THE GOLDEN RATIO

by Nikhat Parveen, UGA

Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of Gold; the second we may name a Precious jewel. Kepler (1571- 1630)

In everyday life, we use the word “proportion” either for the comparative relation between parts of things with respect to size or quantity or when we want to describe a harmonious relationship between different parts. In mathematics, the term “proportion” is used to describe an equality of the type: nine is to three as six is to two. The Golden Ratio provides us with an intriguing mingling, it is claimed to have pleasingly harmonious qualities.

The first clear definition of what has later become known as the Golden Ratio was given around 300 B.C by the founder of geometry as a formalized deductive system , Euclid of Alexandria.

In Euclid’s words:

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

If the ratio of the length AC to that of CB is the same as the ratio of AB to AC, then the line has been cut in extreme and mean ratio, or in a Golden Ratio.

The Golden Ratio is thus the ratio of the larger sub segment to the smaller. If the whole segment has length 1 and the larger sub segment has length x (Figure 2), then

Thus X is a solution of the quadratic equation

X2= 1–X   or   X2+x-1=0

This equation has two solutions

X1= (-1+5) / 2 ≈ 0.618 and X2 = (-1- 5) / 2 ≈ - 1.618

The length X must be positive, so

X = (1+5) / 2 ≈ 1.61803 or Φ (phi)

The American mathematician Mark Barr used the Greek letter Phi (Φ) to represent the golden ratio, using the initial letter of the Greek Phi Dias who used the golden ratio in his sculptures.

Luca Pacioli (also written as Paccioli) wrote a book called De Divina Proportione (The Divine Proportion) in 1509. It contains drawings made by Leonardo da Vinci of the 5 Platonic solids. It was probably Leonardo (da Vinci) who first called it the sectio aurea (Latin for the golden section).

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