The Fibonacci Series
Fibonacci. There's no known
authentic portrait of him.
Leonardo of Pisa (
1170-1250), also known as Fibonacci, wrote books of problems in mathematics, but is best known by laypersons for the sequence of numbers that carries his name:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597.
This sequence is constructed by choosing the first two numbers (the "seeds" of the sequence) then assigning the rest by the rule that each number be the sum of the two preceding numbers. This simple rule generates a sequence of numbers having many surprising properties, of which we list but a few:
- Take any three adjacent numbers in the sequence, square the middle number, multiply the first and third numbers. The difference between these two results is always 1.
- Take any four adjacent numbers in the sequence. Multiply the outside ones. Multiply the inside ones. The first product will be either one more or one less than the second.
- The sum of any ten adjacent numbers in the sequence equals 11 times the seventh one of the ten.
The Fibonacci sequence is but one example of many sequences with simple recursion relations.
The Fibonacci sequence obeys the recursion relation P(n) = P(n-1) + P(n-2). In such a sequence the first two values are arbitrarily chosen. They are called the "seeds" of the sequence. When 0 and 1 are chosen as seeds, or 1 and 1, or 1 and 2, the sequence is called the Fibonacci sequence. The sequence formed from the ratio of adjacent numbers of the Fibonacci sequence converges to a value of 1.6180339887. called "phi", whose symbol is ø or φ. Sometimes the Greek letter "tau", τ, is used.
A striking feature of this sequence is that the reciprocal of φ is 0.6180339887. which is φ - 1. Put another way, φ = 1/φ + 1. This is true whatever two seed integers you use to start the sequence, this result depends only on the recursion relation you use, not the choice of seeds. Therefore there are many different sequences that converge to φ. They are called "generalized Fibonacci sequences".
The ratio φ = 1.6180339887. is called the "golden ratio" or "golden mean". A rectangle that has sides in this proportion is called the "golden rectangle", and it was known to the ancient Greeks. The golden rectangle is the basis for generating a curve known as the "golden spiral", a logarithmic spiral that is fairly well-matched to some spirals found in nature, and this fact is the source of much of the popular and mystical interest in this mathematical subject.
Note: Writers on this subject sometimes concentrate on φ and some on 1/φ as the ratio of interest. This is no "big deal" for when you have a ratio of two values, say A and B, which is a comparison of their sizes, the reciprocal of the ratio A to B is just the ratio of B to A.
It's easy to invent other interesting recursion relations. Some have been interesting enough to mathematicians that they carry the names of their originators.
The next-best known is one of the Lucas sequences:
1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199. It has the seed values 1 and 3, and the same recursion relation as the Fibonacci series. The ratio of adjacent values approaches φ for large values. Other examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Jacobsthal numbers, and a superset of Fermat numbers. See: Lucas Sequence. All are mathematically interesting, so why is it that only the Fibonacci numbers get all the attention from Fibonacci fanatics?
How about a different recursion relation, say P(n) = P(n-2) + P(n-3)? With three seed numbers 0, 1, 1 we get the series 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9. The seeds, together with the recursion relation, uniquely define the sequence. The ratio of successive terms P(n+1)/P(n) converges to 1.3247295. whose reciprocal is 0.7545776665. [Note that its reciprocal is not one smaller than itself, contrary to what you might have expected. However, this ratio and its reciprocal do add to nearly 2 (Approximately 2.08 if you are fussy.) Isn't that close enough to count as "mystical"?
Typically, for all of these sequences, the first few values of the ratios of successive numbers seem to have no consistent pattern, but farther along to larger values they converge to ratios that are nearly constant, and after about n = 30 have settled down to values constant to about 10 decimal places.
Fibonacci Foolishness.A search of the internet, or your local library, will convince you that the Fibonacci series has attracted a lunatic fringe of Fibonacci fanatics who look for mysticism in numbers and in nature. You will find fantastic claims:
- The "golden rectangle" is the "most beautiful" rectangle, and was deliberately used by artists in arranging picture elements within their paintings. (You'd think that they'd always use golden rectangle frames, but they didn't.)
- The patterns based on the Fibonacci numbers, the golden ratio and the golden rectangle are those most pleasing to human perception.
- Mozart used φ in composing music. (He liked number games, but there's no good evidence that he ever deliberately used φ in a musical composition.)
- The Fibonacci sequence is seen in nature, in the arrangement of leaves on a stem of plants, in the pattern of sunflower seeds, spirals of snail's shells, in the number of petals of flowers, in the periods of planets of the solar system, and even in stock market cycles. So pervasive is the sequence in nature (according to these folks) that one begins to suspect that the series has the remarkable ability to be "fit" to most anything!
- Nature's processes are "governed" by the golden ratio. Some sources even say that nature's processes are "explained" by this ratio.
Of course much of this is patently nonsense. Mathematics doesn't "explain" anything in nature, but mathematical models are very powerful for describing patterns and laws found in nature. I think it's safe to say that the Fibonacci sequence, golden mean, and golden rectangle have never, not even once, directly led to the discovery of a fundamental law of nature. When we see a neat numeric or geometric pattern in nature, we realize we must dig deeper to find the underlying reason why these patterns arise.