Number Sequences

On-Line Encyclopedia of Integer Sequences

Listen to some sequences

Uses of the Fibonacci Number System and the Golden Section span nature architecture, and art.

The Fibonacci Number System is a sequence of numbers gerated by adding the two previous numbers in the sequence.

0,1,1,2,3,5,8,13,21,34,55,89,144,233,377....

The Fibonacci Spiral


Creates a Golden Rectangle
89/55=1.6181818


The Rabbit Problem.
Fibonacci hypothosised about how rabbits would breed in perfect circumstances.

He started with simple rules:
1)Rabbits can breed a month after birth
2)Each 'pair' of rabbits consists of one male and one female
3)At the beginning of each month, the rabbits will mate.

As it goes...
1)At the end of the first month the first pair mates.(one pair)
2)At the end of the second month the female has one pair of rabbits.(two pairs)
5)At the end of the third month the female has another pair of rabbits.(three pairs)
6)At the end of the fourth month the female has another pair of rabbits. In addition the female born two months ago begins to produce rabbit pairs(five pairs)

The Fibonacci patterns are found throughout nature.










How is the Fibonacci Number systen related to the Golden Number?

The Golden Number is derived from the ratio of two sucessive numbers and divide each by the number before it. This gives us this sequence:

1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538...

What emerges is the Golden Ratio or the Golden Number - 1.618034. This is an aproximation - we can say 1.6. The amazing thing is that as we continue to process the numbers in this manner the results hover around 1.6.

We find this in ancient Greek architecture. The Golden rectangle is a rectangle in the proportions of 1:1.618 or 0.618:1. Ths proportion is seen in the Parthenon in Athens.



A credit card is a Golden Rectangle.


The Golden Ratio is supposedly the most pleasing to the eye.




The Golden String

The Golden String is connected to the Fibonacci Sequence The best example is by looking at the rabbit issue.

Here we use N for new rabbits and M for mature. By rebuilding the family tree we are building the Golden string.

1)N
2)M
3)M N
4)M N M
5)M N M M N
6)M N M M N M N M
or
10110101

and on into Infinity.

How Does It Sound?

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