The Golden Ratio: The Story of Phi, the World's Most Astonishing Number - Mario Livio (2003)
4. THE SECOND TREASURE
Geometry has two great treasures; one is the Theorem of Pythagoras;
the other, the division of a line into extreme and mean ratio.
The first we may compare to a measure of gold;
the second we may name a precious jewel.
—JOHANNES KEPLER (1571-1630)
There is no doubt that anybody who grew up in a western or mideastern civilization is a pupil of the ancient Greeks, when it comes to mathematics, science, philosophy, art, and literature. The phrase of the German poet Goethe—“of all peoples the Greeks have dreamt the dream of life the best”—is only a small tribute to the pioneering efforts of the Greeks in branches of knowledge that they invented and denominated.
However, even the accomplishments of the Greeks in many other fields pale in comparison with their awe-inspiring achievements in mathematics. In the span of only four hundred years, for example, from Thales of Miletus (at ca. 600 B.C.) to “the Great Geometer” Apollonius of Perga (at ca. 200 B.C.), the Greeks completed all the essentials of a theory of geometry.
The Greek excellence in mathematics was largely a direct consequence of their passion for knowledge for its own sake, rather than merely for practical purposes. A story has it that when a student who learned one geometrical proposition with Euclid asked, “But what do I gain from this?” Euclid told his slave to give the boy a coin, so that the student would see an actual profit.
The curriculum for the education of statesmen at the time of Plato included arithmetic, geometry, solid geometry, astronomy, and music—all of which, the Pythagorean Archytas tells us, fell under the general definition of “mathematics.” According to legend, when Alexander the Great asked his teacher Menaechmus (who is reputed to have discovered the curves of the ellipse, the parabola, and the hyperbola) for a shortcut to geometry, he got the reply: “O King, for traveling over the country there are royal roads and roads for common citizens; but in geometry there is one road for all.”
Into this intellectual milieu enter Plato (428/427 B.C.-348/347 B.C.), one of the most influential minds of ancient Greece and of western civilization in general. Plato is said to have studied mathematics with the Pythagorean Theodorus of Cyrene, who was the first to prove that not just but also numbers like and up to are irrational. (No one knows for sure why he stopped at 17, but clearly he did not have a general proof.) Some researchers claim that Theodorus also may have used a line cut in the Golden Ratio to provide what may be the easiest proof of incommensurability. (The idea is essentially the same as that presented in Appendix 2.)
As Plato states in The Republic, mathematics was an absolute must in the education of all state leaders and philosophers. Accordingly, the inscription over the entrance to his school (the Academy) read: “Let no one destitute of geometry enter my doors.” The historian of mathematics David Eugene Smith describes this in his book Our Debt to Greece and Rome as the first college entrance requirement in history. Plato's admiration for mathematics also shows when he speaks with some envy on the attitude toward mathematics in Egypt, where “arithmetical games have been invented for the use of mere children, which they learn as a pleasure and amusement.”
In considering the role of Plato in mathematics in general, and in relation to the Golden Ratio in particular, we have to examine not just his own purely mathematical contributions, which were not very significant, but the effects of his influence and encouragement on the mathematics of others of his and of later generations. To some extent, Plato may be considered as one of the first true theoreticians. His theoretical inclinations are best exemplified by his attitude toward astronomy, where, rather than observing the stars in their motions, he advocates to “leave the heavens alone” and concentrate on the more abstract heaven of mathematics. The latter, according to Plato, is merely represented by the actual stars, in the same way that the abstract entities of a point, a line, and a circle are represented by geometrical drawings. Interestingly, in his outstanding book A History of Greek Mathematics (1921), Sir Thomas Heath writes: “It is difficult to see what Plato can mean by the contrast which he draws between the visible broideries of heaven [the visible stars and their arrangement], which are indeed beautiful, and the true broideries which they only imitate and which are infinitely more beautiful and marvelous.”
As a theoretical astrophysicist myself, I must say that I feel quite a bit of affinity with some of the sentiments expressed by Plato's underlying motif. The distinction here is between the beauty of the cosmos itself and the beauty of the theory that explains the universe. Let me clarify this with a simple example, the principle of which was first discovered by the famous German painter Albrecht Dürer (1471-1528).
