Transcript Fibonacci Numbers - Oldham Sixth Form College

Fibonacci Numbers
Nature’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...,
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
8/5 = 1·6,
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:
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.
So exact fractions are FRUITless!
The number must be irrational.
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?