The Golden String
The golden string is also called the Infinite Fibonacci Word or the Fibonacci Rabbit sequence. There is another way to look at Fibonacci's Rabbits problem that gives an infinitely long sequence of 1s and 0s called the Golden String:-
1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 ...
This string is a closely related to the golden section and the Fibonacci numbers.
Fibonacci Rabbit Sequence
See show how the golden string arises directly from the Rabbit problem and also is used by computers when they compute the Fibonacci numbers. You can hear the Golden sequence as a sound track too.
The Fibonacci Rabbit sequence is an example of a fractal - a mathematical object that contains the whole of itself within itself infinitely many times over.
Fibonacci sequence
Fibonacci is perhaps best known for a simple series of numbers, introduced in Liber abaci and later named the Fibonacci numbers in his honour.
The series begins with 0 and 1. After that, use the simple rule:
Add the last two numbers to get the next.
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,...
You might ask where this came from? In Fibonacci's day, mathematical competitions and challenges were common. For example, in 1225 Fibonacci took part in a tournament at Pisa ordered by the emperor himself, Frederick II.
It was in just this type of competition that the following problem arose:
Beginning with a single pair of rabbits, if every month each productive pair bears a new pair, which becomes productive when they are 1 month old, how many rabbits will there be after n months?
It is called the Fibonacci series after Leonardo of Pisa or (Filius Bonacci), alias Leonardo Fibonacci, born in 1175, whose great book The Liber Abaci (1202) , on arithmetic, was a standard work for 200 years and is still considered the best book written on arithmetic. It was the principal means of demonstrating and introducing the enormous advantages of the Hindu Arabic system of numeration over the Roman System. Leonardo's reputation amongst scholars was deservedly great. It was so outstanding that King Frederick II, visiting Pisa in 1225, held a public competition in mathematics to test Leonardo's skill and he was the only one able to answer the questions (Huntley 158)
One of the most spectacular examples of the Fibonacci Series in nature is in the head of the sunflower.
Scientists have measured the number of spirals in the sunflower head. They found, not only one set of short spirals going clockwise from the centre, but also another set of longer spirals going anti clockwise, These two beautiful sinuous spirals of the sun flower head reveal the astonishing double connection with the Fibonacci series,
The New Scientist Dec. 81 featured on its cover a daisy head showing the double spiral. The article then went on to discuss the Fibonnaci series, showing other examples of the double spirals in nature and comparing them to computer generated double spirals. The writers also postulated an explanation for the way plants' growth illustrates the Fibonacci series in the position and spacing of the leaves (Phylotaxis.)
The Fibonacci sequence
In Chapter 12 of the Liber Abbaci (pages 283-4 of [2]), Fibonacci states the problem which involves the famous sequence with which his name is irrevocably linked (Quot paria coniculorum in uno anno ex uno paro germinentur):
A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?
Since Fibonacci's time his sequence (which was not important to him) has generated nearly as many research papers as it has hypothetical rabbits! For the history of the work on the sequence until about 1920 we might consult Dickson [1]. Since the foundation of the Fibonacci Association in San Jos�, California, in the 1960's, and the production of the Fibonacci Quarterly (a journal devoted to the study in integers with special properties) there has been an orgy of creativity in which this author has shamelessly participated.
It is not my intention to detail properties of the Fibonacci numbers and of the related Lucas numbers which are well-documented elsewhere. For readable information on these fascinating aspects of Mathematics, see Hoggatt [7], Jarden [11], Lucas [12] and Vorobev [15]. Readily accessible material on the Fibonacci numbers may be found in past issues of The Australian Mathematics Teacher in articles by Guest [6], Horadam [8]-[10] and MacDonald [14].
Current research in modern extensions of Fibonacci's ideas is to be found in issues of the Fibonacci Quarterly [3] which began in 1963.
Liber Quadratorum
Dedicated to Frederick II (�3), this brilliant book was written by Fibonacci after his Liber Abbaci. In it, Fibonacci shows his mathematical prowess in solving Diophantine problems (named after Diophantus, the fourth century A.D.Greek scholar who lived in Alexandria). His problems involve second degree equations in two or more unknowns, the solutions for which are required to be given as integers or rational numbers. Such problems are not easy, particularly when one wishes to generalise solutions and when the algebraic symbolism Fibonacci needed was not at his disposal.
The Liber Quadratorum, a book with less immediate influence and narrower scope than the Liber abaci, was nevertheless an even greater masterpiece, more original and involving subtler reasoning, a systematically arranged, well-conceived collection of theorems many of them written by Leonard [Fibonacci], others discovered from Indian or Arabic sources, but using proofs which were the product of Leonard's own ingenuity.
McClenon [13] says:
The usual method of proof employed is to reason upon general numbers which Leonardo represents by line segments. He has . . . no algebraic symbolism, so that each result of a new operation (unless it be a simple addition or subtraction) has to be represented by a new line. But for one who had studied the 'geometric algbra' of the Greeks, as Leonardo had, in the form in which the Arabs used it, this method offered some of the advantages of our symbolism; and at any rate it is marvelous with what ease Leonardo keeps in his mind the relation between two lines and with what skill he chooses the right road to bring him to the goal he is seeking.
Fibonacci obtains results which are equivalent to:
It is claimed [13] (5*) that this last result should be called "Leonardo's Theorem" as it was not definitely stated in any earlier work. Results (ii) and (iii) are attributed by Proclus to Pythagoras and Plato respectively.
Fibonacci was stimulated to compile his Liber Quadratorum by consideration of one of the problems proposed to him by John of Palermo (�3):
PROBLEM 6. Find a square number which, being increased or diminished by 5, gives a square number.
An extension of PROBLEM 6 is:
Find a number which, being added to, or subtracted from, a square number, leaves in either case a square number.
McClenon [13] remarks that Fibonacci's proof of this is "so very ingenious and original" but long and that "It is not too much to say that this is the finest piece of reasoning in number theory of which we have any record, before the time of Fermat".
The following problems of Fibonacci are left to the reader (see [13] for others):
PROBLEM 7. Find a number of the form 4xy(x + y)(x - y) which is divisible by 5, the quotient being a square. (Fibonacci calls a number of this form a congruum.)
PROBLEM 8. Solve in rational numbers the pair of equations
x2 + x = u2, x2 - x = v2.
Fibonacci obtains x = 3 1/5, y = 9 3/5, z = 28 4/5. Better still, he obtains integer solutions x = 35, y = 144, z = 360.
Fibonacci goes on to say [13]:
And not only can three numbers be found in many ways by this method but also four can be found by means of four square numbers, two of which in order, or three, or all four added together make a square number . . . I found these four numbers, the first of which is 1295, the second 4566 6/7, the third 11417 1/2, and the fourth 79920.
The MS of Liber Quadratorum ceases fairly abruptly at this point. The rest is silence.