You can put together six pentagons (Figure 19) to make one large pentagon, with five holes in the shape of Golden Triangles (isosceles triangles with a ratio of side to base of ö ). Six of these pentagons, in turn, go together to make an even larger (and more holey-looking) pentagon, and so on indefinitely.
I think everyone will agree that the obtained shape (Figure 19) is extremely beautiful. But this shape contains an additional mathematical appeal, which is in the simplicity of the underlying principle of its construction. This is, I believe, that mathematical heaven to which Plato was referring.
There is little doubt that Plato's guidance was far more important than his direct contributions. A text attributed to Philodemus from the first century reads: “Great progress in mathematics [was achieved] during that time, with Plato as the director and problem-giver, and the mathematicians investigating them zealously.”
Nevertheless, Plato himself certainly had an intense interest in the properties of numbers and of geometrical figures. In Laws, for example, he suggests that the 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 can still be divided by 12; (c) it has 59 divisors, including all the whole numbers from 1 to 12 (except for 11, but 5,038, which is very close to 5,040, is divisible by 11). The choice of this number and its properties allow Plato to develop his socioeconomic vision. For example, the state's land is divided into 5,040 lots, with 420 such lots constituting the territory of each of twelve “tribes.” The people of the state themselves are divided into four social categories: free citizens and their wives and children, their slaves, resident aliens, and a diverse population of visiting aliens. In elections to the Council, members from all four property categories vote for ninety members from each class.
Another number often associated with Plato is 216. Plato mentions this number in The Republic in a rather obscure passage that alludes to the fact that 216 is equal to 6 cubed, with 6 being one of the numbers representing marriage (since it is the product of the female 2 and the male 3). Plato, himself a pupil of the Pythagoreans, was also aware of the fact that the sum of the cubes of the sides of the famous 3-4-5 Pythagorean triangle is equal to 216.
Plato and the Golden Section are linked mainly through two areas that were particularly close to his heart: incommensurability and the Platonic solids. In Laws, Plato expresses his own feeling of shame for having learned about incommensurable lengths and irrational numbers relatively late in his life, and he laments the fact that many of the Greeks of his generation were still not familiar with the existence of such numbers.
Plato recognizes (in Hippias Major) that just as an even number may be the sum of either two even or two odd numbers, so can the sum of two irrationals be either irrational or rational. Since we already know that ö is irrational, a rational straight line (e.g., of unit length) divided in a Golden Section provides an illustration of the latter case, although Plato may not have known this fact. Some researchers maintain that Plato had a direct interest in the Golden Section. They point out that when Proclus Diadochus (ca. 411-485) writes (in A Commentary on the First Book of Euclid's Elements): “Eudoxus… multiplied the number of theorems which Plato originated concerning the ‘section,’ ” he may be referring to Plato's (and Eudoxus) association with the Golden Section. This interpretation, however, has been a matter of serious controversy since the second half of the nineteenth century, with many researchers concluding that the word “section” probably has nothing to do with the Golden Section but rather is referring to the section of solids or to the general sectioning of lines. Nevertheless, there is little doubt that much of the groundwork leading to the definition and understanding of the Golden Ratio was carried out during the years just prior to the opening of Plato's Academy in 386 B.C. and throughout the period of the Academy's operation. The key figure and driving force behind the geometrical theorems concerning the Golden Ratio was probably Theaetetus (ca. 417 B.C.-ca. 369 B.C.), who according to the Byzantine collection Suidas “was the first to construct the five so-called solids.” The fourth-century mathematician Pappus tells us that Theaetetus was also the one to have “distinguished the powers which are commensurable in length from those which are incommensurable.” Theaetetus was not attached to the Academy directly, but he surely had some informal connections with it.
In Timaeus, Plato takes on the immense task of discussing the origin and workings of the cosmos. In particular, he attempts to explain the structure of matter using the five regular solids (or polyhedra), which had been investigated already to some extent by the Pythagoreans and very thoroughly by Theaetetus. The five Platonic solids (Figure 20) are distinguished by the following properties: They are the only existing solids in which all the faces (of a given solid) are identical and equilateral, and each of the solids can be circumscribed by a sphere (with all its vertices lying on the sphere). The Platonic solids are the tetrahedron (with four triangular faces; Figure 20a), the cube (with six square faces; Figure 20b), the octahedron (with eight triangular faces; Figure 20c), the dodecahedron (with twelve pentagonal faces; Figure 20d), and the icosahedron (with twenty triangular faces; Figure 20e).
