when we suddenly recognize the face of an old friend at a party where we were convinced we hardly know anyone. You may also have a similar emotional response when you go to an art exhibition and, upon turning a corner, find yourself suddenly facing one of your favorite paintings. The entire notion of a ”surprise party” is in fact based on the pleasure and gratification many of us feel when confronted with such unexpected appearances. Mathematics and the Golden Ratio in particular provide a rich treasury of such surprises.

Imagine that we are trying to determine the value of the following unusual expression that involves square roots that go on forever:

How would we even go about finding the answer? One rather c.u.mbersome way could be to start by calculating (which is 1.414...), then to calculate (which is 1.414...), then to calculate and so on, hoping that the subsequent values will converge rapidly to some number. But there may be a shorter, more elegant method of calculation. Suppose we denote the value we are seeking by and so on, hoping that the subsequent values will converge rapidly to some number. But there may be a shorter, more elegant method of calculation. Suppose we denote the value we are seeking by x. x. We therefore have We therefore have

Now let us square both sides of this equation. The square of x x is is x x, and the square of the expression on the right-hand side simply removes the outermost square root (by the definition of a square root). We therefore obtain

However, note that because the second expression on the right-hand side goes on to infinity, it is actually equal to our original x. x. We therefore obtain the quadratic equation We therefore obtain the quadratic equation x = x = 1+ 1+x. But this is precisely the equation that defines the Golden Ratio! We therefore found that our endless expression is actually equal to . But this is precisely the equation that defines the Golden Ratio! We therefore found that our endless expression is actually equal to .

Let us now look at a very different type of never-ending expression, this time involving fractions:

This is a special case of mathematical ent.i.ties known as continued fractions continued fractions, which are used quite frequently in number theory. How would we compute the value of this continued fraction? Again, we could in principle truncate the series of 1s at successively higher points, hoping to find the limit to which the continued fraction converges. Based on our previous experience, however, we could at least start by denoting the value by x. x. Thus, Thus,

Note, however, that because the continued fraction goes on forever, the denominator denominator of the second term on the right-hand side is in fact identical to of the second term on the right-hand side is in fact identical to x x itself. We therefore have the equation itself. We therefore have the equation

Multiplying both sides by x x2, we get x = x x = x+1, which is again the equation defining the Golden Ratio! We find that this remarkable continued fraction is also equal to . Paul S. Bruckman's poem ”Constantly Mean” refers to this property as well: Expressed as a continued fraction, It's one, one, one,..., until distraction; In short, the simplest of such kind (Doesn't this really blow your mind?) Because the continued fraction corresponding to the Golden Ratio is composed of ones only, it converges very slowly. The Golden Ratio is, in this sense, more ”difficult” to express as a fraction than any other irrational number-it is the ”most irrational” among irrationals.

From never-ending expressions let us now turn our attention to the Golden Rectangle in Figure 26 Figure 26. The lengths of the sides of the rectangle are in a Golden Ratio to each other. Suppose we cut off a square from this rectangle (as marked in the figure). We will be left with a smaller rectangle that is also a Golden Rectangle. The dimensions of the ”daughter” rectangle are smaller than those of the ”parent” rectangle by precisely a factor . We can now cut a square from the daughter Golden Rectangle and we will be left again with a Golden Rectangle, the dimensions of which are smaller by another factor of . Continuing this process ad infinitum, we will produce smaller and smaller Golden Rectangles (each time with dimensions ”deflated” by a factor ). If we were to examine the ever-decreasing (in size) rectangles with a magnifying gla.s.s of increasing power, they would all look identical. The Golden Rectangle is the only only rectangle with the property that cutting a square from it produces a similar rectangle. Draw two diagonals of any mother-daughter pair of rectangles in the series, as in rectangle with the property that cutting a square from it produces a similar rectangle. Draw two diagonals of any mother-daughter pair of rectangles in the series, as in Figure 26 Figure 26, and they will all intersect at the same point. The series of continuously diminis.h.i.+ng rectangles converges to that never-reachable point. Because of the ”divine” properties attributed to the Golden Ratio, mathematician Clifford A. Pickover suggested that we should refer to that point as ”the Eye of G.o.d.”

