### Monday, August 05, 2013

5/8/13

Hello, and welcome toFibonacci Monday.

(if you looked at that date and went "Oh my word, so it is", you can skip the next paragraph)

Fibonacci numbers are numbers in the sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610...

where each number is the sum of the two numbers immediately before it. So for example 2+3=5, 3+5=8 and, most importantly for today, 5+8=13. A mathematician would say F(n)=F(n-1)+F(n-2), but you don't need to worry about that. The Fibonacci sequence goes on forever, eventually hitting numbers like 1597, 75025 and 2971215073. It's not an ordinary sequence where the gaps between the numbers are the same each time, and the terms aren't easily described by a simple formula, which makes it intriguing to mathematicians. And it turns out to have a surprising number of connections to the natural world, including the arrangement of leaves on a stem, the spiral of florets on a sunflower and the breeding of rabbits. Especially the rabbits.

The sequence was named after Leonardo Fibonacci, an 13th century mathematician from Pisa, who in 1202 wrote a book about arithmetic called Liber Abaci. The book brought together lots of ideas about place value and calculation, many originating in Hindu and Arabic volumes written centuries earlier. In chapter three Leonardo discussed several mathematical problems, including one in particular about the size of a population of rabbits. Imagine this pair of rabbits, said Leonardo, breeding once a month to produce two offspring. Rabbits take two months to come to maturity, so for the first two months there's only one pair. Then in month three a new pair is born, so you have two pairs, then in month four another making three. In month five the original pair have another, but their first children also have a pair, so the population's up to five. In month six all the pairs alive two months ago have another, and all the pairs alive last month are still alive, making eight. And so it continues, until by year ten the entire planet's overrun by bloody rabbits.

I prefer to think instead of climbing steps. Suppose there's this infinite staircase, and you can climb either one step or two steps at a time. How many different ways are there to reach each step? There's only one way to reach ground level, and there's only one way to reach the first step. But there are two ways to get to step two (either direct from the ground or another step up from step one), then three ways to reach step three (a big step up from the first step, or one of the two ways to get to step two plus a single step). And it continues thus...

Ground:1 way

Step 1:step up from ground = 1 way

Step 2:step up from ground, or step up from 1st step = 1+1 = 2 ways

Step 3:step up from 1st step, or step up from 2nd step = 1+2 = 3 ways

Step 4:step up from 2nd step, or step up from 3rd step = 2+3 = 5 ways

Step 5:step up from 3rd step, or step up from 4th step = 3+5 = 8 ways

Step N:step up from (N-2)th step, or step up from (N-1)th step = F(N-2)+F(N-1) = F(N)

Whatever, you end up with this simple yet intriguing sequence whose numbers model all sorts of natural patterns. Count the spirals on a pine cone and you'll usually get a Fibonacci number, one if you count one way, and the next number up if you count the other. Flowers very frequently have a Fibonacci number of petals - for example delphiniums have 8, asters have 21 and daisies muster (approximately) 55 or 89. It's all something to do with efficient means of packing things together, and that's how sunflowers reveal the pattern too. Bees have a Fibonacci number of ancestors, because females need two parents but males only one. And then there's the rabbits, but only mathematical rabbits because of course real-life rabbits aren't immortal, don't inbreed, don't always have one male and one female offspring, and don't stick to such rigid schedules.

Another natural phenomenon is the decimal you get if you divide a Fibonacci number by the previous Fibonacci number. 3÷2=1.5, 5÷3=1.6666, 8÷5=1.6, 13÷8=1.625, etc. As the sequence continues that decimal gets closer and closer to 1.61803398874..., a number mathematicians call the golden ratio (and which is actually a root of the quadratic equation x^{2}-x-1=0). The golden ratio can be found in the spiral of the shell of a snail, and is also used by architects and artists to create rectangles that are attractive to the human eye. The Parthenon in Greece, for example, is about 1.618 times as long as it is tall, giving it that pleasing classical feel... a fact known long before Leonardo Fibonacci wrote his book.

Don't let it worry you, but other Fibonacci sequences are available. The main Fibonacci sequence starts with 1 and 1 (or, more properly, with 0 and 1). But you could start with 1 and 3, and then you'd get a completely different list (1, 3, 4, 7, 11, 18, 29 etc). Or you could start with 1 and 4 (1, 4, 5, 9, 14, 23 etc) or 7 and 13 (7, 13, 20, 33, 53, etc). But you don't get anything new if you start with 1 and 2 because that's just the original sequence shifted along a place, ditto 2 and 3 (shifted two places). Only the 1 and 1 version is biologically pertinent. Accept no imitations.

So anyway, it may not be proper maths, but today is a Fibonacci date. Today's 5/8/13, and 5+8=13. It's not something to get unduly excited about, unless you like numbers, but it is a rarity. We've only had four Fibonacci dates so far this century, that's 1/1/02, 1/2/03, 2/3/05 and 3/5/08. Today we've reached the fifth - 5/8/13 - and that's your lot. The next Fibonacci date ought to be 8/13/21, but there isn't a 13th month, so the whole sequence judders to a grinding halt. Except in America, that is, where their date notation always sticks the month before the day. Today's no Fibonacci date in America, because they've already had theirs this year on 8th May (5/8/13). But they've got another one on 13th August 2021, which they'll write as 8/13/21, and that's their very last.

I note that the good people of Scotland are celebrating today's Fibonacci date with a bank holiday, which is nice, and a most appropriate tribute. For the rest of us it's hard to know how best to celebrate - maybe a pizza for dinner, or even some rabbit. You could raise a glass just after half past nine this evening, because that'll be 5/8/13 21:34. You could listen to this Radio 4 programme with Melvyn Bragg, if you have 45 minutes to spare. But unless you move to America, or plan to live into the 22nd century, today's your lot. Happy Fibonacci Monday, everyone, and I hope it all adds up for you.