Plato combined the ideas of Empedocles (ca. 490-430 B.C.), that the four basic elements of matter are earth, water, air, and fire, with the “atomic” theory of matter (the existence of indivisible particles) of Democritus of Abdera (ca. 460 B.C.-ca. 370 B.C.). His “unified” theory suggested that each of the four elements corresponds to a different kind of fundamental particle and is represented by one of the Platonic solids. We should realize that while the details have, of course, changed considerably, the basic idea underlying Plato's theory is not that different from John Dalton's formulation of modern chemistry in the nineteenth century. According to Plato, Earth is associated with the stable cube, the “penetrating” quality of fire with the pointy and relatively simple tetrahedron, air with the “mobile” appearance of the octahedron, and water with the multifaceted icosahedron. The fifth solid, the dodecahedron, was assigned by Plato (in Timaeus) to the universe as a whole, or in his words, the dodecahedron is that “which the god used for embroidering the constellations on the whole heaven.” This is the reason why the painter Salvador Dali decided to include a huge dodecahedron floating above the supper table in his painting “Sacrament of the Last Supper” (Figure 5 on page 9).
The absence of a fundamental element to be associated with the dodecahedron was not accepted by all of Plato's followers, some of whom postulated the existence of a fifth element. Aristotle, for example, took the aether, the material of heavenly bodies which he assumed to permeate the entire universe, to be the cosmic fifth essence (“quintessence”). He posited that by pervading all matter, this fifth essence ensured that motion and change could occur, in accordance with the laws of nature. The idea of a substance that pervades all space as a necessary medium for the propagation of light continued to hold until 1887, when a famous experiment by American physicist Albert Abraham Michelson and chemist Edward Williams Morley showed that this medium does not exist (nor is it required by the modern theory of light). Basically, the experiment measured the speed of two beams of light launched in different directions. The expectation was that because of Earth's motion through the aether, the speeds of the two beams would be different, but the experiment proved categorically that they were not. The result of the Michelson-Morley experiment set Einstein on the road to the theory of relativity.
In a surprising turn of events, in 1998 two groups of astronomers discovered that not only is our universe expanding (a fact already discovered by astronomer Edwin Hubble in the 1920s), but that the expansion is accelerating.This discovery came as a total shock, since astronomers naturally assumed that, due to gravity, the expansion should be slowing down. In the same way that a ball thrown upward on Earth continuously slows down because of gravity's pull (and eventually reverses its motion), the gravitational force exerted by all the matter in the universe should cause the cosmic expansion to decelerate. The discovery that the expansion is speeding up rather than slowing down suggests the existence of some form of “dark energy” that manifests itself as a repulsive force, which in our present-day universe overcomes the attractive force of gravity. Physicists are still struggling to understand the source and nature of this “dark energy.” One suggestion is that this energy is associated with some quantum mechanical field that permeates the cosmos, a bit like the familiar electromagnetic field. Borrowing from Aristotle's invisible medium, this field has been dubbed “quintessence.” Incidentally, in Luc Besson's 1997 science fiction movie The Fifth Element, the “fifth element” of the title was taken to be the life force itself—that which animates the inanimate.
Plato's theory was much more than a symbolic association. He noted that the faces of the first four solids could be constructed out of two types of right-angled triangles, the isosceles 45°-90°-45° triangle and the 30°-90°-60° triangle. Plato went on to explain how basic “chemical reactions” could be described using these properties. For example, in Plato's “chemistry,” when water is heated by fire, it produces two particles of vapor (air) and one particle of fire. In a chemical reaction formulation, this may be written as or, in balancing the number of faces involved (in the Platonic solids that represent these elements, respectively): 20 = 2×8+4. While this description clearly does not conform with our modern understanding of the structure of matter, the central idea—that the most fundamental particles in our universe and their interactions can be described by a mathematical theory that possesses certain symmetries—is one of the cornerstones of today's research in particle physics.
