The Golden Ratio and
The Fibonacci Numbers

The Golden Ratio (φ) is an irrational number with several curious properties. It can be defined as that number which is equal to its own reciprocal plus one:  φ = 1/φ + 1.  Multiplying both sides of this same equation by the Golden Ratio we derive the interesting property that the square of the Golden Ratio is equal to the simple number itself plus one:  φ2 = φ + 1.  Since that equation can be written as φ2 - φ - 1 = 0, we can derive the value of the Golden Ratio from the quadratic equation, , with a = 1, b = -1, and c = -1:  . The Golden Ratio is an irrational number, but not a transcendental one (like π), since it is the solution to a polynomial equation.

This gives us either 1.618 033 989 or -0.618 033 989. The first number is usually regarded as the Golden Ratio itself, the second as the negative of its reciprocal -- or we can use the second in its own right, as the number "τ," for which there will be a use below. The Golden Ratio can also be derived from trigonometic functions:  φ = 2 sin 3π/10 = 2 cos π/5; and 1/φ = 2 sin π/10 = 2 cos 2π/5. The angles in the trigonometric equations in degrees rather than radians are 54o, 36o, 18o, and 72o, respectively.

The Golden Ratio seems to get its name from the Golden Rectangle, a rectangle whose sides are in the proportion of the Golden Ratio. The theory of the Golden Rectangle is an aesthetic one, that the ratio is an aesthetically pleasing one and so can be found spontaneously or deliberately turning up in a great deal of art. Thus, for instance, the front of the Parthenon can be comfortably framed with a Golden Rectangle.

How pleasing the Golden Rectangle is, how often it really does turn up in art, and whether it does really frame the front of the Parthenon, may be largely a matter of interpretation and preference. The construction of a Golden Rectangle, however, is an interesting exercise in the geometry of the Golden Ratio, as seen at right. Beginning with a unit square, first the square is bisected, then a diagonal is drawn in the semi-square. The length of the diagonal can be calculated with the Pythagoran Theorem, based on a triangle that is .5 on one side and 1 on the other. The diagonal is therefore √(1 + .52) = √(1.25) = 1.118033989. If we merely add .5, this produces the Golden Ratio, 1.6l8033989. Thus, if we extend the side of the unit square and draw a circle with a radius of the diagonal and its center at the midpoint on the unit side, the circle will intersect the side at a point that will be 1.6l8033989 units from the corner of the square. In effect, this gives us an equation for the Golden Ratio:  φ = .5 + √1.25. We can easily turn this into the previous equation, however, just by muliplying the numbers so as to get integers:  φ = 1 / 2 + 2√1.25 / 2 = (1 + 2√1.25) / 2 = (1 + √(4 * 1.25)) / 2 = (1 + √5) / 2.

Whether or not the Golden Ratio or the Golden Rectangle are of aesthetic significance, the ratio does turn out to have considerable significance in problems of natural symmetry. For instance, a surface can be completely and symmetrically tiled with triangles, squares, and hexagons, but not with pentagons. Periodic five-fold symmetry does not occur in nature.

However, in the early 70's Roger Penrose discovered that a surface could be completely tiled in an asymetrical and non-periodic way with just two shapes, called "kites" and "darts" -- "Penrose tiles" -- as seen at right. Within this tiling, however, there can be small areas of five-fold symmetry. Multiple decagons, some of which from a distance can look like pentagons, can occur [note].

Now, it just so happens, that, given a large enough area, the ratio of kites to darts is just the Golden Ratio. Why would this happen? Well, a complete circular angle (360o) divided by five is 72o, which occurred above as one of the angles whose trigonometric function is the Golden Ratio. More conspicuously, the very irrationality of the Golden Ratio is an artifact of the square root of five:  . If the Golden Ratio turns up in examples of five-fold symmetry, it may well be because the number itself is fundamentally related to the number five.

