The Fibonacci Numbers

Background on Leonardo of Pisa

The Fibonacci Series, named after Leonardo of Pisa or Leonardo of Fibonacci, has become so well known that few serious high school students would nowadays be unaware of its properties. At the same time, few would know that Leonardo was a mathematician of the l3th century, that he produced a series of mathematical treatises which were published (in manuscript copies of course in that period) in the years right after 1202.

"Liber Abaci" which now prints out to some 450 pages, and includes methods for all calculations with whole numbers and fractions, extraction of square and cube roots, proportion, chain rule, finding of proportional parts, averages, progressions and calculation of compound interest.

"De Practica Geometriae", is a work for the surveyor and civil engineer, with much material drawn from the Roman agrimensores, whose Latin texts he must have had before him. He includes a chapter on basic trigonometry, his "sinus versus arcus"; and goes on to find a square number which remains so after the addition of 5, an old problem which the Arab mathematicians has started work on. In the "Liber Quadratorum" from the same period, Leonardo gives an account of his method in this problem.

In the "Flos" or Flower of Mathematics, he turns to the solution of a cubic equation: x cube + 2 x sq + 10 x = 20, returning an approximative value for which he does not indicate his method, not to be discovered some three centuries later!

Put all this together with the present wide interest in the Fibonacci series, which was possibly the least important part of his work at the time, and you have an impressive picture of a 12th century mathematical mind at work setting the sub-structure for modern mathematical thinking. Equally curious is the fact that his personal history, and the dates of the publication of his works, are largely unknown, as well as the reason for his nickname Bigallone, or "blockhead".


The Fibonacci Series can easily be described as a series of whole numbers which progresses by adding the previous number to present one to make the next number in the series. So if we start by adding 2 to 1, = 3, then 3 +2=5 and 5+3 = 8....... and the familiar series unfolds.

We thus have ratios of 2/1, 5/3, 13/8, 21/13, 34/21.........and so forth.
2/1 = 2
5/3= 1.666666
13/8= 1.625
21/13= 1.61538.

Half an hour with the calculator and one sees that we are approaching the number of the Golden Mean, which is 1.618034, closer and closer at each step. The higher we go the closer we get to the GM number, the approximate discrepancy being constantly smaller and small, either overshooting or falling short on either side of 1.618034........

For larger numbers the discrepancy between the Fibonacci Series and the Golden Mean proportion is so small, that many people confuse the two series, or even think that they are the same thing. But in the world of mathematics, "nearly the same" is not a good description, and I think it is important to outline and try to understand the essential difference between these two very important numerical series, which constantly turn up in statistics, biology, art and perhaps even in the process of cell division.

Let us look at the problem from a modern point of view:

If s and y are two numbers in this series, then x/y will be practically the same as the next ratio, which is y/(x+y).

Now x/y will differ from the next ratio by a very small amount, which we can call s, thus:

x/y = y/(x+y) + s

x/y = 1/(x/y + 1) + s

Now write r in place of x/y for sake of convenience

r = 1/(r + 1) + s

Multiply both sides by r+1, getting:

r*r + r = 1 + s(r+1) or r*r + r - 1 = s(r+1).

Now r has to be between 1/2 and 1, after the Fibonacci sequence has gotten underway, so the product on the right, s(r+1), is no bigger than 2s ---- still a small number, which we can call z. So we have

r*r + r - (1-z) = 0

Apply the quadratic formula to this: r = (-1 +/- sqrt(1 + 4(1-z)))/2.

If z is small, as you can see by experiment, this delivers a value close to r = (-1 +/- sqrt 5)/2,

Therefore we are dealing with a matter of Approach to a Limit, in which the Fibonacci series is constantly striving to reach the point at which it coincides with the GM ratio. The way that ratio homes in on the GM is not only interesting, but involves concept of a limiting value. Our small value r dances around, alternately overshooting and undershooting the GM, but with the error constantly decreasing.

But one small point may be of interest: the particular numbers 2, 3, 5, 8 etc. are not important to the process -- you can launch itwith say x=2.14, y= 1.732, then apply the same rule of formation (each successive term is the sum of the two preceding), and you will still be moving in the direction of the GM. The above reasoning isn't sensitive to the initial values.

