1, 1, 2 , 3, … The Fibonacci sequence

If you look up at the title of this article, you may find something striking. The sum of the last two previous numbers add up to the next number. This was discovered by Indians a long time ago but it was published by Leonardo Pisano Bogollo (a.k.a. Fibonnaci -son of Bonnaci) around the 11th century CE.

The first 20 numbers

Ok…(draws big breath) here goes!

1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765..!

If you’ve been observant, notice every 3rd number is a multiple of 2, every 4th number is a multiple of 3 and that every 5th number is divisible by 5.

You may be thinking…OK. 1+1=2 and 1+2=3, what’s so amazing about the Fibonacci sequence?

φ and Fibonacci!

φ is called the Golden Ratio and it is irrational, roughly equalling 1.61803398874989484820… – it is irrational. If we get the first 10 consecutive numbers of the Fibonacci and divided them by their smaller neighbour, watch what happens!

A	B	B/A
1	1	1
2	1	2
3	2	1.5
5	3	1.6666666…7
8	5	1.6
13	8	1.625
21	13	1.6154 (4.d.p.)
34	21	1.61905 (5.d.p)
55	34	1.61765 (5.d.p.)
89	55	1.6181818181818181818…

You see?

The lower down the table, the closer we get to φ (1.61803398874989484820…). In fact, if we get 6765 (term 20)/4181 (term 19) you get 1.61803396… which is close but no matter how low down we go, we’ll never exactly get φ because it’s irrational (the long part divided by the short part is also equal to the whole length divided by the long part).

A picture of the great Italian mathematician, Fibonacci

Nature and Maths

The fibonnaci sequence can be represented as a spiral…

….seen this before?

Sunflowers love this spiral and their cells always turn x.618 times to make such wonders. Plants also like it because it evenly spreads out sunlight to all the leaves and makes rainwater go quickly to be absorbed by the roots. You might see this in an aloe vera plant (which, by the way, does wonders for your skin and body!).