Why is a Raven like a Writing Desk?

Here's a riddle for you: Why is a Raven like a Writing Desk? Actually I don't know. Although you just might want to read www.straightdope.com/columns/read/1173/why-is-a-raven-like-a-writing-desk .

Here's another riddle: Why is a rabbit population like a sequence of drum beats? And why are both like a special rectangle?

Give up? They are all representations of the Fibonacci sequence which is made up of the numbers 1, 1, 2, 3, 5, 8, 13, ...

Fibonacci was a mathematician who lived in Europe in the Middle Ages. He travelled to north Africa where he learned from Arab scholars how to calculate with Hindu numerals (i.e., 0, 1, 2, 3, ...9). Try the following calculation: LVII x CMIX. Too hard? Its just 57 x 909. However he is most famous for the following problem:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? www-groups.dcs.st-and.ac.uk/~history/Biographies/Fibonacci.html

Answer: Suppose there were 8 pairs of rabbits in August and 13 pairs in Sept. In October all of September's rabbits are still alive, and all of August's rabbits produce new offspring. Hence the number for October is 13 + 8 = 21. This explains why the term-to-term rule for generating new Fibonacci numbers is T_n+1 = T_n + T_n-1
But what do rabbits have to do with drum beats? Fibonacci invented the rabbit problem for his students in 1202 AD. But two Indian mathematicians have a prior claim: Gopāla, before 1135 AD and Hemachandra, about 1150 AD. Let a long beeaat = 2 short beats. Then: B_n+1 = B_n + B_n-1

	1 beat	2 beats	3 beats	4 beats
	S	L	L + S = LS	LS+S = LSS
		SS	SS +S = SSS	SSS+S= SSSS
			S +L = SL	SL+S= SLS
				L+L =LL
				SS+L=SSL
total	1	2	3	5

The next number of beats equals, FIRSTLY all the beats 'alive' in the column previously - add one short beat to the end of these sequences- plus, SECONDLY all of the beats 'alive' two columns previously - add one long beat to the end of those sequences. All 5 beats of length four can become beats of length five by putting a short beat at the end. All 3 beats of length three can also become beats of length five by putting a long beat at the end. Consequently, there must be a total of 8 beats of length five!  Indian poets and drummers speak of 'Hemachandra numbers', not 'Fibonacci numbers.' http://www.sju.edu/~rhall/mathforpoets.pdf

There are many other Fibonacci (or Hemachandra) mysteries to explore. Here is just one more. Draw a square of length 1.

Boring! Draw another square of the same size beside it.

A little better. Continue with a square of length 2 above those.

Eventually.

Nature and artists can create spirals and other amazing patterns from these golden rectangles.

the golden spiral below is available at http://demonstrations.wolfram.com/LogarithmicSpiral/

You can download a free Mathematica player for this applet (and hundreds of others) here:

http://www.wolfram.com/products/player/download.cgi

Are you surprised that the same mathemartics underlies such different topics as population biology, drumming and visual art? Galileo wasn't. To him it was no coincidence, because: "Mathematics is the language with which God has written the universe."