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By Root 759 0

work on geometry, we find a definition of a certain division of a line into two unequal parts (an earlier definition, in terms of areas, appears in book II). According to Euclid, if a line AB is divided by a point C (figure 62) in such a way that the ratio of the lengths of the two segments (AC/CB) is equal to the whole line divided by the longer segment (AB/AC), then the line is said to have been divided in “extreme and mean ratio.” In other words, if AC/CB AB/AC, then each one of these ratios is called the “extreme and mean ratio.” Since the nineteenth century, this ratio is popularly known as the golden ratio. Some easy algebra can show that the golden ratio is equal to

(1 + √5) / 2 = 1.6180339887…

The first question you may ask is why did Euclid even bother to define this particular line division and to give the ratio a name? After all, there are infinitely many ways in which a line could be divided. The answer to this question can be found in the cultural, mystical heritage of the Pythagoreans and Plato. Recall that the Pythagoreans were obsessed with numbers. They thought of the odd numbers as being masculine and good, and, rather prejudicially, of the even numbers as being feminine and bad. They had a particular affinity for the number 5, the union of 2 and 3, the first even (female) and first odd (masculine) numbers. (The number 1 was not considered to be a number, but rather the generator of all numbers.) To the Pythagoreans, therefore, the number 5 portrayed love and marriage, and they used the pentagram—the five-pointed star (figure 63)—as the symbol of their brotherhood. Here is where the golden ratio makes its first appearance. If you take a regular pentagram, the ratio of the side of any one of the triangles to its implied base (a/b in figure 63) is precisely equal to the golden ratio. Similarly, the ratio of any diagonal of a regular pentagon to its side (c/d in figure 64) is also equal to the golden ratio. In fact, to construct a pentagon using a straight edge and a compass (the common geometrical construction process of the ancient Greeks) requires dividing a line into the golden ratio.

Figure 62

Plato added another dimension to the mythical meaning of the golden ratio. The ancient Greeks believed that everything in the universe is composed of four elements: earth, fire, air, and water. In Timaeus, Plato attempted to explain the structure of matter using the five regular solids that now bear his name—the Platonic solids (figure 65). These convex solids, which include the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron, are the only ones in which all the faces (of each individual solid) are the same, and are regular polygons, and where all the vertices of each solid lie on a sphere. Plato associated each of four of the Platonic solids with one of the four basic cosmic elements. For instance, the Earth was associated with the stable cube, the penetrating fire with the pointy tetrahedron, air with the octahedron, and water with the icosahedron. Concerning the dodecahedron (Figure 65d), Plato wrote in Timaeus: “As there still remained one compound figure, the fifth, God used it for the whole, broidering it with designs.” So the dodecahedron represented the universe as a whole. Note, however, that the dodecahedron, with its twelve pentagonal surfaces, has the golden ratio written all over it. Both its volume and its surface area can be expressed as simple functions of the golden ratio (the same is true for the icosahedron).

Figure 63

Figure 64

History therefore shows that by numerous trials and errors, the Pythagoreans and their followers discovered ways to construct certain geometrical figures that to them represented important concepts, such as love and the entire cosmos. No wonder, then, that they, and Euclid (who documented this tradition), invented the concept of the golden ratio that was involved in these constructions, and gave it a name. Unlike any other arbitrary ratio, the number 1.618…now became the focus of an intense and rich history of investigation,