Fibonacci Sequence and the Golden Ratio
Fibonacci sequence  golden
ratio  examples
Fibonacci Sequence
This is a sequence of numbers named after Leonardo of Pisa, who was known
as Fibonacci.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
x_{0} = 0
x_{1} = 1
x_{n} = x_{n1} + x_{n2}
Interestingly, the sum of any ten consecutive Fibonacci numbers is always
divisible by 11. The number of elevens in these sums are themselves the
Fibonacci numbers from 8 upward.
0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 = 88 = 11 x
8
1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 11
x 13
1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 = 231 = 11
x 21
Golden Ratio
The higher the Fibonacci numbers, the closer the ratio between
two consecutive numbers is to the golden ratio, φ (the Greek
letter phi). In other words, 34 / 21 is closer to φ than 21 / 13
is. (This actually happens with any similarly additive sequence except
one starting with 0, 0.) The ratio of Fibonacci numbers F_{25001} and
F_{25000},
each of which is over 5,000 digits long, gives a value of φ accurate
to over 10,000 decimal places.
However, φ is actually an irrational number – a number which
can't be exactly defined by an ordinary fraction (ie, by dividing one
number by another – which means it never ends, like π or √2.
The value of φ can, however, be found using the square root of 5:
φ = ( 1 + √5 ) / 2
φ is called the golden ratio because it conforms to the following
rules:
φ = ( a + b ) / a = a / b

1 / φ = φ  1
1  φ = 1 / φ
φ^{2} = φ + 1
(or more generally) φ^{n+1}
= φ^{n} + φ^{n1} 
φ is about 1.6180339 and 1/φ is thus about 0.6180339. (BTW,
this order of accuracy is reached within the first 20 Fibonacci numbers.)
Examples
Strangely enough, the golden ratio also appears
in a regular (meaning "even") five pointed star (known
as a pentagram). The coloured line segments are in golden ratio
lengths relative to each other. Each intersection of edges breaks
other edges in proportions of the golden ratio. 

A Fibonacci spiral is made from using the Fibonacci numbers to
make quartercircle curves with radii defined by the Fibonacci numbers,
and linked in a spiral. The spiral at the right uses the Fibonacci
numbers 1, 1, 2, 3, 5, 8, 13, 21, 34. 

The Fibonacci spiral is a good approximation to
an ideal logarithmic spiral based on the golden ratio. (Note that
not all logarithmic spirals are related to the golden ratio.) In
the diagram, the Fibonacci spiral is green and the logarithmic spiral
is red. Where they overlap is yellow.
They are obviously very close, and the Fibonacci spiral is much
easier to draw than a true logarithmic spiral.


