Part 2 (1/2)

Figure 20 In Timaeus 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 circ.u.mscribed by a sphere (with all its vertices lying on the sphere). The Platonic solids are the tetrahedron (with four triangular faces; Figure 20a Figure 20a), the cube (with six square faces; Figure 20b Figure 20b), the octahedron (with eight triangular faces; Figure 20c Figure 20c), the dodecahedron (with twelve pentagonal faces; Figure 20d Figure 20d), and the icosahedron (with twenty triangular faces; Figure 20e Figure 20e).

Plato combined the ideas of Empedocles (ca. 490430 B.C. 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. B.C.ca. 370 B.C. 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 a.s.sociated 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 a.s.signed by Plato (in Timaeus) Timaeus) to the universe as a whole, or in his words, the dodecahedron is that ”which the G.o.d 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” ( to the universe as a whole, or in his words, the dodecahedron is that ”which the G.o.d 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). on page 9).

The absence of a fundamental element to be a.s.sociated 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 a.s.sumed 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 s.p.a.ce 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. accelerating. This discovery came as a total shock, since astronomers naturally a.s.sumed 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 a.s.sociated 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 This discovery came as a total shock, since astronomers naturally a.s.sumed 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 a.s.sociated 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, the ”fifth element” of the t.i.tle was taken to be the life force itself-that which animates the inanimate.

Plato's theory was much more than a symbolic a.s.sociation. 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 = 28+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 quant.i.ty which we call linear momentum linear momentum (equaling the product of the ma.s.s 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 pa.s.sing of time, the quant.i.ty 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. (equaling the product of the ma.s.s 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 pa.s.sing of time, the quant.i.ty 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 (16191655) science-fiction novel A Voyage to the Moon: with Some Account of the Solar World 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, , 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 and a volume of 5 and a volume of 5 3 3/(6 2 ). Similarly, an icosahedron with a unit length edge has a volume of 5 5 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. . The tetrahedron is selfreciprocating-joining the four centers of the tetrahedron's faces makes another tetrahedron.

Figure 21 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 Hippias Major reads: ”All that is beautiful is difficult.” In ”On the Failure of the Oracles,” the Greek historian Plutarch (ca. 46ca. 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. ” reads: ”All that is beautiful is difficult.” In ”On the Failure of the Oracles,” the Greek historian Plutarch (ca. 46ca. 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. ”

Figure 22 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 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 ( (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 a.s.sociations 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. 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 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. 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 a.s.sistants 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 Konigsmark, 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 Figure 23. The Golden Ratio is a.s.sumed 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 Der Goldne Schnitt (The golden section; published in 1884), Zeising claims that the height of the facade 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 (The golden section; published in 1884), Zeising claims that the height of the facade 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 Le Nombre d'or (The golden number; appeared in 1931). Other authors, such as Miloutine Borissavlievitch in (The golden number; appeared in 1931). Other authors, such as Miloutine Borissavlievitch in The Golden Number and the Scientific Aesthetics of Architecture The Golden Number and the Scientific Aesthetics of Architecture (1958), while not denying the presence of in the Parthenon's design, suggest that the temple owes its harmony and beauty more to the regular rhythm introduced by the repet.i.tion of the same column (a concept termed the ”law of the Same”). (1958), while not denying the presence of in the Parthenon's design, suggest that the temple owes its harmony and beauty more to the regular rhythm introduced by the repet.i.tion of the same column (a concept termed the ”law of the Same”).

Figure 23 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 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; 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 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 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 (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) 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 . Other researchers are also skeptical about phi's role in the Parthenon's design. Christine Flon notes in The World Atlas of Architecture 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.” (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 inst.i.tuted by the Greeks in the fourth century B.C. B.C., that work culminated in the publication of Euclid's Elements Elements, in around 300 B.C. B.C. Indeed, from a perspective of logic and rigor, the Indeed, from a perspective of logic and rigor, the Elements Elements was long thought to be an apotheosis of certainty in human knowledge. was long thought to be an apotheosis of certainty in human knowledge.

EXTREME AND MEAN RATIO.

In 336 B.C. 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. 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. B.C., and among his first actions was the establishment of the equivalent of a university (known then as the Museum) in Alexandria. This inst.i.tution 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). Elements (Stoi-chia). In spite of Euclid being a ”best-selling” author (only the Bible sold more books than In spite of Euclid being a ”best-selling” author (only the Bible sold more books than Elements Elements until the twentieth century), his life is so veiled in obscurity that even his birthplace in unknown. Given the contents of the until the twentieth century), his life is so veiled in obscurity that even his birthplace in unknown. Given the contents of the Elements 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 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 Division of Figures, Optics, Phaenomena, and Data. Optics Data. Optics contains some of the earliest studies of perspective. contains some of the earliest studies of perspective.