To Plato, the complex phenomena that we observe in the universe are not what really matters; the truly fundamental things are the underlying symmetries, and those are never changing. This view is very much in line with modern thinking about the laws of nature. For example, these laws do not change from place to place in the universe. For this reason, we can use the same laws that we determine from laboratory experiments whether we study a hydrogen atom here on Earth or in a galaxy that is billions of light-years away. This symmetry of the laws of nature manifests itself in the fact that the quantity which we call linear momentum (equaling the product of the mass of an object and the speed, and having the direction of the motion) is conserved, namely, has the same value whether we measure it today or a year from now. Similarly, because the laws of nature do not change with the passing of time, the quantity we call energy is conserved. We cannot get energy out of nothing. Modern theories, which are based on symmetries and conservation laws, are thus truly Platonic.
The original fascination of the Pythagoreans with polyhedra may have originated from observations of pyrite crystals in southern Italy, where the Pythagorean school was located. Pyrite, commonly known as fool's gold, often has crystals with a dodecahedral shape. However, the Platonic solids, their beauty, and their mathematical properties continued to captivate the imagination of people for centuries after Plato, and they turn up in the most unexpected places. For example, in Cyrano de Bergerac's (1619-1655) science-fiction novel A Voyage to the Moon: with Some Account of the Solar World, the author uses a flying machine in the form of an icosahedron to escape from prison in a tower and to land on a sunspot.
The Golden Ratio, ö , plays a crucial role in the dimensions and symmetry properties of some Platonic solids. In particular, a dodecahedron with an edge length (the segment along which two faces join) of one unit has a total surface area of - ö and a volume of 5 ö 3/(6 − 2 ö ). Similarly, an icosahedron with a unit length edge has a volume of 5 ö 5/6.
The symmetry of the Platonic solids leads to other interesting properties. For example, the cube and the octahedron have the same number of edges (twelve), but their number of faces and vertices are in terchanged (the cube has six faces and eight vertices and the octahedron eight faces and six vertices). The same is true for the dodecahedron and icosahedron; both have thirty edges, and the dodecahedron has twelve faces and twenty vertices, while it is the other way around for the icosahedron. These similarities in the sym metries of the Platonic solids allow for interesting mappings of one solid into its dual or reciprocal solid. If we connect the centers of all the faces of a cube, we obtain an octahedron (Figure 21), while if we connect the centers of the faces of an octahedron, we obtain a cube. The same procedure can be applied to map an icosahedron into a dodecahedron and viceversa, and the ratio of the edge lengths of the two solids (one embedded in the other) that are obtained can again be expressed in terms of the Golden Ratio, as . The tetrahedron is selfreciprocating—joining the four centers of the tetrahedron's faces makes another tetrahedron.
While not all the properties of the Platonic solids were known in antiquity, neither Plato nor his followers failed to see their sheer beauty. To some extent, even the initial difficulties in constructing these figures (until methods using the Golden Ratio were found) could be taken as their attributes. After all, the last sentence in Hippias Major reads: “All that is beautiful is difficult.” In “On the Failure of the Oracles,” the Greek historian Plutarch (ca. 46-ca. 120) writes: “A pyramid [a tetrahedron], an octahedron, an icosahedron, and a dodecahedron, the primary figures which Plato predicates, are all beautiful because of the symmetries and equalities in their relations, and nothing superior or even like to these has been left for Nature to compose and fit together. ”
As noted above, the icosahedron and the dodecahedron are intimately related to the Golden Ratio, in more ways than one. For example, the twelve vertices of any icosahedron can be di vided into three groups of four, with the vertices of each group lying at the corners of a Golden Rectangle (a rectangle in which the ratio of length to width is the Golden Ratio). The rectangles are perpendicular to each other, and their one common point is the center of the icosahedron (Figure 22). Similarly, the centers of the twelve pentagonal faces of the dodecahedron can be divided into three groups of four, and each of those groups also forms a Golden Rectangle.
The close associations between some plane figures, such as the pentagon and the pentagram, and some solids, such as the Platonic solids, and the Golden Ratio lead to the inescapable conclusion that the Greek interest in the Golden Ratio probably started with attempts to construct such plane figures and solids. Most of this mathematical effort occurred around the beginning of the fourth century B.C. There exist, however, numerous claims that the Golden Ratio is embodied in the architectural design of the Parthenon, which was built and decorated between 447 and 432 B.C., under the rule of Pericles. Can these claims be verified?
