Replies

1. 
   roninfante | Dec 30, 2002 04:50pm | #1

   If you set A = 1 in Uncle Dunc's reply and then solve for B you will get the Golden Ratio (1 + sqrt(5))/2. 

   Ron

1. 
   Leverone | Dec 30, 2002 07:03pm | #3

   Thanks very much for the help.

2. 
   UncleDunc | Dec 30, 2002 10:03pm | #4

   OK, I admit it. I'm the guy who flunked algebra twice and finally took it pass/fail. I've always solved this iteratively. Start with 1.6, invert and add 1. Repeat until you have as many decimal places as you like. It takes five or six iterations on my calculator to get 3 decimal places.

   Your way looks quicker if I could figure out the algebra. I get as far as B^2 - B = 1, then I get stuck. Can you help me out?

   The Golden Mean can also be approximated by any two adjacent Fibonacci numbers. The higher you go up the series, the closer the approximation. 144 / 233 = 1.61806 is the first pair that gives three correct decimal places.

1. 
   roninfante | Jan 06, 2003 05:15pm | #5

   Uncle Dunc,

   I just got your message this morning because I take long Christmas breaks. I don't know how to express formula with this editor so what I say will be more wordy than really required.

   First, your remark about the ratios of successive Fibonacci numbers is absolutely correct. The limit of these ratios is the Golden mean. since the convergence is rapid, you can get a good approximation fairly quickly.

   The point I mentioned is that the Golden mean is the root of a quadratic equation, the one you wrote. The quadratic formula gives the roots, one root is positive while the other is negative. the positive root is the Golden ratio (1 + sqt(5))/2. Your calculator will provide you more than enough decimal points for any practical need.

   Ron

1. 
   UncleDunc | Jan 06, 2003 06:30pm | #6

   Thanks, Ron. Quadratic equation was the hint I needed. The first time I flunked algebra it was because I was unable to derive the quadratic formula. :)

1. 
   roninfante | Jan 06, 2003 09:16pm | #7

   Uncle Dunc,

   The secret to deriving the quadratic formula is the "completing the square" trick. I thought it was so neat that I've never forgotten it and that was more than 45 years ago.

   That being said.... there is some question as to the extent the Golden mean has been consciously used by artists and designers. Indeed there is a minor industry devoted to the question. My own feeling is that Golden rectangles look good and if the ancients didn't use them that doesn't change the fact that they are pleasing.

   Ron

2. 
   Leverone | Dec 30, 2002 07:03pm | #2

   Thanks.  Nice to know my memory isn't all shot.