Fibonacci series

Les Schaffer schaffer at
Sat Feb 17 10:46:53 MST 2001

1.) from The Universal History of Numbers: from prehistory to the
invention of the computer, by Georges Ifrah (pp. 361-362):

    Considering that the Greeks had invented such an instrument, the
    next logical step would have been their discovery of the
    place-value system and zero, through eliminating the columns of
    the instrument. This would have provided them with the fully
    operational counting system that we use today.

    However, the Greeks did not bother themselves with such practical

    India: The True Birthplace of Our Numerals
    The real inventors of this fundamental discovery, which is no less
    important than such feats as the mastery of fire, the development
    of agriculture, or the invention of the wheel, writing or the
    steam engine, were the mathematicians and astronomers of Indian
    civilization: scholars who, unlike the Greeks, were concerned with
    practical applications and who were motivated by a kind of passion
    for both numbers AND NUMERICAL CALCULATIONS [my emphasis].

    There is a great deal of evidence to support this fact, and even
    the Arabo-Muslim scholars themselves have often voiced their

    Evidence from Europe which supports the claim that modern
    numeration originated in India

    10. In 1202, Leonard of Pisa (known as Fibonacci), after voyages
    that took him to the Near East and Nothern Africa, and in
    particular to Bejaia (now in Algeria), wrote a tract on arithmetic
    entitled Liber Abaci ("a tract about the abacus"), in which he
    explains the following [greek omitted] "My father was a public
    scribe of Bejaia, where he worked for his country in Customs,
    defending the interests of Pisan merchants who made their fortune
    there. He made me learn how to use the abacus when i was still a
    child because he saw how i would benefit from this in later
    life. In this way i learned the art of counting using the nine
    Indian figures ...

    The nine Indian figures are as follows:
        9 8 7 6 5 4 3 2 1

    That is why, with these nine numerals, and with this sign 0,
    called zephirum in Arab, ONE WRITES ALL THE NUMBERS ONE WISHES
    [emphasis mine].

later in the book (p.588):

    The spread of "algorism" was given renewed impetus from the start
    of the thirteenth century by a great Italian mathematician,
    Leonard of Pisa, better known by the name of Fibonacci. He visited
    Islamic North Africa and also traveled in the Middle East. He met
    Arabic arithmeticians and learned from them their numeral system,
    the operational techniques, the rules of algebra and the
    fundamentals of geometry. This education was what underlay the
    treatise that he wrote in 1202 and which was to become the
    algorist's bible, the Liber Abaci. Despite its title, Fibonacci's
    treatise (which assisted greatly the spread of Arabic numerals and
    the development of algebra in Western Europe) has no connection
    with Gerbert's abacus or the arithmetical course-books of that
    tradition -- for it lays out the rules of written computation
    using both the zero and the rule of position. Presumably Fibonacci
    used "abacus" in his title in order to ward off attacks from the
    practical abacists who effectively monopolized the world of
    accounting and clung very much to their counters and ruled
    tables. At all events, from 1202 the trend began to swing in
    favour of the algorists, and we can thus mark the year as the
    beginning of the democratization of number in Europe.

    Resistance to the new methods was not easily overcome, however,
    and many conservative counting-masters continued to defend the
    archaic counter-abacus and its rudimentary arithmetical

    Professional arithmeticians, who practiced their art on the
    abacus, constituted a powerful caste, enjoying the protection of
    the Church. They were inclined to keep the secrets of their art to
    themselves; they necessarily saw algorism, which brought
    arithmetic within everyone's grasp, as a threat to their

    Knowledge, though it may now seem rudimentary, brought power and
    privilege when it represented the state of the art, and the
    prospect of seeing it shared seemed fearful, perhaps even
    sacrilegious, for its practitioners. But there was another, more
    properly ideological reason for European resistance to Indo-Arabic

    Even whilst learning was reborn in the West, the Church maintained
    a climate of dogmatism, of mysticism, and of submission to the
    holy scriptures, through doctrines of sin, hell and the salvation
    of the soul. Science and philosophy were under ecclesiastical
    control, were obliged to remain in accordance with religious dogma,
    and to support, not to contradict, theological teachings.