THE VIRGIN'S PLACE
The Parthenon (“the virgin's place” in Greek) was built on the Acropolis in Athens as a temple sacred to the cult of Athena Parthenos (Athena the Virgin). The architects were Ictinus and Callicrates, and Phidias and his assistants and students were charged with supervising the sculptures. Sculptured groups ornamented the pediments terminating the roof at the eastern and western ends. One group depicted the birth of Athena and the other the contest between Athena and Poseidon.
Somewhat deceptive in its simplicity, the Parthenon remains one of the finest architectural expressions of the ideal of clarity and unity. On September 26, 1687, Venetian artillery hit the Parthenon directly, during an attack on the Ottoman Turks who held Athens at the time and who used the Parthenon as a powder magazine. While the damage was extensive, the basic structure remained intact. In describing this event, General Königsmark, who accompanied the field commander, wrote: “How it dismayed His Excellency to destroy the beautiful tem ple which had existed three thousand years!” Numerous attempts have been made, especially since the end of the Turkish control (in 1830), to discover some mathematical or geometrical basis supposedly employed to achieve the Parthenon design's high perfection. Most books on the Golden Ratio state that the dimensions of the Parthenon, while its triangular pediment was still intact, fit precisely into a Golden Rectangle. This statement is usually accompanied by a figure similar to that in Figure 23. The Golden Ratio is assumed to feature in other dimensions of the Parthenon as well. For example, in one of the most extensive works on the Golden Ratio, Adolph Zeising's Der Goldne Schnitt (The golden section; published in 1884), Zeising claims that the height of the façade from the top of its tympanum to the bottom of the pedestal below the columns is also di vided in a Golden Ratio by the top of the columns. This statement was repeated in many books, such as Matila Ghyka's influential Le Nombre d'or (The golden number; appeared in 1931). Other authors, such as Miloutine Borissavlievitch in The Golden Number and the Scientific Aesthetics of Architecture (1958), while not denying the presence of ö in the Parthenon's design, suggest that the temple owes its harmony and beauty more to the regular rhythm introduced by the repetition of the same column (a concept termed the “law of the Same”).
The appearance of the Golden Ratio in the Parthenon was seriously questioned by University of Maine mathematician George Markowsky in his 1992 College Mathematics Journal article “Misconceptions about the Golden Ratio.” Markowsky first points out that invariably, parts of the Parthenon (e.g., the edges of the pedestal; Figure 23) actually fall outside the sketched Golden Rectangle, a fact totally ignored by all the Golden Ratio enthusiasts. More important, the dimensions of the Parthenon vary from source to source, probably because different reference points are used in the measurements. This is another example of the number-juggling opportunity afforded by claims based on measured dimensions alone. Using the numbers quoted by Marvin Trachtenberg and Isabelle Hyman in their book Architecture: From Prehistory to Post-Modernism (1985), I am not convinced that the Parthenon has anything to do with the Golden Ratio. These authors give the height as 45 feet 1 inch and the width as 101 feet 3.75 inches. These dimensions give a ratio of width/height of approximately 2.25, far from the Golden Ratio of 1.618… Markowsky points out that even if we were to take the height of the apex above the pedestal upon which the series of columns stands (given as 59 feet by Stuart Rossiter in his 1977 book Greece), we still would obtain a width/height ratio of about 1.72, which is closer to but still significantly different from the value of ö. Other researchers are also skeptical about phi's role in the Parthenon's design. Christine Flon notes in The World Atlas of Architecture (1984) that while “it is not unlikely that some architects… should have wished to base their works on a strict system of ratios … it would be wrong to generalize.”
So, was the Golden Ratio used in the Parthenon's design? It is difficult to say for sure. While most of the mathematical theorems concerning the Golden Ratio (or “extreme and mean ratio”) appear to have been formulated after the Parthenon had been constructed, considerable knowledge existed among the Pythagoreans prior to that. Thus, the Parthenon's architects might have decided to base its design on some prevailing notion for a canon for aesthetics. However, this is far less certain than many books would like us to believe and is not particularly well supported by the actual dimensions of the Parthenon.
