The above image depicts a close representation of the Fibonacci 'Golden Rectangle' which has an aspect ratio of the Fibonacci Number. The shown rectangle deviates less than 3 parts in 1600 by the ratio of 8th/7th term. Ratios up to the first 15 terms is shown; the 15th term asymptotes to within 3 parts in 1.618 million of the Fibonacci Number.
Another interesting aspect is that the (Fib) Number is the only Number whose reciprocal is equal to the Number minus 1. Solving this "word problem", 1/Fib = Fib-1, allows for easy solution of the Fibonacci Number without need for SUMming an infinite series. Applying the quadratic solution: Fib = (1+(5)½)/2 or the reciprocal, 1/Fib = (1-(5)½)/2.
Interesting if not surprising, While the ratio if the F(n+1)/F(n) term converges on the Fibonacci ratio, the F(n+2)/F(n) term converges on Fibonacci squared and F(n+3)/F(n) on Fibonacci cubed, etc. I hear the stock market Guru's discussing "buy and sell Fibonacci Retracement Points" that are multiples of Fibonacci ratios like 55/89= 61.8% or 34/89=38.2%. Who am I to snicker just a bit?
Fibonacci shows up in unusual places. A 5 pointed symmetrical star (shown above) made by 5 lines forming 15 line segments exhibits the Fibonacci for the ratios of segment lengths The center is a pentagon with 5 each triangles forming the 5 points, each triangle being an isosceles triangle with two 72 degree angles at the base and 36 degrees at the point vertex. The ratio of the base length to the side is the Fibonacci Number.
An interesting equation shows up in a Jan-Feb 2010 American Scientist Article P-85 written by David T. Kung. The article references an equation submitted to a high school calculus teacher, Don Joffray, by a former student that finds the nth Fibonacci term: Fib(n) = ((1+(5)½)n -(1-(5)½ )n)/(2n*(5)½); especially cool having occurred in a high school calculus class.
It turns out that the relativistic mass of an object in-falling from infinity to the Schwarzschild radius (event horizon) is increased by Fibonacci times rest mass at the Event Horizon. See Above Chart. For more detailed explanation goto the following Link; [Main Menu] and follow path: index.html/BH_0.html/ Relativistic_Corrections.html