Some basics about the Golden Ratio and the Fibonacci Series.
Clint Thomas, June/July, 2004
1. Phi as defined by the Golden Proportion: If two numbers (or line segments with lengths), a and b, satisfy the relation (a + b)/b = b/a then that ratio is defined as Phi, a unique, irrational number. Thus, we have two equations involving Phi:
Solving (2) for b Y b = a N, and substituting in (1), N = (a + aN)/aN = (1 + N)/N Y N2 = 1+N or
Equation (3), then becomes the defining relationship for Phi and leads directly to some of the ratio’s interesting properties, including its numerical value. And also shows a+b can have any value but must be partitioned according to the Golden Proportion. This quadratic equation, (3), can be solved via the quadratic formula which yields the roots:
or the irrational number values, N = 1.618033988.... and - 0.618033988...
Phi is usually given the value of the positive root and often rounded to
The relationship in (4) can be used to derive some interesting properties of N. For example with some algebraic manipulation (including ‘rationalizing the denominator’) one can show that
1/N = N - 1 (or, easier, use the numerical value and a pocket calculator to show that 1/1.618034 = 0.618034 ).
2. Obtaining the Fibonacci series from the Golden Section definition of Phi:
Starting with (3) in the form N2 = 1 + N, one can multiply each power of N by N to form the next power and reusing (3) to render the powers in the form Nn = un + vnN . Thus:
N1 = 1 N
N2 = 1 + 1 N
N3 = (1 + N)N = N + N2 = N + (1 + N) = 1 + 2N
N4 = (1 + 2 N)N = N + 2 N2 = N + 2(1 + N) = 2 + 3N
N5 = (2 + 3 N)N = 2 N + 3 N2 = 2 N + 3(1 + N) =3 + 5N
N6 = (3 + 5 N)N = 3 N + 5N2 = 3 N + 5(1 + N) = 5 + 8N
N7 = (5 + 8 N)N = 5 N + 8N2 = 5 N + 8(1 + N) = 8 + 13N
N8 = (8 + 13 N)N = 8 N + 13 N2 = 8 N + 13(1 + N) = 13 + 21N
Etc.
The series of integers formed by the coefficients of N in the series of powers of N, viz.,
1, 1, 2, 3, 5, 13, 21.... (which can be continued indefinitely) are the famous Fibonacci series.
3. Obtaining the Golden Ratio from the Fibonacci series:
The above is a valid development-–obtaining the Fibonacci series from the Divine Proportion using only simple algebra–-but historically, I guess, things went the other way around. The Fibonacci series can be developed from the simple rule for defining terms whereby each is the sum of the previous two and starting with any two numbers. Fibonacci started with a story about breeding rabbits and used the integers 1 and 1. His rule can be symbolized by expressing the nth term in the series as Sn = Sn-1 + Sn-2, .Thus 1, 1, 2, 3, 5, 8, 13, 21, etc.–the Fibonacci series.
Using the terms of this series to form another one consisting of the ratios of successive members, that is, Sn+1/ Sn (expressing each to six decimal places)
1/1 = 1.000000 (< Φ)
2/1 = 2.000000 (> Φ)
3/2 = 1.500000 (< Φ)
5/3 = 1.666667 (> Φ)
8/5 = 1.600000 (< Φ)
13/8 = 1.625000 (> Φ)
21/13 = 1.615385... (< Φ)
34/21 = 1.619048... (> Φ)
55/34 = 1.617647... (< Φ)
89/55 = 1.618182..... etc. (> Φ but getting closer)
And, remembering the value of N is (1 + %5)/2 =1.618034 (rounded to six decimal places) one can see that members of the series above alternate between more than and less than N but get closer to it as the series continues. Proving that the limit (approached by members of the series as the number of members grows to infinity) is precisely N is straightforward, but requires math slightly beyond simple algebra. Fibonacci never did it but Atalay does in his notes.
4. Golden Rectangles and the Logarithmic Spiral
(If you need to, click on sketches to enlarge.)
Consider a rectangle, one unit by N . Such a shape is termed a Golden Rectangle. If we partition a Golden Rectangle by forming a square in one end, in this case a one by one square, the remaining part, 1 by N - 1, is another Golden Rectangle. This is so because from equation (6) above,
N -1 = 1/N and so the ratio of the sides of the new rectangle are 1/(1/N) = N . The new rectangle can be partitioned into a square and still another new rectangle which will also be a Golden Rectangle. Etc.
Or, we can build an array of rectangles in the other direction, getting larger ones instead of smaller. If we take our original, 1 x N, Golden Rectangle and build a N by N square on the bottom, we create another, larger Golden Rectangle. It’s Golden because its dimensions are N by N + 1. But by Equation 3 above N + 1 = N2 and N2/N = N.
Let’s continue this process several more times, adding squares to the right side, the top, and the left side to make an array of five Golden Rectangles. The process could be extended indefinitely, of course, getting larger Golden Rectangles rotated in a positive (counterclockwise) sense.
This is fun, but what’s the point? Well, our array of Golden Rectangles has some interesting properties. The ratio of the dimensions of successive rectangles is a constant and is exactly N. You can work this out re-using Eq. (3) repeatedly. This means the ratio of the areas of successive rectangles will be N2 . If you play the same game starting with a different shape rectangle, the ratios will not be constant. If you like playing with numbers, try starting with, say, a 1 by 2 rectangle. The ratios will vary. But, you know what? If you continue to build the array the ratios will get closer and closer to a constant value, even starting with a 1 x 2 rectangle. And, of course, what is the value the ratios will approach? You guessed it, N.! What we’re really doing of course, is dealing with the Fibonacci series in a geometrical way.