Whether or not the Golden Ratio features in the Parthenon, what is clear is that whichever mathematical “programs” concerning the Golden Ratio were instituted by the Greeks in the fourth century B.C., that work culminated in the publication of Euclid's Elements, in around 300 B.C. Indeed, from a perspective of logic and rigor, the Elements was long thought to be an apotheosis of certainty in human knowledge.
EXTREME AND MEAN RATIO
In 336 B.C., twenty-year-old Alexander (the Great) of Macedonia succeeded to the throne and, by a sequence of brilliant victories, conquered most of Asia Minor, Syria, Egypt, and Babylon and became ruler of the Persian Empire. A few years before his death at the young age of thirty-three, he founded what became the greatest monument to his name—the city of Alexandria near the mouth of the Nile.
Alexandria was located at the crossroads of three great civilizations: Egyptian, Greek, and Jewish. Consequently, it became an extraordinary intellectual center that lasted for centuries and the birthplace of such remarkable achievements as the Septuagint (meaning “translation of the 70”)—the Greek translation of the Old Testament, traditionally attributed to seventy-two translators. The translation was begun in the third century B.C., and the work progressed in several stages over about a century.
After Alexander's death, Ptolemy I gained control over Egypt and the African dominions by 306 B.C., and among his first actions was the establishment of the equivalent of a university (known then as the Museum) in Alexandria. This institution included a library, which, following an immense gathering effort, was reputed to hold at one time 700,000 books (some confiscated from unlucky tourists). The first staff of teachers at the Alexandria school included Euclid, the author of the best-known book in the history of mathematics—the Elements (Stoi-chia). In spite of Euclid being a “best-selling” author (only the Bible sold more books than Elements until the twentieth century), his life is so veiled in obscurity that even his birthplace in unknown. Given the contents of the Elements, it is very likely that Euclid studied mathematics in Athens with some of Plato's students. Indeed, Proclus writes about Euclid: “This man lived in the time of the first Ptolemy … he is then younger than the pupils of Plato, but older than Eratosthenes and Archimedes.”
The Elements, a thirteen-volume work on geometry and number theory, is so colossal in its scope that we sometimes tend to forget that Euclid was the author of almost a dozen other books, covering topics from music through mechanics to optics. Only four of these other treatises survived to the present day: Division of Figures, Optics, Phaenomena, and Data. Optics contains some of the earliest studies of perspective.
Few will disagree that the Elements is the greatest and most influential mathematical textbook ever written. A story has it that when Abraham Lincoln wanted to understand the true meaning of the word “proof in the legal profession, he started to study the Elements in his cabin in Kentucky. The famous British logician and philosopher Bertrand Russell describes in his Autobiography his first encounter with Euclid's Elements (at age eleven!) as “one of the great events of my life, as dazzling as first love.”
The picture of the author that emerges from the pages of the Elements is that of a conscientious man, respectful of tradition, and very modest. Nowhere does Euclid attempt to take credit for work that was not originally his. In fact, he claims no originality whatsoever, in spite of the fact that it is very obvious that he contributed many new proofs, totally rearranged the contents contributed by others to entire volumes, and designed the whole work. Euclid's scrupulous fairness and modesty gained him the admiration of Pappus of Alexandria, who around A.D. 340 composed an eight-volume work entitled Collection (Synagoge), which provides an invaluable record of many aspects of Greek mathematics.
In the Elements, Euclid attempted to encompass most of the mathematical knowledge of his time. Books I to VI deal with the plane geometry we learn in school and which has become synonymous with Euclid's name (Euclidean geometry). Of these books, I, II, IV, and VI discuss lines and plane figures, while Book III presents theorems related to the circle, and Book V gives an extensive account of the work on proportion originated by Eudoxus of Cnidus (ca. 408-355 B.C.). Books VII to × deal with number theory and the foundations of arithmetic. In particular, irrational numbers are elaborated on in Book X, the contents of which are mostly the work of Theaetetus. Book XI provides the basis for solid geometry; Book XII (mostly describing the work of Eudoxus) proves the theorem for the area of the circle, and Book XIII (due mostly to Theaetetus) demonstrates the constructions of the five Platonic solids.
