The
Fibonacci numbers are important in the run-time analysis
of Euclid's algorithm to determine the greatest common divisor
of two integers.
The Fibonacci numbers occur in a formula about the diagonals of Pascal's
triangle (binomial coefficient).
In
music Fibonacci numbers are sometimes used to determine tunings,
and, as in visual art, to determine the length or size of
content or formal elements.
An interesting use of the Fibonacci sequence is for converting miles
to kilometers. For instance, if you want to know how many kilometers 8 miles
is, take the Fibonacci number (8) and look at the next one (13). 8 miles is about
13 kilometers. This works because it so happens that the conversion factor between
miles and kilometers (1.609) is roughly equal to 
(1.618).
Even more amazing is a surprising relationship to
magic squares. Magic squares are arrangements of numbers in a square pattern
in which all the rows, columns, and diagonals add up to the same number. The
simplest is the 3x3 pattern shown below:
If
one substitutes for these numbers the corresponding Fibonacci
number, a new "magic square" is produced in which
the sum of the products of the three rows is equal to the
sum of the products of the three columns!
Fibonacci
(Leonardus Pisanus de filiis Bonaccii) 
The "greatest
European mathematician of the middle ages", his full
name was Leonardo of Pisa, or Leonardo Pisano in Italian
since he was born in Pisa (Italy), the city with the famous
Leaning Tower, about 1175 AD. Strangely, Fibonacci is best
remembered by the sequence which bears his name but which,
ironically, he treated only lightly!
Little is known about Leonardo's life beyond the few
facts given in his writings. During Leonardo's boyhood his father, Guglielmo,
an official of the Republic of Pisa, was appointed consul over the community
of Pisan merchants in the North African port of Bugia (now Bejaïa,
Algery). He intended for Leonardo to become a merchant and so arranged for his
instruction in calculational techniques. Leonardo was sent to study calculation
with an Arab master. He later went to Egypt, Syria, Greece, Sicily, and Provence,
where he studied different numerical systems and methods of calculation.
Around 1200, Fibonacci returned to Pisa where, for at least the next
twenty-five years, he worked on his own mathematical compositions. The five works
from this period which have come down to us are:
• Liber
Abaci, ~1202, 1228. (The Book of Calculating). An
encyclopaedia of thirteenth-century mathematics, both theoretical
and practical. One of the problems in this book involves the
famous sequence 1, 2, 3, 5, 8, 13, ... 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?". Actually,
it was much later (~ 1870) that Edouard Lucas named this famous
series of numbers after Fibonacci.
You
can acquire the English translation of 'Liber Abaci' here.
• De Practica Geometriae, ~1220. (Practice of Geometry).
A book on geometry based on Euclid's "Elements" and "On Divisions".
• Flos, ~1225. (Flower). In this short work (the title
of which might suggest that algebra is the "flower of Mathematics"),
Fibonacci describes inter alia two of the 'Diophantine problems' he
worked on at the court of the Emperor Frederick II.
• Epistola ad Magistrum Theodorum, ~1225. A letter to
Master Theodorus, the imperial philosopher to the court of the Hohenstaufen emperor
Frederick II.
• Liber Quadratorum, ~1225. (The Book of Squares). His
largest work, a number-theoretical book concerned with the simultaneous solution
of quadratic equations in two or more variables.
These however are not the only books written by Fibonacci. Other
works known to have existed include the Di minor guisa, a book
for commercial arithmetic.
So
great was Fibonacci's reputation as a mathematician as a
result of these works that Frederick II summoned him for
an audience when he was in Pisa around 1225.
After 1228, virtually nothing is known of Fibonacci's life, except
that by decree the Republic of Pisa awarded the "'serious and learned Master
Leonardo Bigollo' (discretus et sapiens) a yearly salarium of 'libre XX denariorem'
in addition to the usual allowances". This stipend rewarded Fibonacci for
his pro bono advising to the Republic on matters involving accounting
and related mathematical matters.
His
various names
Leonardo
Pisano called himself Fibonacci [pronounced fib-on-'ah-chee],
short for fillio Bonacci (or Bonaccii), which
means "Bonaccio's son" in old Italian (Fi' = "son"; Bonacci(o) + i = "Bonaccio
+ 's"), since his father's name was Guglielmo Bonaccio. Fi'-Bonacci
is like the English name of John-son or the
Scottish name Mac-Donald.
His father's name was most probably a nickname with the ironical
meaning of a 'good, stupid fellow', while to Leonardo himself another nickname, Bigollo or Bigollone ('loafer,
wanderer', cf. the Italian word bighellone), has been given.
(chrysodromos,
lit. the "golden course") is a sequence of numbers
first created by the Italian mathematician Leonardo di Pisa,
or Pisano, known also under the name Fibonacci in 1202. It
is a deceptively simple series, but its ramifications and
applications are nearly limitless.
5/2
+ 0.5 ~ 1.618...), or Golden Ratio,
is intimately related to Fibonacci numbers. The closed form
of Fn is:
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