nfn φn = fnφ + f n-1

    = φn-1 + φn-2
    = φn+1 - φn-1
    = φn+2 - φn+1

-10-55φ-10 = 89 - 55φ = φ-11 + φ-12 = φ-8 - φ-9
-934φ-9 = 34φ - 55 = φ-10 + φ-11 = φ-7 - φ-8
-8-21φ-8 = 34 - 21φ = φ-9 + φ-10 = φ-6 - φ-7
-713φ-7 = 13φ - 21 = φ-8 + φ-9 = φ-5 - φ-6
-6-8φ-6 = 13 - 8φ = φ-7 + φ-8 = φ-4 - φ-5
-55φ-5 = 5φ - 8 = φ-6 + φ-7 = φ-3 - φ-4
-4-3φ-4 = 5 - 3φ = φ-5 + φ-6 = φ-2 - φ-3
-32φ-3 = 2φ - 3 = φ-4 + φ-5 = φ-1 - φ-2
-2-1φ-2 = 2 - φ = φ-3 + φ-4 = 1 - φ-1
-11φ-1 = φ - 1 = φ-2 + φ-3 = φ - 1
001 = 0 + 1 = φ-1 + φ-2 = φ - φ-1
11φ = φ + 0 = 1 + φ-1
21φ2 = φ + 1 = φ + 1
32φ3 = 2φ + 1 = φ2 + φ = φ5 - φ4
43φ4 = 3φ + 2 = φ3 + φ2 = φ6 - φ5
55φ5 = 5φ + 3 = φ4 + φ3
68φ6 = 8φ + 5 = φ5 + φ4
713φ7 = 13φ + 8 = φ6 + φ5
821φ8 = 21φ + 13 = φ7 + φ6
934φ9 = 34φ + 21 = φ8 + φ7
1055φ10 = 55φ + 34 = φ9 + φ8
1189φ11 = 89φ + 55 = φ10 + φ9
12144φ12 = 144φ + 89 = φ11 + φ10
13233φ13 = 233φ + 144
14377φ14 = 377φ + 233
15610φ15 = 610φ + 377
16987φ16 = 987φ + 610
171597φ17 = 1597φ + 987 = φ18 - φ16
182584φ18 = 2584φ + 1597
194181φ19 = 4181φ + 2584
206765φ20 = 6765φ + 4181
Another connection of the Golden Ratio to partial symmetries in nature is through the Fibonacci Numbers (fn). This is a number series where each member is simply the sum of the previous two numbers. Thus, if we begin with 1 and 1, these add up to 2, but then 1 and 2 add up to 3, 3 and 2 add up to 5, 5 and 3 add up to 8, etc. If we began the series with 1 and 1, this would also work, and the series could be extended below zero when we ask what number added to 0 will produce 1, which is then 1. The number added to 1 that will produce 0 is -1. It happens, then, that below zero, we find the same Fibonacci Numbers, but they alternate as positive and negative. The Fibonacci Numbers from the minus eleventh to the twentieth (with zero as the "zeroth") are listed in the table at right. It turns out to be convenient to count the Fibonaccis beginning with f1 = 1.

Now, the Fibonacci Numbers turn up in nature. The spirals discernible in the head of a daisy consist of individual "florets" that count up as Fibonacci Numbers. This is significant here because the ratio between any two successive Fibonacci Numbers approaches a limit as the numbers get larger, and that limit is the Golden Ratio. Thus, 6765/4181 (the 20th and 19th Fibonaccis) is 1.618033963, which only differts from the Golden Ratio by 0.000000025.

The table at right illustrates an interesting way in which the Fibonacci Numbers occur naturally in relation to the Golden Ratio. This is based on the property of the Golden Ratio already noted above, that φ2 = φ + 1. Similarly, any power of the Golden Ratio can be broken down into the sum of smaller powers, such as φ5 = φ4 + φ3. Because of this, any power of the Golden Ratio can be ultimately reduced to the sum of an integer and an interger multiple of the Golden Ratio. An example of going through this whole process for a large power is given in a footnote. Curiously, all of those integers turn out to be Fibonacci Numbers. Thus, φ7 is equal to 13φ + 8. This can be generalized, so that for every power of φ:
φn = fnφ + f n-1.

