Fibonacci NumbersNature’s Mathematics

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …

Where did the series come from?

Fibonacci investigated rabbits in 1202.The question he posed was this:

‘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. How many rabbits will there be after ‘n’ months?’

How Many?

Other appearances in nature:

Shells

Other appearances in nature:

Flowers (number of petals)

Other appearances in nature:

Flowers (seed heads)

Coneflower: right spiral 55, closer to centre 34 spirals.

Other appearances in nature:

Pine Cones

Golden Ratio

We find a useful number Phi coming from Fibonacci numbers

Taking the numbers in the series: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 ...

This value converges to Phi,

Phi ( ) = 1·61803 39887 49894 … = (√5 +1)/2

We also use phi = Phi -1 = 0. 61803 …

Note phi=Phi/1

Phi in Nature

All leaves and petals on every type of flower are arranged with 0·61803… (phi) leaves/ petals per turn. This is because it is the optimal number for maximum space, rain, and light per leaf. But why is this optimal?

Why Phi?

If turn per seed was 0.5:

So exact fractions are FRUITless!The number must be irrational.

If turn per seed was 0.6:Because 0·6=3/5 so every 3 turns will have produced exactly 5 seeds and the sixth seed will be at the same angle as the first.

Why Phi?

So how about pi turns per seed:

Or e turns per seed:

- pi(3·14159..) – 7 arms turns-per-speed = 0·14159 close to 1/7=0·142857... - e(2·71...) - 7 arms turns-per-seed = 0·71828... (a bit more than 5/7=0·71428..). It is a little more, so the "arms" bend in the opposite direction to that of pi's (which were a bit less than 1/7).These rational numbers are called rational approximations to the real number value.

Why Phi?So the best turn-per-seed angle would be one that never

settles down to a rational approximation for very long. The mathematical theory is called CONTINUED FRACTIONS.The simplest such number is that which is expressed as a=1+1/(1+1/(1+1/(...)

a is just 1+1/a, or a2 = a+1.

To solve: a2 - a – 1 = 0

If we use c.t.s.: (a – 1/2)2 – 5/4 = 0

We then know: (a – 1/2) = +/- √(5/4)

Giving us the two numbers with this property:

a = (√5 +1)/2, -(√5-1)/2

which are Phi and -phi

Any Questions?