The Common Vein

Lev, if there were a God, not only would he be a mathematician, but he would be an Italian Mathematician. How else can one explain that the Fibonacci numbers relate to sunflower spirals, artichoke patterns and pineapple fruitlets, rabbit populations, golden ratios, financial trading, project duration estimates, musical themes, body ratios, file compression, parallel computing, search techniques and now?courtesy of you, cricket duck analysis!A former Australian prime minister once referred to a favourable budget report as ?a beautiful set of numbers?. Rather, I think the phrase should be reserved for the Fibonacci series.

For anyone interested, the series begins with two 1s, i.e, 1,1.
The next number is obtained by adding the previous two in the series and so we get 2. This formula is repeated ad infinitum so the next one would be 1 + 2 giving us 3 and the next 2+3 yielding 5 and so on.

Try this at home!

You will get?

1,1,2,3,5,8,13,21?..

An interesting feature of this is that if you keep going and then take the ratio of successive numbers, i.e. divide each number by the one before the result approaches a limit ? the value being 1.618033989

This is known as the golden ratio and has very appealing aesthetic qualities. If you draw a line of length 1.618033989 and get people to put a mark on the line that gives the most eye pleasing proportional relationship between the two segments so created, they will on average plump for a point which divides the line 1 to 0.618033989. Furthermore, the whole line is to the longer segment as the longer segment is to the shorter segment. There is a magician at work here!

The Parthenon contains golden ratios. You will find it in Salvador Dali paintings. It is thought that most human bodies reflect this ratio. For example, the nose on a face is located according to this proportion which also describes other lengthts in relation to that of the body.

Party trick!!
To find the nth item in the list do the following on your calculator or spreadsheet.

1. Raise the Golden ratio 1.618033989) to the nth power (multiply it by itself n times)
2. Take only the fractional part (.618033989) and do the same thing.
3. Subtract the result in 2 from that in 1.
4. Divide the result by the square root of 5.

Bingo ? you have it. You don?t have to generate the sequence one by one to get to the nth one. Just use the procedure above (I can see you all rushing for your calculators now) and impress your friends!

There are deep philosophical issues here. Why does mathematics describe nature so well? Did we create the mathematics to do so or did we discover it? Put another way, did we write mathematics into nature or do we read it from her? If the latter, who put it there? Does this mean that there is such a thing as a number independent of the human mind? Is mathematics universal so that this will work in non-humans? minds also or would they have a different mathematics? Discuss.