Golden Ratio φ

The Golden Ratio

The topic of φ will be discussed in several posts. This post will show basic definition, derivation and some interesting relationships. The ratio φ is one of the most ubiquitous number, in that it is found in nature, architecture art, and music. It even has found its way into the “junk” sciences and inserted into many science fiction plots, though usually as “techno-babble”. This ratio is old. First reviled to mathematicians more than 2,500 years ago. Attributions have been made to Pythagoras (c.570BC-c.495BC) and Euclid(c. 325–c.265BC). Euclid certainly was not responsible, since the Parthenon, which made extensive architectural use of the ratio and was started in 447BC, under the supervision of the sculptor Pericles. What is the Golden Ratio? It starts with a “what if”.:

Two line segments of length a and b such that the ratio of

(1) Golden Ratio φ

Figure-1-PHI Golden Ratio φ This relationship establishes Golden Ratio φ

(2) Golden Ratio φ

Let b=1 then:

(3) