21

The Golden Ratio

Suppose you are being asked to divide a line segment into two parts such that the whole segment is to the longer part as the longer part is to the shorter. The Greeks were greatly intrigued by this seemingly simple problem, but exactly why is not quite clear: perhaps it was posed by an anonymous scholar as an exercise to his students, or it may have arisen from the challenge of constructing a regular pentagon with straightedge and compass (see the next chapter). Whatever its origins, this particular division of a line segment into two parts became known as the golden section (sectio aura in Latin). The ratio between the lengths of the two parts is called the golden ratio and is usually denoted by the Greek letter ϕ (phi), although some authors denote it by τ (tau).

Let the line segment be of unit length (figure 21.1). Denoting the length of the longer part by x, the problem leads to the equation

which, after rearranging, yields the quadratic equation x² + x − 1 = 0. This equation has two solutions, one positive and one negative; but since x stands for length, it cannot be negative. Using the familiar quadratic formula and taking only the positive solution, we get

or about 0.618. The golden ratio ϕ, by definition, is 1/x. A little arithmetic manipulation will show that , which, you will notice, differs from x by exactly 1. Thus, the decimal value of ϕ is about 1.618.

The number ϕ enjoys many interesting properties, some quite surprising. We have already noticed that ϕ = 1 + 1/ϕ. Multiplying both sides of this equation by ϕ results in ϕ² = ϕ + 1. Multiplying again by ϕ gives us ϕ ³ = ϕ² + ϕ = (ϕ + 1) + ϕ = 2ϕ + 1. This process can be repeated; replacing ϕ² by ϕ + 1 at each step and collecting like terms, we get the sequence:

The coefficients in these expressions turn out to be none other than the Fibonacci numbers! In the previous chapter we saw that the ratio of two consecutive Fibonacci numbers approaches the golden ratio as we move higher up in the sequence; now we have a second example of how seemingly unrelated mathematical objects may in fact be intimately connected. Who would have expected that the golden section—a purely geometric entity—would have anything to do with the Fibonacci numbers, whose origin is in number theory?

The golden section has also found its way into art and architecture. It has been claimed that the Greeks, in their obsessive quest for aesthetic perfection, have examined rectangles of various proportions and found that the rectangle whose length-to-width ratio is equal to the golden ratio appeared to be the most aesthetically pleasing. They may have used this proportion in the construction of their temples. The famous Parthenon in Athens has the approximate proportions of the golden ratio, but whether it was built specifically with this ratio in mind is the subject of an ongoing debate.¹ Leonardo da Vinci noticed that if a man stretches his hands to their full extent eagle-manner, the ratio of his height to the wingspan of his hands is approximately 1.6—very close to ϕ. And in our own time, someone with a keen eye has noticed that the dimensions of a standard credit card are 8.5 × 5.3 cm, resulting in a length-to-width ratio of 1.60377—within less than 1 percent of ϕ.

But how close is close? Is 3 a close approximation to π? If so, then we can find a connection to π whenever 3 shows up, which is to say, everywhere. Once you accept approximations into the picture, you are opening the door to endless speculation, which is indeed what happened with the golden ratio. Regardless of whether its connections to art and architecture are purely accidental or are anchored in some hidden principle, they have greatly added to the aura of mystique that surrounds this number. No wonder medieval scholars dubbed it the divine proportion.

Plate 21 showcases a sample of the many occurrences of the golden ratio in art and nature. Some of the panels describe scenes mentioned earlier in this chapter. For the remaining panels, we’ll use a “coordinate” notation (x, y) to identify each panel, x standing for the row number (counting from top to bottom) and y for the column (from left to right).

In panel (1, 1) we see a regular pentagon and its associated pentagram, with a smaller pentagon nested inside; as we will see in the next chapter, the golden ratio is the key to constructing a regular pentagon with straightedge and compass.

Panel (2, 1) shows one way of dividing a line segment (here of length 2) in the ratio ϕ:1; the dot on the base line marks the point of division. Panel (4, 2) shows another way; the horizontal line is parallel to the triangle’s base and cuts the two lateral sides at their midpoints.

Panel (2, 3) is rather intriguing. We see an infinite succession of nested square roots, with 1 added to each radical before taking the next, like a Russian Matryoshka doll inside which resides a smaller doll, a still smaller one inside that, and so on. Surprisingly, this infinite expression converges to ϕ.²

Panels (3, 1), (3, 2), and (3, 3) bring the golden ratio into three-dimensional space. Take three identical cards, each of length-to-width ratio equal to ϕ. Cut a slit along the centerline in two of the cards, equal in length to the card’s width. Insert the cards into each other as shown, resulting in a three-dimensional structure with 12 corners. Connect pairs of adjacent corners, and you get an icosahedron, a polyhedron with 20 equilateral triangles as faces—one of the five regular solids we met in chapter 4. Interestingly, the two structures appear on the logo of two organizations devoted to mathematical research and teaching—the stacked cards as the icon of the German DFG Research Center Matheon and the icosahedron on the logo of the Mathematical Association of America.

NOTES:

1. On this subject see Livio, The Golden Ratio.

2. To see this, let us denote the entire expression by x (assuming, of course, that the nested radicals indeed converge to a limit). Since the same x also appears inside the radical, we have the equation . Squaring both sides and rearranging terms, we get the quadratic equation x² − x − 1 = 0, whose positive solution is , the golden ratio (we take only the positive solution, because by definition the symbol is the positive solution of the equation x² = a).