Still in ancient times, Hero (in the first century A.D.), Pappus (in the fourth century), and Proclus (in the fifth century), all of Alexandria, and Simplicius of Athens (in the sixth century) all wrote commentaries on the Elements. A new revision of the work, by Theon of Alexandria, appeared in the fourth century A.D. and served as the basis for all translations until the nineteenth century, when a manuscript containing a somewhat different text was discovered in the Vatican. In the Middle Ages, the Elements was translated into Arabic three times. The first of these translations was carried out by al-Hajjaj ibn Yusuf ibn Matar, at the request of Caliph Harun ar-Rashid (ruled 786-809), who is familiar to us through the stories in The Arabian Nights. The Elements was first made known in Western Europe through Latin translations of the Arabic versions. English Benedictine monk Adelard of Bath (ca. 1070-1145), who according to some stories was traveling in Spain disguised as a Muslim student, got hold of an Arabic text and completed the translation into Latin around 1120. This translation became the basis of all editions in Europe until the sixteenth century. Translations into many modern languages followed.
While Euclid himself may not have been the greatest mathematician who ever lived, he was certainly the greatest teacher of mathematics. The textbook he wrote remained in use practically unaltered for more than two thousand years, until the middle of the nineteenth century. Even the fictional detective Sherlock Holmes, in Arthur Conan Doyle's A Study in Scarlet, claimed that his conclusions, achieved by deduction, were “as infallible as so many propositions of Euclid.”
The Golden Ratio appears in the Elements in several places. The first definition of the Golden Ratio (“extreme and mean ratio”), in relation to areas, is given somewhat obliquely in Book II. A second, clearer definition, in relation to proportion, appears in Book VI. Euclid then uses the Golden Ratio, especially in the construction of the pentagon (in Book IV) and in the construction of the icosahedron and dodecahedron (in Book XIII).
Let me use some very simple geometry to examine Euclid's definition and explain why the Golden Ratio is so important for the construction of the pentagon. In Figure 24, the line AB is divided by point C. Euclid's definition in Book VI of extreme and mean ratio is such that: (larger segment)/(shorter segment) is equal to (whole line/larger segment). In other words, in Figure 24:
How is this line division related to the pentagon? In any regular planar figure (those with equal sides and interior angles; known as regular polygons), the sum of all the interior angles is given by 180 (n − 2), where n is the number of sides. For example, in a triangle n = 3, and the sum of all the angles is equal to 180 °. In a pentagon n = 5, and the sum of all the angles is equal to 540 °. Every angle of the pentagon is therefore equal to 540/5 = 108 °. Imagine now that we draw two adjacent diagonals in the pentagon, as in Figure 25a, thus forming three isosceles (with two equal sides) triangles. Since the two angles near the base of an isosceles triangle are equal, the base angles in the two triangles on the sides are 36 ° each [half of (180° − 108°)]. We therefore obtain for the angles of the middle triangle the values 36-72-72 (as marked in Figure 25a). If we bisect one of the two 72-de-gree base angles (as in Figure 25b), we obtain a smaller triangle DBC with the same angles (36-72-72) as the large one ADB. Using very elementary geometry, we can show that according to Euclid's definition, point C divides the line AB precisely in a Golden Ratio. Furthermore, the ratio of AD to DB is also equal to the Golden Ratio. (A short proof is given in Appendix 4.) In other words, in a regular pentagon, the ratio of the diagonal to the side is equal to ö. This fact illustrates that the ability to construct a line divided in a Golden Ratio provides at the same time a simple means of constructing the regular pentagon. The construction of the pentagon was the main reason for the Greek interest in the Golden Ratio. The triangle in the middle of Figure 25a, with a ratio of side to base of ö, is known as a Golden Triangle; the two triangles on the sides, with a ratio of side to base of 1/ ö , are sometimes called Golden Gnomons. Figure 25bdemonstrates a unique property of Golden Triangles and Golden Gnomons—they can be dissected into smaller triangles that are also Golden Triangles and Golden Gnomons.
The association of the Golden Ratio with the pentagon, fivefold symmetry, and the Platonic solids is interesting in itself and, indeed, was more than sufficient to ignite the curiosity of the ancient Greeks. The Pythagorean fascination with the pentagon and the pentagram, coupled with Plato's interest in the regular solids and his belief that the latter represented fundamental cosmic entities, prompted generations of mathematicians to labor on the formulation of numerous theorems concerning ö. Yet the Golden Ratio would not have reached the level of almost reverential status that it eventually achieved were it not for some truly unique algebraic properties. In order to understand these properties, we need first to find the precise value of ö.