There’s more. If you draw one of the diagonals of Rectangle 5, you’ll find it also contains a diagonal of Rectangles 3 and 1. (And of Rectangles 7, 9, 11, etc. if we had drawn them.) And, if you draw a diagonal of Rectangle 4, you’ll find it contains a diagonal of Rectangle 2. (And of 6, 8, 10, etc.) The intersection of these two diagonals has been called ‘the eye of God’. It lies in exactly the same relative position in each of the Rectangles in the array. Many claim that this point has been used by famous artists over the centuries, probably intuitively, to position some feature of their art--the dominant eye of the subject, for example. For what follows, we need a coordinate system and we’re going to place its origin at this Golden Point.
What we have is an array of rectangles that rotate and grow. What single word encompasses both of these motions, grow and rotate? Why, it’s ‘spiral’.
There are a number of mathematical curves that ‘spiral’. The one of interest here is the ‘equiangular spiral,’ also known as the ‘logarithmic spiral’. The simplest form of the equation controlling the logarithmic spiral uses polar coordinates. But we want to plot in Cartesion coordinates and use these parametric equations.
Here x and y are coordinates of points on the spiral, e is the base of natural logarithms (hence the name ‘logarithmic spiral’), 2 (theta) is an angle and the independent variable, and a and b are constants whose value control the shape and rate of growth of the spiral. With the proper choice of a and b, a logarithmic spiral can be inscribed perfectly in our array of Golden Rectangles. (If you deal with these equations with numbers, the angle has to be expressed in radians of course.)
The logarithmic spiral has properties that give it a prominent role in both nature and human endeavors. Without dwelling on those roles, what are some of its mathematical properties?
If we draw radii, lines from the origin to points any where on the logarithmic spiral, they make the same angle with the spiral, that is, with the tangent to the curve at the intersection point. (The constant, b, above is the cotangent of this angle.) It is this fact that gives our spiral its other name, ‘equiangular spiral.’
The spiral has no ends; it continues forever in both directions. This is a familiar property because it is possessed by a mathematical line. A line is considered to continue to infinity in both directions. In the real world, we deal with line segments. But while the logarithmic spiral continues in both directions forever, at its inner end it curls into itself more and more tightly. But, if you were to magnify a piece of the disappearing spiral, it has the same shape and would fit exactly a portion of the spiral at some other location. It is this sameness, we’re told, that guides the Nautilus snail to add, as it grows, rooms of the same shape, only larger, and hence in a logarithmic spiral pattern.
All the occurrences of the Golden Ratio, logarithmic spirals, etc. in nature probably result from function even if that function isn’t obvious. But what about mankind’s use of these relations? I’ve be unable to find out what banking entity decided on the shape of credit cards. Standardization is necessary if they’re all to work in the same card readers. But measure as carefully as you can the sides of a credit card and compute the ration. It’ll be very close to the Golden Ratio. Was this somebody’s conscious decision or is it just a ‘pleasing shape’? What I believe is that we’ve been observing nature for so many eons that a taste for such shapes and relationships is inbred. Of course. That’s it. We each have a Golden Ratio gene!
5. Fibonacci Numbers and Pythagorean Triplets
Fibonacci Numbers pop up often in Number Theory fundamentals. For example there is an algorithm for generating Pythagorean Triplets and this algorithm establishes that there are an infinite number of such triplets.
You'll recall that a Pythagorean triplet is a set of three natural numbers,
x, y and z, that satisfy the famous Pythagorean Theorem:
x2 + y2
= z2 . 3, 4 and 5 are the members of
the best know such triplet.
The Algorithm: If a, b, c, d are any four, consecutive
Fibonacci Numbers, then a Pythagorean Triplet is formed when
x = ad, y = 2bc and z = b2 + c2 . It's
easy to demonstrate (with a pocket calculator) the correctness of the algorithm
with an example:
13, 21, 34, 55 are four consecutive Fibonacci Numbers. The algorithm gives
x = 13x55 = 715
y = 2x21x34 = 1428
z = 212 + 342 = 1597
Sure enough, 7152 + 14282 = 2,550,409 whose square root is indeed 1597
Proving the algorithm in general only requires some algebra, albeit messy. One procedure is to note that if a and b are any two consecutive Fibonacci Numbers then the next two are a+b and a+2b. The Pythagorean Triplet can be expressed, using the algorithm above, as:
x = a(a + 2b)
y = 2b(a + b)
z = b2 + (a+b)2
{So, we have another algorithm for forming Pythagorean Triplets using any two consecutive Fibonacci Numbers.)
Forming expressions for x2 + y2 and for z2 will yield two identical expressions in a and b, but showing they're identical involves such things as squaring trinomials. Anyway, the expressions turns out to be:
4b4 + 8ab3 + 8ab3 + 8a2b2 + 4a3b + a4
The sketches above were done in AutoCAD. The spiral is truly logarithmic and is sketched by an AutoLISP routine. The sketch fitting the curve to the array of Golden Rectangles, however, is not yet perfect but it can be. CT