Because, as we saw above, there are two solutions for the quadratic equation for φ, it is also the case that τn = fnτ + f n-1. If we subtract one equation from the other, we get an interesting result:

φn = fnφ + f n-1
-(τn = fnτ + f n-1)
φn - τn = fn(φ - τ)
or fn = (φn - τn)/(φ - τ). If we go back to the quadratic equation,
φ - τ = (1 + √5)/2 - (1 - √5)/2 = √5. Thus we get a general equation for Fibonacci Numbers:  fn = (φn - τn)/√5. In using this equation, care must be taken that τ is actually a negative number, and so becomes negative or positive depending on the power to which it is raised. This is the "Binet" formula, named after the mathematician Jacques Binet.

While we see that the Fibonacci series emerges naturally in the evaluation of the powers of the Golden Ratio, this does not necessarily make it clear why the ratio of the members of the Fibonacci series should approach the Golden Ratio as a limit. As it happens, the connection can be illustrated through the technique of Continued Fractions, which is a device for reducing non-repeating decimals to fractions, i.e. to ratios of integers. The technique for reducing repeating decimals to fractions has been discussed elsewhere. With non-repeating decimals, the integer part of the number is successively removed, and the reciprocal is taken of the remaining decimal, producing a new integer, which is then removed, and the process repeated. This can be continued until the desired accuracy is attained or the capacity of the calculator is exceeded -- since I would assume that most people today would be using a calculator to get the reciprocals (it is not a very convenient procedure otherwise). Once enough integers are obtained, the corresponding fraction with all the embedded fractions can then be solved for a simple integer fraction.

For instance, at right is a continued fraction for the ratio between the the length of the lunar (synodic) month (29.530588 days), and the length of the solar (tropical) year (365.24219878 days). This tells us the number of lunar months per solar year; and, in integer form, the fractional part would tell us how many extra lunar months (more than 12 per year) would need to be added in a certain period of solar years to approximate the true ratio. The true ratio is 12.368267058. Removing the 12 and then successively taking the reciprocal and removing the integer part again gives us the integers, after 12, 2, 1, 2, 1, 1, and 17 (at least). Successive approximations can be made by stopping at each new integer. Thus, with only 2, we would have the approximation 12+1/2. Stopping with the next 1, 1 gets added to the 2, and the next approximation is 12+1/3. Adding in the next number, a 2, produces a fraction of 3/8, which is historically very important because a popular ancient device for approximating the lunar calendar to a solar year was to add 3 extra months every 8 years. More important, however, was the fraction two steps further. Adding 7 lunar months every 19 years was a device adopted for the Babylonian Calendar. The rule was inherited and used even today by the Jewish Calendar and for the Christian reckoning of Easter. The next fraction in the series, 123/334, is too large to have been practical for calendrical purposes. (The continued fraction for π is given in a footnote.)

The Golden Ratio has the unique property that its reciprocal always produces the same decimal and the reciprocal of the decimal will always produce the integer 1. This means that the continued fraction can be constructed without bothering with a calculator! The continued fraction uniquely only has 1's in it. This also means that the successive fractions can be generated without consulting the diagram. For each fraction, we add 1 and then flip it over (make the reciprocal) for each new fraction. Thus, the first number is 1, producing the fraction 1/1. That is its own reciprocal. To this is added 1 (1/1), resulting in the fraction 2/1. The reciprocal of that is 1/2, our second fraction. To that is added 1 again (now 2/2), resulting in the fraction 3/2. The reciprocal of that is 2/3, our third fraction. To that is added 1 again (now 3/3), resulting in the fraction 5/3. The reciprocal of that is 3/5, our fourth fraction. As this continues, we might notice that the procedure generates fractions that all consist of successive Fibonacci Numbers! This is why ratios of Fibonacci numbers approximate the Golden Ratio, they are all solutions to the unique continued fraction for the Golden Ratio!

I have now received from Devin Chalmers in Juneau, Alaska, a derivation of φ directly from a definition of the Fibonacci Series (equation 1). Equation 2 moves the definition over one in the series; then we divide both sides by a(n) in equation 3, which we manipulate with a bit of algebra. The limit of both ratios in equation 4 should be the same number, so we replace them with the variable φ in equation 5. This is equivalent to equation 6, which, as we've seen above, can be plugged into the quadratic equation to get equation 7.

