1016 words - 4 pages

The golden ratio is approximately 1.6. The mysteriously aesthetically pleasing ratio is the relation "a+b is to a as a is to b" or a+b/b=a/b=φ. It is denoted as the Greek letter φ or Φ (lower case, upper case respectively, upper case most often used as reciprocal). The letter is pronounced "phi". The golden ratio is found prominently in art, nature, and architecture. Throughout the centuries countless mathematicians have spent countless hours with the golden ratio and all its applications. It can be found in the great pyramid of Giza, the Parthenon and the Mona Lisa. It is prominent in human and animal anatomy, it can be found in the structure of plants, and even the DNA molecule exemplifies the ratio 1.6. The golden ratio also has applications in other mathematical equations such as logarithmic spirals and the Fibonacci numbers.Before we can begin to discuss the application of the golden ratio we must examine how we translate "a+b is to a as a is to b" into the real, usable number 1.6. Phi is an irrational number, so it's impossible to calculate exactly, but we can calculate a close approximation. As previously stated, the basic equation for phi is a+b/a=a/b=φ. So if a/b=φ, then a=bφ. Now returning to our previous equation, a+b/a=φ, we can substitute a for bφ. After substituting we have bφ+b/bφ=bφ/b. Dividing out by b gives us φ+1/φ=φ. Rearranging yields the quadratic equation φ2-φ-1=0. Therefore via previous knowledge of the general form of a quadratic equation (ax2+bx+c=0) we can extrapolate the following values for our phi equation: a=1, b=-1, c=-1. Substitute these numbers in the quadratic function: x=[-b+/-√(b2-4ac)]/2a and you get φ=[1+/-√5]/2. This allows us to find the roots of the equation; φ=1.618 033 989 (commonly stated 1.6) and φ=-0.618 033 989 ("τ" related to Fibonacci numbers). The positive root is considered the golden ratio and the negative considered the negative reciprocal. The golden ratio is best illustrated by the golden rectangle. To create a golden rectangle, one begins with a unit square. The length of the line from the midpoint of one side of the square to an opposing corner is the radius of a circle centered at the midpoint of the side used. By continuing the arc one can define the height of the golden rectangle. You can see that φ=(√5)/2+1/2, or φ=[1+√5]/2; the golden ratio.Now that we've established what the golden ratio is and how to calculate it, we can look at the many different applications found in the real world. Architects have used the golden ratio in buildings for thousands of years. The earliest known example of the golden ratio being used in architecture is the great pyramid of Giza (built c. 2800 B.C.). If you take the cross section of the great pyramid you get a right angle triangle. This triangle is commonly known as the Kepler triangle, or The Triangle of Price. It is special because it...

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