## Tuesday, 25 September 2012

### Fibonacci

Leonardo da Pisa (otherwise known as Fibonacci) was an Italian mathematician born around 1770. Fibonacci is known for popularising the Arabic numerals (1, 2, 3 etc.) in Christian Europe, but is probably most famous for the number sequence he published in his book, Liber Abaci. The Fibonacci Sequence. This sequence of numbers was actually already known to Indian mathematicians as early as the 6th Century, but Fibonacci was the one who introduced it to the West.

Fibonacci originally came across the sequence when trying to solve a problem to do with the growth rate of rabbit populations. His problem was this:

Let a pair of baby rabbits into a fenced off garden. After a month they will mature into adult rabbits. Every month after that they will produce another pair of rabbits. If every pair of rabbits follows this model, and assuming none of the rabbits die, how many rabbits will there be after a year?

Obviously this theoretical model was created based on some very crude assumptions. For starters, the rabbits never die (which clearly wouldn't happen). Also it assumes that rabbits mate for life, they consistently produce another pair every month who also mate for life, and only take a month to mature into rabbits who can reproduce. These assumptions meant that he wasn't accurately predicting the growth rates of rabbits, but he did produce a sequence that is still used today, in science and in nature!

This image shows how the model plays out when you apply the limitations and run it through. The orange rabbits are the babies, and the white rabbits are when they have matured. When an arrow points from a white pair to a white pair, it shows the same pair of rabbits. The sequence Fibonacci ended up with was 1, 1, 2, 3, 5, 8, 13, 21... and so on. This is the famous Fibonacci Sequence. It is so famous because it has some interesting properties.

• The most obvious property is that every term in the sequence is the sum of the previous two. For example, 1+1=2, 1+2=3, 2+3=5 and so on.
• When you add the terms of the sequence (1+1=2, 1+1+2=4, 1+1+2+3=7) the answers create a new sequence that is similar to the original. When you add n terms of the Fibonacci Sequence together, the total equals the (n+2)th (for want of a better description...) term minus 1. For example, if you add the first 5 terms of the Fibonacci Sequence, the answer is the 7th term of the Fibonacci Sequence minus 1.
Original  1   1   2   3   5   8   13   21   34   55...
New                       2   4   7   12   20   33   54...

• When you add the squares of the Fibonacci numbers, you get another sequence. It just so happens there is an easy way to work out the sum up to the nth term of the squared sequence by using the Fibonacci Sequence. Say you want to work out the 5th term of the sum of squares, you take the 5th and 6th Fibonacci numbers and multiply them together. It comes out to the same number. In this case, 5x8=40.
Original                1   1   2   3       5     8       13       21      34        55...
Squares                1   1   4   9       25   64     169     441    1156    3025...
Sum of Squares    1   2   6   15    40    104   273    714     1870   4895

When you find the ratio between successive Fibonacci numbers, the ratios soon approach the number known as the Golden Ratio. The Golden Ratio is an irrational number (meaning the decimals don't end, and don't repeat in a pattern) that equals 1.618 to 3dp. The Golden Ratio (depicted by the Greek letter phi φ) can be shown as a rectangle, where the ratio of a:b is the same as the ratio of a+b:a (referring to the sides in the picture below).

When you draw the Fibonacci numbers as squares with side lengths corresponding to the number in the sequence, you end up with a picture like this.

Ignore the random line, I didn't count it correctly... Anyway, presenting the Fibonacci numbers in this way shows that it is a good approximation for the Golden Rectangle. The Golden Rectangle was thought to be the most pleasing rectangle by ancient scholars, and is used by many artists and architects. For example, the Parthenon in Athens, Greece, follows the Golden Rectangle, as does Leonardo da Vinci's Mona Lisa.

If you join the corners of the Golden Rectangle with a curved line, you end up with a spiral, known as (surprise, surprise) the Golden Spiral. The Fibonacci spiral is also a good approximation to this.

The Golden Spiral is found everywhere in nature. For example, the patterns on many different shells, such as snail shells, follows the same spiral. And of course, the picture that started off all of this research of mine:

That's it from me on Fibonacci. I've been trying to write this blog post for so long now, I think it's about time I actually finished it off and posted it.

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