Figure 26 If you did not find it mind-boggling that all of these diverse mathematical circ.u.mstances lead to , take a simple pocket calculator and I will show you an amazing magic trick. Choose any two numbers (with any number of digits) and write them one after the other. Now, using the calculator (or in your head), form a third number, by simply adding together the first two (and write it down); form a fourth number by adding the second number to the third; a fifth number by adding the third to the fourth; a sixth number by adding the fourth to the fifth and so on, until you have a series of twenty 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, phi. I shall return to this trick and its explanation in Chapter 5.

TOWARD THE DARK AGES.

In his definition in the Elements Elements, Euclid was interested primarily in the geometrical interpretation of the Golden Ratio and in its use in the construction of the pentagon and some Platonic solids. Following in his footsteps, Greek mathematicians in the next centuries produced several new geometrical results involving the Golden Ratio. For example, the ”Supplement” to the Elements Elements (often referred to as Book XIV) contains an important theorem concerning a dodecahedron and an icosahedron that are circ.u.mscribed by the same sphere. The text of the ”Supplement” is attributed to Hypsicles of Alexandria, who probably lived in the second century (often referred to as Book XIV) contains an important theorem concerning a dodecahedron and an icosahedron that are circ.u.mscribed by the same sphere. The text of the ”Supplement” is attributed to Hypsicles of Alexandria, who probably lived in the second century B.C. B.C., but it is believed to contain theorems by Apollonius of Perga (ca. 262190 B.C. B.C.), one of the three key figures (together with Euclid and Archimedes) of the Golden Age of Greek mathematics (from about 300 to 200 B.C. B.C.). Developments concerning the Golden Ratio become more spa.r.s.e after that and are a.s.sociated mainly with Hero (in the first century A.D. A.D.), Ptolemy (in the second century A.D. A.D.), and Pappus (in the fourth century). In his Metrica Metrica, Hero provided approximations (often without offering a clue of how they were obtained) for the areas of the pentagon and the decagon (the ten-sided polygon) and for the volumes of dodecahedrons and icosahedrons.

Ptolemy (Claudius Ptolemaus) lived around A.D. A.D. 100 to 179, but virtually nothing is known about his life, except that he did most of his work in Alexandria. Based on his own and previous astronomical observations, he developed his celebrated geocentric model of the universe, according to which the Sun and all the planets revolved around Earth. While fundamentally wrong, his model did manage to explain (at least initially) the observed motions of the planets, and it continued to govern astronomical thinking for some thirteen centuries. 100 to 179, but virtually nothing is known about his life, except that he did most of his work in Alexandria. Based on his own and previous astronomical observations, he developed his celebrated geocentric model of the universe, according to which the Sun and all the planets revolved around Earth. While fundamentally wrong, his model did manage to explain (at least initially) the observed motions of the planets, and it continued to govern astronomical thinking for some thirteen centuries.

Ptolemy synthesized his own astronomical work with that of other Greek astronomers (in particular Hipparchos of Nicaea) in an encyclopaedic, thirteen-volume book, H Mathmatik Syntaxis H Mathmatik Syntaxis (The mathematical synthesis). The book later became known as (The mathematical synthesis). The book later became known as The Great Astronomer. The Great Astronomer. However, ninth-century Arab astronomers referred to the book invoking the Greek superlative ”Megist” (”the greatest”) but prefixing it with the Arabic identifier of proper names, ”al.” The book thereby became known, to this day, as the However, ninth-century Arab astronomers referred to the book invoking the Greek superlative ”Megist” (”the greatest”) but prefixing it with the Arabic identifier of proper names, ”al.” The book thereby became known, to this day, as the Almagest. Almagest. Ptolemy also did important work in geography and wrote an influential book ent.i.tled Ptolemy also did important work in geography and wrote an influential book ent.i.tled Guide to Geography. Guide to Geography.

In the Almagest Almagest and the and the Guide to Geography Guide to Geography, Ptolemy constructed one of the earliest equivalents of a trigonometric table for many angles. In particular, he calculated lengths of chords connecting two points on a circle for various angles, including the angles 36, 72, and 108 degrees, which, as you recall, appear in the pentagon and are therefore closely a.s.sociated with the Golden Ratio.

