The Swirly Spirals of Nature's Numbers

             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 rabbit-breeding 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 mind-boggling 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 square-side 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.