Because of my love of spirals, I got interested in the Fibonacci sequence. Now, I’m no mathematician, so don’t get too excited, but I did a bit of research, and this is what I found out.
Leonardo Bonacci (c. 1170 – c. 1250), known as Fibonacci, was the son of a wealthy Italian merchant. During travels with his father, he came across the Hindu-Arabic numeral system in Algeria – this is the counting system we use today, with symbols 0-9 for numbers and positional notation (place values showing ones, tens, hundreds etc.). He promoted this system in 1202 in a book called Liber Abaci.
Liber Abaci also outlined a problem involving the hypothetical growth of a population of rabbits. The solution was a sequence of numbers which were later named ‘Fibonacci sequence’ by the 19th-century number theorist Édouard Lucas. It should be noted that although Fibonacci’s Liber Abaci contains the earliest known description of the sequence outside of India, the sequence had been noted by Indian mathematicians as early as the sixth century. Good on you, Fibonacci (and Lucas), sorry ’bout that Indian mathematicians!
In the Fibonacci sequence of numbers, each number is the sum of the previous two numbers. Fibonacci’s problem considered the growth of an idealized rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
OK, so that’s all sounding a bit like ‘two trains are heading in opposite directions at 53km per hour, at a gradient of 14 degrees – so, what is the driver’s name?” however:
At the end of the first month, they mate, but there is still only 1 pair.
At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.
So the sequence looks like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 which can be shown in tiles in this fashion:
From this, comes the glorious Fibonacci spiral – which is an approximation of the ‘golden spiral‘ created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling.
Fibonacci sequences appear in nature – branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, artichoke flowers, an uncurling fern all follow the sequence. Apparently some claims of Fibonacci numbers or ‘golden sections’ in nature (the breeding of rabbits in Fibonacci’s own unrealistic example, the seeds on a sunflower, the spirals of shells, and the curve of waves) are poorly substantiated – pretty sure the above looks just like a shell, though, so I’m running with it! And there, my friends, endeth the lesson – if you’d like to know more, a Google search will lead you down many rabbit warrens on the interesting Fibonacci spiral.