Pentagons and The Golden Ratio

        Fibonacci numbers play an important role in regular pentagons.  Pentagons are abundant in nature, for many flowers bloom in pentagons, also the insides of many fruits and vegetables are pentagonal in shape.  For example, the sand dallar, sea urchin, and starfish are all pentagons found in the sea.  The cross section of apples and the hoya blosson are other examples.
 
 



    When the length of the sides of a regular pentagon is a Fibonacci number, the length of the diagonals of the pentagon are the next Fibonacci nuber in sequence.  Furthermore, the diagonals of this pentagon separate each other into two adjacent Fibonacci numbers.
 


    We learned in class that the ratio of the diagonal length to the side length is called "The Golden Ratio."  And if we take the ratio of any two succesive numbers in the Fibonacci Sequence and divide each by the number before it, the following is obtained.....

1/1=1    2/1=2    3/2=1.5    5/3=1.66    8/5=1.6    13/8=1.625    21/13=1.61538.............

Now, if we plot the ratios on a graph we can see that this approaches the Golden Ratio!!!!!  The value of the Golden Ratio is approximately 1.61804 or more exactly (1+sqrt(5)) /2.
 
 






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