Few will disagree that the Elements 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 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 Elements in his cabin in Kentucky. The famous British logician and philosopher Bertrand Russell describes in his in his cabin in Kentucky. The famous British logician and philosopher Bertrand Russell describes in his Autobiography Autobiography his first encounter with Euclid's his first encounter with Euclid's Elements Elements (at age eleven!) as ”one of the great events of my life, as dazzling as first love.” (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 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 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. A.D. 340 composed an eight-volume work ent.i.tled 340 composed an eight-volume work ent.i.tled Collection (Synagoge) Collection (Synagoge), which provides an invaluable record of many aspects of Greek mathematics.

In the Elements Elements, Euclid attempted to encompa.s.s 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. 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. 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. Elements. A new revision of the work, by Theon of Alexandria, appeared in the fourth century A new revision of the work, by Theon of Alexandria, appeared in the fourth century A.D. A.D. and served as the basis for all translations until the nineteenth century, when a ma.n.u.script containing a somewhat different text was discovered in the Vatican. In the Middle Ages, the and served as the basis for all translations until the nineteenth century, when a ma.n.u.script containing a somewhat different text was discovered in the Vatican. In the Middle Ages, the Elements 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-Ras.h.i.+d (ruled 786809), who is familiar to us through the stories in 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-Ras.h.i.+d (ruled 786809), who is familiar to us through the stories in The Arabian Nights. The Arabian Nights. The The Elements Elements was first made known in Western Europe through Latin translations of the Arabic versions. English Benedictine monk Adelard of Bath (ca. 10701145), 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. was first made known in Western Europe through Latin translations of the Arabic versions. English Benedictine monk Adelard of Bath (ca. 10701145), 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 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 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). 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).

Figure 24 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 Figure 24, the line AB AB is divided by point is divided by point C. 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 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 Figure 24: AC/CB=AB/AC.

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) polygons), the sum of all the interior angles is given by 180 (n 2), where 2), where n n is the number of sides. For example, in a triangle is the number of sides. For example, in a triangle n = n = 3, and the sum of all the angles is equal to 180 . In a pentagon 3, and the sum of all the angles is equal to 180 . In a pentagon n = 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 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 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 Figure 25a). If we bisect one of the two 72-de-gree base angles (as in Figure 25b Figure 25b), we obtain a smaller triangle DBC DBC with the same angles (36-72-72) as the large one with the same angles (36-72-72) as the large one ADB. ADB. Using very elementary geometry, we can show that according to Euclid's definition, point Using very elementary geometry, we can show that according to Euclid's definition, point C C divides the line divides the line AB AB precisely in a Golden Ratio. Furthermore, the ratio of precisely in a Golden Ratio. Furthermore, the ratio of AD AD to to DB 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 . This fact ill.u.s.trates 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 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 . This fact ill.u.s.trates 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 Figure 25a, with a ratio of side to base of , is known as a Golden Triangle; Golden Triangle; the two triangles on the sides, with a ratio of side to base of 1/ , are sometimes called the two triangles on the sides, with a ratio of side to base of 1/ , are sometimes called Golden Gnomons. Golden Gnomons. Figure 25b Figure 25b demonstrates a unique property of Golden Triangles and Golden Gnomons-they can be dissected into smaller triangles that are also Golden Triangles and Golden Gnomons. demonstrates a unique property of Golden Triangles and Golden Gnomons-they can be dissected into smaller triangles that are also Golden Triangles and Golden Gnomons.

Figure 25 The a.s.sociation 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 ent.i.ties, prompted generations of mathematicians to labor on the formulation of numerous theorems concerning . 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 .

Examine again Figure 24 Figure 24, and let us take the length of the shorter segment, CB CB, to be 1 unit and the length of the longer one, AC AC, to be x x units. If the ratio of units. If the ratio of x x to 1 is the same as that of to 1 is the same as that of x x+1 (length of the line AB) AB) to to x x, then the line has been cut in extreme and mean ratio. We can easily solve for the value, x x, of the Golden Ratio. From the definition of extreme and mean ratio

Multiplying both sides by x x, we get x = x x = x+1, or the simple quadratic equation