Fibonacci... ugh. SEquences and Series were my least favorite!
OK. Here' goes.
The Fibonacci series had something to do with rabbits.. and their breeding habits. I don't remember the exact situation, but I can find it.
Anyway... the Fibonacci series is the sequence of numbers such that each new number in the series is the sum of the two numbers before it. You have to start out with the first 2 being given though - so the series is:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc.
Notice that 0+1=1, 1+1=2, 1+2=3, 2+3=5, 3+5=13, 5+13=18, etc.
The golden mean comes in because that's what the series of the ratios of consecutive numbers converges to... so the sequence 1/1, 2/1. 3/2, 5/3, 13/5, 18/13, etc. eventually converges to the golden mena. Don't remember what that value is... but I bet I can find out.
The golden mean is a ratio that comes into play in much of the natural world... harmonics in music are separated by it in some way, and the greeks liked to use it in their architecture. I do recall that the golden mean is one that satisfies the following situation:
Start out with a rectangle. Let the length of the long edge be L and the length of the short edge be S (yes... for "L"ong and "S"hort). Then let M = L/S (that is, we'll define the value "M" to be the ratio of the long side to the short side). So... If you set your rectangle up right, then if you cut off a portion of the rectangle that is SxS, then what's left over will now be a rectangle that has the dimensions S by (L-S). Helps if you draw a picture... cut a perfect square off of the end of the rectangle and look at what's left over.
Then we now have a new ratio M' = S/(L-S).
It's possible to set it up so that M = M'... this means that even if you keep cutting off squares, the ratio of the long side to the short side always stays the same. There's only one way to do this so it works... and it just happens that the ratio is... you guessed it... the golden mean.
Do I pass yet... or am I going to have to dig out my math books?