Consider for a moment the Real Number System:

This entity contains all the numbers everyone is comfortable with -- zero and all the other positive and negative integers, and the common fractions (including finite decimals). It also contains the spookier numbers, such as sqrt 5, pi, etc., which are typically "gotten" by infinite processes.

If you have a sequence of numbers that get bigger and bigger forever but always remain smaller than some fixed number, then there exists a number to which your sequence "converges" -- i.e., gets infinitely close to. Example (from Fibonacci) 1/2, 3/5, 8/13, 21/34,... -- each one is bigger than the one before, but less than the fixed number 1. Ipso facto, it converges to some number. It happens, in this case, that we can give a formula for that number, but even if we couldn't, we have license to talk about it with confidence and produce useful approximations to it. This general area is captioned Limits of Sequences.


What then is the essential difference between the Golden Mean ratio, which has an exact number 1.618034...., and the Fibonacci Series which pointstoward that number but never quite reaches it?

The GOLDEN MEAN PROPORTION is founded, in one way or another on the quadratic equation, in its form (sqrt 5 +-1)/2. The sqrt 5 is essential for the GM calculation and characterizes this operation, whether it derives mathematically from the Quadratic Equation ax sq + bx + c =0 ----- or geometrically from the hypotenuse of a 1 : 2 rectangle ----- or from a series of approximations which produce the only number 1.618034, which as its inverse 1/1.618034 gives. 618034. With the sqrt 5, you get an exact proportion which is stable over the whole range of numbers. Note that the difference between these two numbers is exactly '1'.

The FIBONACCI SERIES or SEQUENCE is an arithmetic series, which deals with whole numbers in an additive sequence. But since as a whole number series, it lacks the one tool which the GM ratio has (sqrt 5) it functions as an approximative series. Closer and closer to the goal, like Achilles trying forever to pass the slow moving turtle, it never quite reaches the exact number of the GM proportion.

If the "fault" in the Fibonacci calculation is inexactitude based on its arithmetic calculations, the value of this series is the applicability to situations which deal with whole numbers, like Leonardo's famous calculation for the number of rabbits which can be generated from a single pair in a hypothetical year of breeding. The rabbit must be treated as a digit unit, since there are no "fractional rabbits" to be found in hutches anywhere, so the Fibonacci series suits such statistical calculations admirably.

But if a calculation is desired for expanding or contracting a ratio form a given point, for example calculating a GM ratio for the rectangular frame of a painting, or for the height/width ratio of a doorway, or internal relationships among parts of the Parthenon, then the GM calculation fulfills an entirely different function. It can start with any number of any size, and produce a second number which exactly fulfills the conditions of the ratio 1: (1).618034.

These are practical considerations pertaining to actual use in calculations. In ideal terms, we find the arithmetically derived Fibonacci series remarkably similar to the GM Ratio, it is similar in its developed shape, and always getting nearer at the upper limit. But it is never exactly the same, either in final degree of exactitude, or in the method by which it is constructed. In last analysis, it seems it is the difference between the linear, arithmetic way of thinking with numbers of the Fibonacci method which sets it apart from the GM ratio, which is geometric in concept and as such involves itself automatically with the use of square numbers.

What seems most interesting is that two procedures operating on entirely different bases, will come so close to each other in the final analysis.

For a detailed exposition of the GOLDEN MEAN or GOLDEN PROPORTION, see my Essay on the Golden Mean

But there is always more to say, many scholarly heads have considered these matters over the course of centuries, and at this point I must refer the curious reader to the brilliant website of Mr. Ron Knott, of the University of Surrey, UK: A Virtual Library on the Fibonacci Numbers Here is detailed information on almost everything that anyone has at any time written on the Fibonacci Series, assembled by a professional in the fields of mathematics, engineering and mathematical education. If my discussion is perhaps insufficient as anything more than a start-off, Mr Knott's website will take you authoritatively along the rest of the way.

William Harris
Prof. Em. Middlebury College