Examine again Figure 24, and let us take the length of the shorter segment, CB, to be 1 unit and the length of the longer one, AC, to be x units. If the ratio of x to 1 is the same as that of x+1 (length of the line AB) to x, then the line has been cut in extreme and mean ratio. We can easily solve for the value, x, of the Golden Ratio. From the definition of extreme and mean ratio
Multiplying both sides by x, we get x = x+1, or the simple quadratic equation
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 ö 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] button. Do you see something surprising? Now enter the number again, and this time hit the [1/x] button. 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/ ö.
Paul S. Bruckman of Concord, California, published in 1977 in the journal 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:
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 ö to the 4,599th decimal place. (The result was published in the 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 ö to the 2,000th decimal place:
Intriguing as they are, you may think that the properties of ö 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.
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 cumbersome way could be to start by calculating (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 x. We therefore have
Now let us square both sides of this equation. The square of x is 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. We therefore obtain the quadratic equation x = 1+x. But this is precisely the equation that defines the Golden Ratio! We therefore found that our endless expression is actually equal to ö.
Let us now look at a very different type of never-ending expression, this time involving fractions:
This is a special case of mathematical entities known as 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. Thus,
Note, however, that because the continued fraction goes on forever, the denominator of the second term on the right-hand side is in fact identical to x itself. We therefore have the equation
Multiplying both sides by x2, we get x = x+1, which is again the equation defining the Golden Ratio! We find that this remarkable continued fraction is also equal to ö. 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. 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 ö. 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 ö. Continuing this process ad infinitum, we will produce smaller and smaller Golden Rectangles (each time with dimensions “deflated” by a factor ö ). If we were to examine the ever-decreasing (in size) rectangles with a magnifying glass of increasing power, they would all look identical. The Golden Rectangle is the 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 Figure 26, and they will all intersect at the same point. The series of continuously diminishing 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 God.”
If you did not find it mind-boggling that all of these diverse mathematical circumstances lead to ö , 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, 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 (often referred to as Book XIV) contains an important theorem concerning a dodecahedron and an icosahedron that are circumscribed by the same sphere. The text of the “Supplement” is attributed to Hypsicles of Alexandria, who probably lived in the second century B.C., but it is believed to contain theorems by Apollonius of Perga (ca. 262-190 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.). Developments concerning the Golden Ratio become more sparse after that and are associated mainly with Hero (in the first century A.D.), Ptolemy (in the second century A.D.), and Pappus (in the fourth century). In his 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. 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, Hē Mathēmatikē Syntaxis (The mathematical synthesis). The book later became known as The Great Astronomer. However, ninth-century Arab astronomers referred to the book invoking the Greek superlative “Megistē” (“the greatest”) but prefixing it with the Arabic identifier of proper names, “al.” The book thereby became known, to this day, as the Almagest. Ptolemy also did important work in geography and wrote an influential book entitled Guide to Geography.
In the Almagest and the 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 associated 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; ca. 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 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 (538-594) 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 Āryabhata (476-ca. 550). In his best-known book, entitled Āryabhattya, 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 surpass 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 (786-833) established in Baghdad the Beit al-hikma (House of wisdom), which operated in a similar fashion 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-Khwārizmī and Abu Kamil Shuja in the ninth century and Abu'l-Wafa in the tenth century.
Mohammed ibn-Musa al-Khwārizmī composed, in Baghdad (at about 825), what is considered to be the most influential algebraic work of the period—Kitāb al-jabr wa al-muqābalah (The science of restoration and reduction). From this title (“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-Khwārizmī's name. 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 thingby ten; it is ten things. 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, and 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 the possible solutions. In his 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 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 Algebra 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 (940-998). 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, 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. 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 compass, in which the angle between the two legs of the compass is kept fixed (known as “rusty compass” constructions). This particular genre was probably inspired by Pappus' Collection but may also represent Abu 'l- Wafa's response to a practical problem—the results with a fixed-angle compass 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.