Fibonacci series
Les Schaffer
schaffer at SPAMoptonline.net
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
concerns.
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
agreement.
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
operations.
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
livelihood.
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
numerals.
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:
http://www.rpi.edu/~eglash/eglash.dir/hit.dir/afch7.dir/afch7.htm
http://www.rpi.edu/~eglash/eglash.dir/hit.dir/afch7.dir/fig7_2.jpg
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
boredom].
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
efficiently.
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)
b
(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:
http://library.thinkquest.org/27890/applications5.html
http://library.thinkquest.org/27890/mainIndex.html
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
Use of the Fibonacci Series in the Bassoon Solo in Bartók's Dance
Suite:
http://idrs.colorado.edu/Publications/Journal/JNL17/JNL17.Ewell.Bartok.html
waskly wabbits:
http://www.ac.wwu.edu/~stephan/webstuff/ratio.fibonacci.html
On Rabbits, Mathematics and Musical Scales:
http://www.bikexprt.com/tunings/fibonaci.htm
THE FIBONACCI SERIES AND THE FINANCIAL MARKETS:
http://www.luckymojo.com/fibonaccimkt.html
fibonacci series and the golden mean:
http://www.indoorooss.qld.edu.au/05studgl/fibonacci/webs/series.html
fractals and fibonacci:
http://math.bu.edu/DYSYS/FRACGEOM2/node7.html
fibonacci bio:
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html
growth patterns:
http://www2.umdnj.edu/pathweb/matrix/gnomonic.htm
the fibonacci series in java bytecodes:
http://www.artima.com/insidejvm/applets/FibonacciForever.html
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 :
http://www.onereed.com/articles/fib.html
fibonacci and quilts:
http://www.bryerpatch.com/exhibit/exhibit_1999.htm
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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