Golden ratio (idea)

I believe the geometric proportion served the Creator as an idea when He introduced the continuous generation of similar objects from similar objects. -- Johannes Kepler

The Golden Ratio can be obtained easily by the proper division of a line segment, instead of constructing the geometrically pleasing rectangles. Simply divide the line segment into two parts of uneven lengths, so that the ratio of the smaller part to the larger part equals the ratio of the larger part to the entire line segment.

A picture is worth a thousand words:

       1         x
   •-------•-----------•
   A       B           C

         AB   BC         1     x
         -- = --    or:  - =  ---
         BC   AC         x    1+x


         x2 - x - 1 = 0

         x = (1 ± √5)/2