2.) from the book: African Fractals: Modern Computing and Indigenous
Design, Ron Eglash, pp86-89:

see especially the "triangular numbers" game from Africa.

in this regard, it is interesting to note that the stacking of
discrete objects (for example, firewood) is most stable in a
triangular configuration. Its very hard to stack a rectangular shaped
set of logs, as anyone who has tried knows. So we can perhaps glimpse
a connection between these triangular numbers, to take one example,
and material practices of a society.

3.) Connection between Fibonacci and golden mean and modern nonlinear
dynamical systems theory: the golden mean, which is the limiting ratio
of consecutive terms in the Fibonacci series is the least "rational"
irrational number, meaning it is the irrational number most difficult
(in a precisely definable mathematical sense) to approximate by the
ratio of two integers.

The great Soviet mathematician Vladamir Arnold (along with colleagues
Kolmogorov and Moser), showed, in a pinnacle of twentieth century math
and classical physics, that "invariant tori" related to these "most
irrational" numbers survive the longest in perturbed oscillatory
systems, providing a means of bounding chaotic diffusion through the
phase space of any particular problem.

Their proofs used a development called "superconvergent" series
solutions, which was effectively the cap stone to the earlier work of
Poincare on the approximation of the motion of three bodies in
celestial mechanics.

These theoretical developments were later confirmed in numerous
numerical simulation schemes, most notably that of Greene (i forget
his first name) who showed graphically how the tori related to the
golden mean was indeed the last to disappear as the strength of
coupling in the oscillatory systems was increased.

[long story, big words help encapsulate the ideas, hopefully more to
entice the reader in a sense of wonder rather than panic or

4.) [For afficiandos] In the textbook Structure and Interpretation of
Computer Programs, by Abelson and Sussman, they use Scheme, a
"tail-recursive" Lisp programming language, widely regarded as
efficient in solving recursion-type computational problems

Fibonacci numbers can be defined _algorithmically_ as a two-term
recursion relation:

   Fib(0) = 0 the "initial conditions"
   Fib(1) = 1          """

   Fib(n) = Fib(n-1) + Fib(n-2)

Abelson and Sussman show on pp. 37-38 how a naive implementation of
this scheme

 (define (fib n)
    (cond ((= n 0) 0)
          ((= n 1) 1)
           (else (+ (fib (- n 1))
                    (fib (- n 2)) ))))

involves a lot of redundant computation: to see this, draw the
computation as a tree structure and count the redundant branches.

They then go on to show that a more efficient scheme for computing
this function is to combine recursion with iteration using three
rather than two "state variables":

  (define (fib n)
    (fib-iter 1 0 n) )

  (define (fib-iter a b count)
    (if (= count 0)
        (fib-iter (+ a b) a (- count 1))))

which requires only numbers of steps scaling linearly with any n, as
opposed to the naive implementation which grows as fast as the
fibonacci numbers themselves.

These kinds of techiques for recognizing the _structure_ of a
computational problem in abstract and/or picturesque terms is
fundamental to a large portion of modern computer science.

5.) Web links. The series is simple enough an idea that of course
there is a ton of web sites covering this.

Here is a sample:

Use of the Fibonacci Series in the Bassoon Solo in Bartók's Dance

waskly wabbits:

On Rabbits, Mathematics and Musical Scales:


fibonacci series and the golden mean:

fractals and fibonacci:

fibonacci bio:

growth patterns:

the fibonacci series in java bytecodes:

interesting factoid: the largest fibonacci number calculatable by long
integer arithmetic: 7540113804746346429L

The Fibonacci Numbers: Connections within the Mathematics and
Calendrical Systems of Ancient Mesoamerica :

fibonacci and quilts:

6.) Since Andrew Austin didn't bother to trim the complete quote of
Lou's post in his own reply in this thread, we feel compelled to
mention the ratio of bytes in Andrew's reply to the total byte count
in the message:

total count:         3019 bytes
andrews contribution: 655 bytes

ratio      = 3019/655 = 4.6091

since the golden mean = 1.6182 == "phi" (traditional)

their difference is then approximately = 3 + phi.

les schaffer

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