Historically people are fascinated by natural constants such as PI, the Pythagoras triples, e - the natural growth rate and such. In that series comes the sequence of numbers identified by a man who is known by his father's name as son of Bonacci, i.e. Fibonacci. He set out to solve the following problem and ended up with a surprisingly powerful sequence of numbers that has many expressions in nature.
"A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced by that pair in a year, if in every month, each pair begets a new pair, which from second month onward becomes productive? Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on."
Fibonacci solved it as follows and arrived at his famous sequence.
1, 1, 2, 3, 5, 8, 13,....and so on. Simply put, each number is the sum of two previous numbers.
There are many significance to this sequence of numbers. First of all the sequence expresses the natural phenomenon of physical growth with numbers. Plant and Animal cell divisions follow the Fibonacci sequence! Looks at the images below, the expressions of Fibonacci sequence.
There are more fascinating facts about the sequence.
Sum of 10 consecutive Fibonacci numbers is equal to 11 times the 7th number!
1+1+2+3+$+8+13+21+34+55 = 143
so is 11*13 = 143
5+8+13+21+34+55+89+144+233+377 = 979
so is 11*89 = 979
Here is another interesting property of the sequence.
Take any four consecutive Fibonacci numbers (a,b,c,d)
Pythagoras triples are ad, (2*bc) and (b2+c2)
Take 1,1,2 and 3 for a,b,c and d respectively,
ad=3, 2*bc =4 and b2+c2 = 5
i.e. 3,4,5 a well known Pythagorean sequence
Take 1,2,3 and 5
ad=5, 2*bc=12 and b2+c2 = 13
i.e. 5,12,13 another well known Pythagorean sequence!
and so on.
That is not all. Here is another interesting property. The ratio of nth number of the sequence to the (n-1)th number converges to the famous "Golden Ratio"
Interesting corollary is that Golden Ratio (Phi) is the number which when added with 1 gives the square of the number! i.e. (Phi)2 = Phi + 1
Fascinating facts indeed!
The book "Taming the Infinite" by Ian Stewart is a good read. Its about the history of Math. The book triggered renewed interest in Fibonacci sequence and a good journey into the fascinating facts around that set of numbers.
"A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced by that pair in a year, if in every month, each pair begets a new pair, which from second month onward becomes productive? Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on."
Fibonacci solved it as follows and arrived at his famous sequence.
1, 1, 2, 3, 5, 8, 13,....and so on. Simply put, each number is the sum of two previous numbers.
There are many significance to this sequence of numbers. First of all the sequence expresses the natural phenomenon of physical growth with numbers. Plant and Animal cell divisions follow the Fibonacci sequence! Looks at the images below, the expressions of Fibonacci sequence.
There are more fascinating facts about the sequence.
Sum of 10 consecutive Fibonacci numbers is equal to 11 times the 7th number!
1+1+2+3+$+8+13+21+34+55 = 143
so is 11*13 = 143
5+8+13+21+34+55+89+144+233+377 = 979
so is 11*89 = 979
Here is another interesting property of the sequence.
Take any four consecutive Fibonacci numbers (a,b,c,d)
Pythagoras triples are ad, (2*bc) and (b2+c2)
Take 1,1,2 and 3 for a,b,c and d respectively,
ad=3, 2*bc =4 and b2+c2 = 5
i.e. 3,4,5 a well known Pythagorean sequence
Take 1,2,3 and 5
ad=5, 2*bc=12 and b2+c2 = 13
i.e. 5,12,13 another well known Pythagorean sequence!
and so on.
That is not all. Here is another interesting property. The ratio of nth number of the sequence to the (n-1)th number converges to the famous "Golden Ratio"
Interesting corollary is that Golden Ratio (Phi) is the number which when added with 1 gives the square of the number! i.e. (Phi)2 = Phi + 1
Fascinating facts indeed!
The book "Taming the Infinite" by Ian Stewart is a good read. Its about the history of Math. The book triggered renewed interest in Fibonacci sequence and a good journey into the fascinating facts around that set of numbers.
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2 comments:
Actually, fibonacci finds a lot of applications. Some years back, my daughter did a maths project for school on fibonacci series and since then, I have been finding out more and more places where fibonacci keeps popping out. You might be surprised to know that the golden ratio is used even in photography. In fact, using golden ratio is one of the basic techniques of getting great photographs!
Yep. That's right. Fibonacci applications are in Art, Music and Architecture. Here is a good article http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html
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