The last great Greek geometer who contributed theorems related to the Golden Ratio was Pappus of Alexandria. In his Collection (Synagoge; Collection (Synagoge; ca. ca. A.D. A.D. 340), Pappus gives a new method for the construction of the dodecahedron and the icosahedron as well as comparisons of the volumes of the Platonic solids, all of which involve the Golden Ratio. Pappus' commentary on Euclid's theory of irrational numbers traces beautifully the historical development of irrationals and is extant in its Arabic translations. However, his heroic efforts to arrest the general decay of mathematics and of geometry in particular were unsuccessful, and after his death, with the overall withering of intellectual curiosity in the West, interest in the Golden Ratio entered a long period of hibernation. The great Alexandrian library was destroyed by a series of attacks, first by the Romans and then by Christians and Muslims. Even Plato's Academy came to an end in 340), Pappus gives a new method for the construction of the dodecahedron and the icosahedron as well as comparisons of the volumes of the Platonic solids, all of which involve the Golden Ratio. Pappus' commentary on Euclid's theory of irrational numbers traces beautifully the historical development of irrationals and is extant in its Arabic translations. However, his heroic efforts to arrest the general decay of mathematics and of geometry in particular were unsuccessful, and after his death, with the overall withering of intellectual curiosity in the West, interest in the Golden Ratio entered a long period of hibernation. The great Alexandrian library was destroyed by a series of attacks, first by the Romans and then by Christians and Muslims. Even Plato's Academy came to an end in A.D. A.D. 529, when the Byzantine emperor Justinian ordered the closing of all the Greek schools. During the depressing Dark Ages that followed, the French historian and bishop Gregory of Tours (538594) lamented that ”the study of letters is dead in our midst.” In fact, the whole enterprise of science was essentially transferred in its entirety to India and the Arab world. A significant event of this period was the introduction of the so-called Hindu-Arabic numerals and of decimal notation. The most important Hindu mathematician of the sixth century was ryabhata (476ca. 550). In his best-known book, ent.i.tled 529, when the Byzantine emperor Justinian ordered the closing of all the Greek schools. During the depressing Dark Ages that followed, the French historian and bishop Gregory of Tours (538594) lamented that ”the study of letters is dead in our midst.” In fact, the whole enterprise of science was essentially transferred in its entirety to India and the Arab world. A significant event of this period was the introduction of the so-called Hindu-Arabic numerals and of decimal notation. The most important Hindu mathematician of the sixth century was ryabhata (476ca. 550). In his best-known book, ent.i.tled ryabhattya ryabhattya, we find the phrase ”from place to place each is ten times the preceding,” which indicates an application of a place-value system. An Indian plate from 595 already contains writing (of a date) in Hindu numerals using decimal place-value notation, implying that such numerals had been in use for some time. The first sign (albeit with no real influence) of Hindu numerals moving West can be found in the writings of the Nestorian bishop Severus Sebokht, who lived in Keneshra on the Euphrates River. He wrote in 662: ”I will omit all discussion of the science of the Indians... and of their valuable methods of calculation which surpa.s.s description. I wish only to say that this computation is done by means of nine signs.”

With the ascendancy of Islam, the Muslim world became an important center for mathematical study. Had it not been for the intellectual surge in Islam during the eighth century, most of the ancient mathematics would have been lost. In particular, Caliph al-Mamun (786833) established in Baghdad the Beit al-hikma (House of wisdom), which operated in a similar fas.h.i.+on to the famous Alexandrian university or ”Museum.” Indeed, the Abbasid empire subsumed any Alexandrian learning that had survived. According to tradition, after having a dream in which Aristotle appeared, the caliph decided to have all the ancient Greek works translated.

Many of the important Islamic contributions were algebraic in nature and touched on the Golden Ratio only very peripherally. Nevertheless, at least three mathematicians should be mentioned: Al-Khwrizm and Abu Kamil Shuja in the ninth century and Abu'l-Wafa in the tenth century.