Thus, while the Golden Ratio may not be as important as other mathematical constants, it does have its claim to fame and does have its own unique properties. And, with one further reflection, we can put it into very serious company indeed. Thus, if we define the Golden Ratio as that number which, when one is substracted, is equal to its own reciprocal ( φ - 1 = 1/φ ), we might ask in turn what number, when multiplied by minus one, is equal to its own reciprocal ( x * -1 = 1 / x ). This turns out to be the imaginary number, i = √-1:   -i = l / i. This is a much more significant and mysterious number than the Golden Ratio -- not bad as a kind of cousin.

Philosophy of Science, Mathematics

Philosophy of Science

Home Page

Copyright (c) 1999, 2003, 2008, 2011, 2012, 2019, 2021, 2023 Kelley L. Ross, Ph.D. All Rights Reserved

The Golden Ratio and The Fibonacci Numbers, Note 1;
The ein Stein

Where it takes two Penrose tiles to tile a surface, mathematicians have been looking for a single irregular tile that would perfectly tile a surface. This was to be called an "einstein," not, as it happens, named after Albert Einstein, but simply the German words ein Stein, which means "one stone."

Such a tile has been found, discovered by David Smith, as I see announced in Science News of April 22, 2023 ("Elusive 'einstein' tile finally found," p.7). The shape is called "the hat." We see that it is made of a single smaller tile, in the shape of a kite. But it is not a Penrose kite. As we see above, Penrose kites can be arranged into a pentagonal shape, while the "hat" kites, as we see at right, can be put together in a hexagonal form.

In the plane below, we can see these hexagons marching across the surface, overlapping the "hat" shapes, sometimes two, sometimes three of them. I think this makes the result seem less mysterious, and rather more regular than what we see above with the Penrose tiling. But it still seems rather remarkable, and perhaps a reminder that mathematics will always remain incomplete.

Return to text

The Golden Ratio and The Fibonacci Numbers, Note 2

Example; reducing a power of the Golden Ratio to a linear quantity
φ8 = φ7 + φ6 = (φ6 + φ5) + φ6 = ((φ5 + φ4) + φ5) + (φ5 + φ4) = 3φ5 + 2φ4 = 3(φ4 + φ3) + 2(φ3 + φ2) = (3φ4 + 3φ3) + (2φ3 + 2φ2) = 3φ4 + 5φ3 + 2φ2 = 3(φ3 + φ2) + 5φ3 + 2φ2 = (3φ3 + 3φ2) + 5φ3 + 2φ2 = 8φ3 + 5φ2 = 8(φ2 + φ) + 5(φ + 1) = (8φ2 + 8φ) + (5φ + 5) = 8φ2 + 13φ + 5 = 8(φ + 1) + 13φ + 5 = (8φ + 8) + 13φ + 5 = 21φ + 13

Return to text

The Golden Ratio and The Fibonacci Numbers, Note 3;
The Continued Fraction for Pi

The continued fraction for π is of some interest because Greek attempts to derive a value for π always gave it in the form of a fraction -- the ancients did not have decimal notation. The best they could do, according to Isaac Asimov ("A Piece of Pi," 1964), was 22/7, which is only the smallest fraction in the series (afer 3/1, of course). Archimedes found that π was smaller than 22/7 and larger than 223/71, but this still wasn't good enough to find the next fraction in the series. It wasn't until the 16th Century that the value 355/113 was discovered. As you can see, this is actually the best that can be done with a relatively simple fraction, and modern mathematicians had new notation and methods for tackling π anyway, without continuing to state it as a fraction at all. The next couple of fractions in the series, however, do produce extraordinarily precise values. 104,348/33,215 differs from π by only 1/3,030,303,030; and 103,993/33,102 differs by only 1/1,724,137,931. By comparison, 355/113 differs from π by 1/3,748,688; 333/106 by 1/12,016; and 22/7 by 1/791. Any desire for integer values of π, across a large range of accuracy, is thus easily met.

Return to text on the Golden Ratio

Text on the Continued Fraction for the Synodic Month

Philosophy of Science

Home Page

Copyright (c) 1999 Kelley L. Ross, Ph.D. All Rights Reserved