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The Golden Ratio and the Fibonacci Sequence

December 27th, 2009 by Math Tricks | No Comments | Filed in Fibonacci Sequence, Golden Rectangle, Math Tricks

So you have seen already a few posts about the golden ratio, which is approximately equal to 1.618. Are there math tricks that will allow you to determine the value for the golden ratio?  Well, since the title of this post is “The Golden Ratio and the Fibonacci Sequence”, then you might have guessed that there is a way to determine the golden ratio with the Fibonacci Sequence.

What the heck is the Fibonacci Sequence?  True, I have not written a post on it yet – but you just wait because there is a LOT to say about it later on – a LOT!!  But for now, let me just give you the sequence:

0,1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

Do you see the pattern?  Each value, starting with the second “1” in the sequence, is simply the sum of the preceding two values.  The general formuls for the Fibonacci sequence is:

F(n) = F(n – 1) + F(n – 2)

So how does this interesting sequence of numbers relate to the golden ratio?  Take any value in the sequence and divide it by the preceding value – what do you get?

For example:

34/21 = 1.619

Looks familiar, eh?

Try it again for a pair farther down the sequence:

233/144 = 1.61806

In fact, this manipulation of the Fibonacci series converges to the golden ratio.

Also, you can perform this manipulation using an “out of frame” Fibonacci series – that is, choose any two consecutive numbers, apply the general formula F(n) = F(n – 1) + F(n – 2) to get a new sequence, and then from the new sequence you will be able to determine an approximation for the golden ratio by following the same procedure as outlined above.  For example, starting with 887 and 888, we get the series:

887, 888, 1775, 2663, 4438, 7101, 11539, 18640, 30179 ,48819, 78998, 127817, 206815, 334632

Notice here that you do not get a very good approximation if you divide 888 by 887.  But as you move down the sequence, the value you obtain gets closer and closer to the golden ratio:

334632/206815 = 1.6180258

A superb example of math tricks in nature!

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Golden Rectangle Dimensions

November 12th, 2009 by Math Tricks | No Comments | Filed in Golden Rectangle, Mathematics Concepts

So you have a single line of length X, and you want to extent the line to height Y such that you produce a golden rectangle.  Simple as pi pie!

This problem can be solved with some simple algebra, and it is useful if you wish to draw a golden rectangle given a line of any length.  For instance, you may want to incorporate golden rectangles into some artwork you are working on, or you may wish to crop photographs such that they are framed within a golden rectangle.

So, given a line of any length, you can break the line into two parts:


It is easy to see that:


From this, you can calculate that the length (A) of the sides of the square part of the golden rectangle is:

A = (A + B)/1.618

So just extend the line into a rectangle with base=(A+B) and height=(A+B)/1.618

Using this type of reasoning, if you have a square with a side of length A, and wished to extend the length to A+B such that the A+B is the length of the base of a golden rectangle, you can determine the length of B very easily:

golden ratio

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