Mohammed ibn-Musa al-Khwrizm composed, in Baghdad (at about 825), what is considered to be the most influential algebraic work of the period-Kitb al-jabr wa al-muqbalah (The science of restoration and reduction). From this t.i.tle (”al-jabr”) comes the word ”algebra” that we use today, since this was the first textbook used in Europe on that subject matter. Furthermore, the word ”algorithm,” used for any special method for solving a mathematical problem using a collection of exact procedural steps, comes from a distortion of al-Khwrizm's name. (The science of restoration and reduction). From this t.i.tle (”al-jabr”) comes the word ”algebra” that we use today, since this was the first textbook used in Europe on that subject matter. Furthermore, the word ”algorithm,” used for any special method for solving a mathematical problem using a collection of exact procedural steps, comes from a distortion of al-Khwrizm's name. The Science of Restoration The Science of Restoration was synonymous with the theory of equations for a few hundred years. The equation required to solve one of the problems presented by al-Khwarizmi bears a close resemblance to the equation defining the Golden Ratio. Al-Khwarizmi says: ”I have divided ten into two parts; I have multiplied the one by ten and the other by itself, and the products were the same.” Al-Khwarizmi calls the unknown shai (”the thing”). Consequently, the first line in the description of the equation obtained (for the above problem) translates to: ”you multiply was synonymous with the theory of equations for a few hundred years. The equation required to solve one of the problems presented by al-Khwarizmi bears a close resemblance to the equation defining the Golden Ratio. Al-Khwarizmi says: ”I have divided ten into two parts; I have multiplied the one by ten and the other by itself, and the products were the same.” Al-Khwarizmi calls the unknown shai (”the thing”). Consequently, the first line in the description of the equation obtained (for the above problem) translates to: ”you multiply thing thing by ten; it is by ten; it is ten things. ten things. The equation one obtains, 10 The equation one obtains, 10x=(10-x)2, is the one for the smaller segment of a line of length 10 divided in a Golden Ratio. The question of whether al-Khwarizmi actually had the Golden Ratio in mind when posing this problem is a matter of some dispute. Under the influence of al-Khwarizmi's work, the unknown was called ”res” in the early algebraic works in Latin, translated to ”cosa” (”the thing”) in Italian. Accordingly, algebra itself became known as ”l'arte della cosa” (”the art of the thing”). Occasionally it was referred to as the ”ars magna” (”the great art”), to distinguish it from what was considered as the lesser art of arithmetic.

The second Arab mathematician who made contributions related to the history of the Golden Ratio is Abu Kamil Shuja, known as al-Hasib al-Misri, meaning ”the Calculator from Egypt.” He was born around 850, probably in Egypt, and died at about 930. He wrote many books, some of which, including the Book on Algebra, Book of Rare Things in the Art of Calculation Book on Algebra, Book of Rare Things in the Art of Calculation, and Book on Surveying and Geometry Book on Surveying and Geometry, have survived. Abu Kamil may have been the first mathematician who instead of simply finding a solution to a problem was interested in finding all all the possible solutions. In his the possible solutions. In his Book of Rare Things in the Art of Calculation Book of Rare Things in the Art of Calculation he even describes one problem for which he found 2,678 solutions. More important from the point of view of the history of the Golden Ratio, Abu Kamil's books served as the basis for some of the books of the Italian mathematician Leonardo of Pisa, known as Fibonacci, whom we shall encounter shortly. Abu Kamil's treatise he even describes one problem for which he found 2,678 solutions. More important from the point of view of the history of the Golden Ratio, Abu Kamil's books served as the basis for some of the books of the Italian mathematician Leonardo of Pisa, known as Fibonacci, whom we shall encounter shortly. Abu Kamil's treatise On the Pentagon and the Decagon On the Pentagon and the Decagon contains twenty problems and their solutions, in which he calculates the areas of the figures and the length of their sides and the radii of surrounding circles. In some of these calculations (but not all), he uses the Golden Ratio. A few of the algebraic problems appearing in contains twenty problems and their solutions, in which he calculates the areas of the figures and the length of their sides and the radii of surrounding circles. In some of these calculations (but not all), he uses the Golden Ratio. A few of the algebraic problems appearing in Algebra Algebra may have also been inspired by the concept of the Golden Ratio. may have also been inspired by the concept of the Golden Ratio.

