Yesterday’s mention of breeding livestock, and the earlier
mention of the nautilus here has put me in mind of ratios and sequences. Let’s
do a little thought experiment: instead of stick insects, let’s substitute
rabbits. Assume I want to start breeding rabbits for fun and profit, so I pop
down to Bob’s Bunny Shop and buy my first pair of rabbits.
Now these
imaginary rabbits don’t reach maturity until they are two months old, so for
the first month I have one pair of rabbits, and for the second month I still
only have one pair. Then, in month three, they produce a pair of baby rabbits,
so now I have two pairs of rabbits. In month four, the first pair produces yet
another pair, so now I have three pairs of rabbits. Month five gets me another
pair from Pair 1 but now the pair born in month three start to breed, so I get
a pair from them too, giving me a total of five pairs.
And so it goes,
throughout the next year, with pairs producing more pairs after a two month
wait. Here’s a little table to illustrate the progress of my rabbitbreeding
programme:
Month 
Generations of Rabbits 



1st

2nd

3rd

4th

5th

6th

Total 
One

1






1

Two

1






1

Three

1

1





2

Four

1

2





3

Five

1

3

1




5

Six

1

4

3




8

Seven

1

5

6

1



13

Eight

1

6

10

4



21

Nine

1

7

15

10

1


34

Ten

1

8

21

20

5


55

Eleven

1

9

28

35

15

1

89

Twelve

1

10

36

56

35

6

144

Super. After twelve months, I
have 144 pairs of rabbits. Not bad for the first year. But look at the totals
column of the right. Do you notice anything about the sequence? Each pair of
consecutive numbers add up to the number below them: 1+1=2, 1+2=3, 2+3=5,
3+5=8, 5+8=13 and so on.
This is called a Fibonacci sequence, after
Leonardo Pisano, also called Leonardo Fibonacci (son of Gugleilmo Bonacci – Fi(lius)Bonacci)
or simply Fibonacci, an Italian mathematician who introduced the Hindu/Arabic
numeral system into Europe, supplanting the system of Roman numerals. In 1202,
he published his Liber Acubi (Book of Calculation), in which he used the
sequence that now bears his name as an example (he did not discover this
sequence) using the hypothetical breeding of rabbits (which I’ve just pinched
above).
There are lots of things you can
do with the Fibonacci sequence, (some of which are mindboggling complicated),
but some of the most aesthetically pleasing are the relationships found in
natural objects. Instead of rabbits, let’s look at squares.
Take a single
square.
Then add another square of the same size. That gives us a rectangle one
unit wide and two units high. Like the bunnies in the table, 1+1 has given us
two.
So add another square, with its sides equal to two units. And now we have
another rectangle of two units by three units, so we can add another square
with sides of three units.
The resulting rectangle measures 3 x 5, so we can
add yet another square of 8 units.
And so it goes, adding squares of increasing
side length.
Then, if you get yourself a compass (try this with a ruler and a
bit of paper), draw an arc in the first square with a radius of one unit.
Add
an arc in the second square, also with a radius of one unit.
In the third
square, draw an arc with a radius of two units, and so on.
The result is a
Fibonacci spiral.
This is a logarithmic spiral, where the proportional increase
in each quarter turn is equal to the squareside radius of the arc. This is the
spiral found in the shells of molluscs, as the growth allows for a larger shell
without altering the shape of the shell. (The Golden Spiral, another
logarithmic spiral, increases by the ratio of phi (Φ) – more of which later –
and so would necessarily alter the shape of a shell).
Once you start looking
for these spirals, you see them everywhere.
Cretaceous Silver Ammonite from Madagascar 
Nautilus shells we have seen, and
the same spiral occurred in their extinct relatives, the ammonites (ammonites
get their name from the similarity of their shape to the ram’s horns of the
Egyptian god. Ammon).
Ammon 
It appears in other shells – this conch shell has one.
As
does this murex shell.
You can find them on your dinner plate – here is a
Romanesco cauliflower.
You can find them in the sky – here are a couple of
spiral galaxies.
They are in pinecones and pineapples.
But, and this is
important, there is nothing mystical or magical about these spirals – they are
entirely natural phenomena. They have not been put there, they have
evolved over millions of years. They are the result of an underlying geometry
in the universe, as nature produces the most economical solution to problems in
terms of effort and materials. The garden is already beautiful enough as it is
– there is no need to go and invent fairies at the bottom of it.
what a lovely, lucid exposition i particularly enjoyed the conclusion.
ReplyDeleteall the best,
d