Cryptography Reference
In-Depth Information
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Leonardo Fibonacci was a medieval mathematician from
the northern Italian city of Pisa. Known to history either
as Leonardo Pisano or simply as Fibonacci, he gained fame
throughout Europe for his
Liber abaci
(“Book of the Abacus”),
which he published in 1202 and which was the first European
work on Hindu-Arabic numerals. He is known to modern
mathematicians mainly because of the Fibonacci sequence,
derived from a problem in the
Liber abaci:
A certain man put a pair of rabbits in a place surrounded
on all sides by a wall. How many pairs of rabbits can be pro-
duced 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?
The resulting number sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34,
55 (Leonardo himself omitted the first term), in which each
number is the sum of the two preceding numbers, is the first
recursive number sequence (in which the relation between
two or more successive terms can be expressed by a formula)
known in Europe. Terms in the sequence were stated in a for-
mula by the French-born mathematician Albert Girard in 1634:
u
n + 2
=
u
n + 1
+
u
n
, in which
u
represents the term and the subscript
its rank in the sequence. The mathematician Robert Simson at
the University of Glasgow in 1753 noted that, as the numbers
increased in magnitude, the ratio between succeeding num-
bers approached the number
a
, the golden ratio, whose value
is 1.6180 . . . , or (1 + 5)/2.
In the 19th century the term Fibonacci sequence was
coined by the French mathematician Edouard Lucas, and
scientists began to discover such sequences in nature; for
example, in the spirals of sunflower heads, in pine cones, in
the regular descent (genealogy) of the male bee, in the related
logarithmic (equiangular) spiral in snail shells, in the arrange-
ment of leaf buds on a stem, and in animal horns.