The last of the Islamic mathematicians I would like to mention is Mohammed Abu'l-Wafa (940998). Abu'l-Wafa was born in Buzjan (in present-day Iran) and lived during the rule of the Buyid Islamic dynasty in western Iran and Iraq. This dynasty reached its peak under the reign of Adud ad-Dawlah, who was a great patron of mathematics, the sciences, and the arts. Abu'l-Wafa was one of the mathematicians who were invited to Adud ad-Dawlah's court in Baghdad in 959. His first major book was Book on What Is Needed from the Science of Arithmetic for Scribes and Businessmen Book on What Is Needed from the Science of Arithmetic for Scribes and Businessmen, and according to Abu'l-Wafa, it ”comprises all that an experienced or novice, subordinate or chief in arithmetic needs to know.” Interestingly, although Abu'l-Wafa himself was an expert in the use of Hindu numerals, all the text of his book is written with no numerals whatsoever-numbers are written only as words, and calculations are done only mentally. By the tenth century, the use of Indian numerals had not yet found application in the business circles. Abu'l-Wafa's interest in the Golden Ratio appears in his other book: A Book on the Geometric Constructions Which Are Needed for an Artisan. A Book on the Geometric Constructions Which Are Needed for an Artisan. In this book, Abu'l-Wafa presents ingenious methods for the construction of the pentagon and the decagon and for inscribing regular polygons in circles and inside other polygons. A unique component of his work is a series of problems that he solves using a ruler (straightedge) and a compa.s.s, in which the angle between the two legs of the compa.s.s is kept fixed (known as ”rusty compa.s.s” constructions). This particular genre was probably inspired by Pappus' In this book, Abu'l-Wafa presents ingenious methods for the construction of the pentagon and the decagon and for inscribing regular polygons in circles and inside other polygons. A unique component of his work is a series of problems that he solves using a ruler (straightedge) and a compa.s.s, in which the angle between the two legs of the compa.s.s is kept fixed (known as ”rusty compa.s.s” constructions). This particular genre was probably inspired by Pappus' Collection Collection but may also represent Abu 'l- Wafa's response to a practical problem-the results with a fixed-angle compa.s.s are more accurate. but may also represent Abu 'l- Wafa's response to a practical problem-the results with a fixed-angle compa.s.s are more accurate.

The work by these and other Islamic mathematicians produced important but only incremental progress in the mathematical history of the Golden Ratio. As is often the case in the sciences, such preparatory periods of slow advancement are necessary to give birth to the next breakthrough. The great playwright George Bernard Shaw once expressed his views on progress by the statement: ”The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.” In the case of the Golden Ratio, the quantum leap had to await the appearance of the most distinguished European mathematician of the Middle Ages-Leonardo of Pisa.

The nine Indian figures are: 9 8 7 6 5 4 3 2 1.With these nine figures, and with the sign 0... any number may be written, as is demonstrated below.-LEONARDO F FIBONACCI ( (CA. 1170s-1240s) With the above words, Leonardo of Pisa (in Latin Leonardus Pisa.n.u.s), also known as Leonardo Fibonacci, began his first and best-known book, Liber abaci Liber abaci (Book of the abacus), published in 1202. At the time the book appeared, only a few privileged European intellectuals who cared to study the translations of the works of al-Khwarizm and Abu Kamil knew the Hindu-Arabic numerals we use today. Fibonacci, who for a while joined his father, a customs and trading official, in Bugia (in present-day Algeria) and later traveled to other Mediterranean countries (including Greece, Egypt, and Syria), had the opportunity to study and compare different numerical systems and methods for arithmetical operations. Upon concluding that the Hindu-Arabic numerals, which included the place-value principle, were far superior to all other methods, he devoted the first seven chapters of his book to explanations of Hindu-Arabic notation and its use in practical applications. (Book of the abacus), published in 1202. At the time the book appeared, only a few privileged European intellectuals who cared to study the translations of the works of al-Khwarizm and Abu Kamil knew the Hindu-Arabic numerals we use today. Fibonacci, who for a while joined his father, a customs and trading official, in Bugia (in present-day Algeria) and later traveled to other Mediterranean countries (including Greece, Egypt, and Syria), had the opportunity to study and compare different numerical systems and methods for arithmetical operations. Upon concluding that the Hindu-Arabic numerals, which included the place-value principle, were far superior to all other methods, he devoted the first seven chapters of his book to explanations of Hindu-Arabic notation and its use in practical applications.

Leonardo Fibonacci was born in the 1170s to a businessman and government official named Guglielmo. The nickname Fibonacci (from the Latin filius Bonacci, son of the Bonacci family, or ”son of good nature”) was most probably introduced by the historian of mathematics Guillaume Libri in a footnote in his 1838 book Histoire des Sciences Mathematique en Italie Histoire des Sciences Mathematique en Italie (History of the mathematical sciences in Italy), although some researchers attribute the first use of Fibonacci to Italian mathematicians at the end of the eighteenth century. In some ma.n.u.scripts and doc.u.m

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