The golden ratio is one of the most important numbers in all of mathematics. Recall that one can construct a rectangle in which the ratio of the longer side to the shorter side is equal to the ratio of the sum of the two sides to the longer side. This ratio is called the golden ratio. To determine this ratio exactly, let us denote the length of the longer side by x and let the smaller side have length 1. Then by our requirement, we must have x:1=(1+x):x.

Multiplying the equation through by x and rearranging terms leads to the quadratic equation x·x − x −1 = 0 , and the quadratic formula then yields two solutions,
1±√5/2. Only one of these solutions is positive, and it is equal to 1.6180….

Some are of the opinion that rectangles whose sides are in the golden ratio are particularly aesthetically pleasing. Indeed, the golden ratio is to be found frequently in architecture, for example in the outlines of ancient Greek temples and in modern buildings as well (for instance, the tower on the Leipzig town hall is divided into sections that correspond to the golden ratio). However, in our daily lives, where we are confronted with pieces of paper in the international standard A series, for which a sheet folded in half has the same proportions as the original, we have become more accustomed to the ratio of the sides being equal to the square root of 2, or 1.414….

The importance of the golden ratio is revealed in the fact that it pops up in one way or another in almost every branch of mathematics. It is certainly no surprise that it plays a role in geometry, since the term was introduced as a result of a geometric problem. But it also appears in situations in which only numbers are involved. For example, consider the famous Fibonacci sequence. It begins 1, 1, and then each succeeding term is the sum of the two that precede it. Therefore, after 1,1, we have 2, 3, 5, 8, 13, 21, and so on. The quotient of two successive terms approaches ever more closely the golden ratio. Even 21/13=1.615… is a good approximation.

Fibonacci numbers pop up in nature, for example, in the arrangement of the seeds of a sunflower. And if you have a tape measure handy, you can search for the golden ratio on your own person. The relationship "distance from elbow to fingertips divided by distance from elbow to wrist" is just one of many possible examples.

However, such results lead one quickly into the realm of speculation. Perhaps the ratio of the number of "bad" characters in Grimm's fairy tales to the number of "good" characters is equal to the golden ratio.

Continued Fractions

The golden ratio is a remarkable number for yet another reason. This time it comes about in a type of approximation. In dealing with numbers that cannot be represented as fractions, it is convenient to replace such a number with a fraction that has a relatively small numerator and denominator and that represents a good approximation to the original number. For example, the fraction 22/7=3.14285… represents a good approximation to the circle number π (Pi), whose value to five decimal places is 3.14159. This approximation was known to the Egyptians 2500 years ago, and such an approximation suffices for many everyday applications.

The best rational approximations to a number are obtained by continued fractions. these are fractions that arise from a rather complicated process. This is how it works.

A continued fraction is written as a finite sequence of natural numbers enclosed in square brackets. The notation is interpreted as follows:

[a_0]= a_0,
[a_0, a_1] = a_0+1/[a_1],
[a_0, a_1, a_2] = a_0+1/[a_1,a_2],
[a_0, a_1, a_2, a_3] = a_0+ 1/[a_1,a_2,a_3],

...

If that seems too abstract, here are a couple of concrete examples:

[3,9] =3 + 1/9 =28/9,
[2,3,5,7] = 2+(1/(3+1/(5+1/7)))=266/115.

If one approximates a number by the best-possible continued fraction, then the larger the numbers in the continued fraction, the better the approximation in general. That is because all the numbers after the first appear in the denominator of the fraction that the continued fraction represents, and the larger the numbers, the faster the denominator grows; therefore, more decimal places of the number being approximated will be accurately represented.

Returning to the golden section, this number has the remarkable property that among all irrational numbers, it is the one that is least well approximated by a continued fraction, in the sense that the numbers in its representation are as small as possible and thus it takes a relatively large number of terms to approximate the number to a given accuracy. Indeed, the best continued fractions for the golden section are [1], [1,1], [1,1,1], [1,1,1,1], and so on. This fact plays an important role in KAM theory. One can conclude from this theory that a vibrating system whose frequency relation is the golden ratio is particularly insensitive to perturbations.

A Riddle

There is a riddle well known to Internet surfers that has an indirect relationship to the Fibonacci numbers. Figure shows a triangle that has been divided into four regions. After rearranging the segments, the same triangle appears, but now a small piece is missing.

Figure 1.1: What happened to the missing piece?

 

The solution to the riddle will be given at the end of this article.

The Pacioli Icosohedron

The Italian mathematician Luca Pacioli (1445-1517) discovered an interesting relationship between the golden ratio and the Platonic solids.

Take three rectangles with sides whose lengths correspond to the golden ratio: the longer side is approximately 1.618 times the shorter side.

These rectangles are then interpenetrated perpendicularly, as depicted in Figure . If they were made out of wood, you would have to do a bit of sawing. And now for the surprise: if you join the vertices of the rectangles, with string for example, you get a Platonic solid, an icosahedron.

Figure 1.2: Pacioli's icosohedron.

 

This is an impressive example of how surprising relationships are to be found between diverse branches of mathematics.

The Solution to the Triangle Puzzle

The reason that a little chunk of space can appear or disappear when the pieces are reassembled is that the figure isn't really a triangle at all. In the top picture of Figure 1.1, what looks like a triangle really has the "hypotenuse" bent inward a bit, while in the lower image, it bulges out a little.You can convince yourself of this: the red triangle rises at a rate of 3/8 = 0.375, while that of the green triangle is 2/5 = 0.4. The numbers appearing in these two fractions, 2,3,5,8, belong to the Fibonacci sequence, and the fact that 2/5 is close to 3/8 is related to the convergence of these fractions drawn from the Fibonacci sequence to the golden ratio.


This is an article from the book "Five-minute mathematics" by Ehrhard Behrends which was published in 2008 by the American Mathematical Society (AMS). It is reproduced here with the kind permission of the AMS.