Download Binet`s formula for Fibonacci numbers

In nature's portfolio of architectural works, the
magnificent shell of the chambered nautilus holds a place of
special distinction. The spiral-shaped shell, with its revolving
interior stairwell of ever-growing chambers, is more than a
splendid piece of natural architecture – it is also a work of
remarkable mathematical creativity.
Humans have imitated nature's wondrous spiral designs
in their own architecture for centuries, all the while trying to
understand how the
magic works. What are
the physical laws that
govern spiral growth in
nature? Why do so many
different and unusual
mathematical concepts
come into play? How do
the
mathematical
concepts and physical
laws mesh together?
What is the source of the
intrinsic
beauty
of
nature's spirals? Trying
to answer these questions
is the goal of this
chapter.
DEF:
Fibonacci numbers
F1 = 1,
F2 = 1,
FN = FN-1 + FN-2, for any N > 2.
Therefore, after the first two numbers (both of them
being equal to 1), each subsequent number in the Fibonacci list
is the sum of the two numbers before.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …
OBS:
The list of Fibonacci numbers is an infinite sequence
and is called Fibonacci sequence. The previous definition is a
recursive one, i.e. each number is defined in terms of the other
(earlier) numbers.
Binet’s formula for Fibonacci numbers
N
N
1 5 
1 5 

 

 2 
 2 


 , for any N a whole number.
FN  
5
OBS: In spite of its rather nasty appearance, this formula has
one advantage over a recursive definition: it gives us an explicit
rule for calculating any Fibonacci number without having to
first calculate the preceding Fibonacci numbers. For this
reason, Binet's formula is called an explicit definition of the
Fibonacci numbers.
Fibonacci Numbers in Nature:
Our interest in Fibonacci numbers stems from their
frequent occurrence in nature. Take flowers, for example.
Consistently, the number of petals in a daisy is a Fibonacci
number, which depends on the variety: 13 for Blue daisies; 21
for English daisies; 34 for Oxeye daisies;
55 for African daisies, and so on. What's
true for daisies is also true for many
other types of flowers (geraniums,
chrysanthemums, lilies, etc.). Fibonacci
numbers also appear consistently in
conifers and seeds. The bracts in a pine
cone spiral in two different directions in
8 and 13 rows; the scales in a pineapple
spiral in three different directions in 8,
13, and 21 rows; the seeds in the center
of a sunflower spiral in 55 and 89 rows.
Exactly why and how this unusual connection between a
purely mathematical concept (Fibonacci numbers) and natural
objects (daisies, sunflowers, pineapples, etc.) occurs is still not
fully understood. But this connection certainly has something
to do with the spiraling way in which these objects grow.
Reminder:
The quadratic equation
a x2 + b x + c = 0
has two solutions given by the quadratic formula
 b  b 2  4ac
 b  b 2  4ac
, x2 
.
x1 
2a
2a
Therefore, the solutions of the equation x2 = x + 1 are
1 5
 1.6180339887...  1.618
2
1 5
 0.6180339887...  0.618
2
DEF:
The Golden Ratio

1 5
 1.618
2
OBS:
Ф2 = Ф + 1
Ф3 = 2Ф + 1
[since Ф3=Ф*Ф2=Ф(Ф+1)=Ф2+Ф=(Ф+1)+Ф=2Ф+1]
Ф4 = 3Ф + 2
[since Ф4=Ф*Ф3=Ф(2Ф+1)=2Ф2+Ф=2(Ф+1)+Ф=3Ф+2]
Ф5 = 5Ф + 3
[since Ф5=Ф*Ф4=Ф(3Ф+2)=3Ф2+2Ф=3(Ф+1)+2Ф=5Ф+3]
Ф6 = 8Ф + 5
[since Ф6=Ф*Ф5=Ф(5Ф+3)=5Ф2+3Ф=5(Ф+1)+3Ф=8Ф+5]
Ф7 = 13Ф + 8 [since Ф7=Ф*Ф6=Ф(8Ф+5)=8Ф2+5Ф=8(Ф+1)+5Ф=13Ф+5]
and so on.
Hence:
ФN = FNФ + FN-1
Example (page 322):
Ratios of consecutive Fibonacci numbers
The magic number that the ratios of two consecutive
Fibonacci numbers approach is Ф = (1 + V5)/2.
Essentially this means that, except for the first few, each
Fibonacci number is approximately equal to Ф times the
preceding one, and the approximation gets better as the numbers
get larger.
The Golden Ration in Art, Architecture, and Design:
As a special number, the golden ratio Ф is considered,
right along with π (the ratio between the circumference and the
diameter of a circle) among the great mathematical discoveries
of antiquity. The golden ratio was known to the ancient
Greeks, who ascribed to it mystical and religious meaning and
called it the divine proportion. The Greeks used the golden
ratio as a benchmark for proportion and scale in art and
architecture. The famous Greek sculptor Pheidias consistently
used the golden ratio in the proportions for his sculptures, as
well as in his design of the Parthenon, perhaps the best-known
building of ancient Greece.
Many others example where the Golden Ratio appear can
be found in worksheets and in the textbook.