The Relation of the Golden Ratio and the Fibonacci SeriesDate: 1/28/96 at 22:52:8 From: Anonymous Subject: The connection between the Golden Ratio and the Fibonacci Series I am trying to figure out why the Golden Ratio and the Fibonacci series are related. I have figured out that the ratio of a number in the Fibonacci series over the previous becomes increasingly closer to the golden ratio, but I have no idea how to relate that occurrence to some mathmatical reason or logic. I would appreciate any help. Thanks. Date: 1/29/96 at 11:23:8 From: Doctor Ethan Subject: Re: The connection between the Golden Ratio and the Fibonacci Series Great question, You have done the sticky part. Now let's look at the beautiful part. You may have noticed that if you have a Golden rectangle and you cut off a square with side lengths equal to the length shorter rectangle side, then what remains is another Golden rectangle. This could go on forever. You can just keep cutting off these big squares and getting smaller and smaller Golden rectangles. The idea with the Fibonacci series is to do the same thing in reverse. Here what you do is start with a square (1 by 1), find the longer side, and add a square of that size to the whole thing to form a new rectangle. So when we start with a 1 by 1 square the longest side is one, so we add another square to it. Now we have a 2 by 1 rectangle Then the longest side is 2, so we connect a 2 by 2 square to our 2 by 1 rectangle to get a 3 by 2 rectangle. This continues, and the sides of the rectangle will always be a successive Fibonacci number. Eventually the rectangle will be very close to a Golden Rectangle. I will do a few steps to let you see it in action: ### # # 1 by 1, so we add. 1 by 1 to get... ### ###### # ## # Now it is 2 by 1, so we add 2 by 2 to get...... ###### ###### # ## # ###### ###### # # Now it is 2 by 3, so we add a 3 by 3 to get....... # # # # # # ###### ############### # ## ## # ####### # # ## # # ## # Now it is 3 by 5, so we would add a 5 by 5 square. # ## # # ## # ############### I hope that you can follow this and it helps. -Doctor Ethan, The Math Forum |
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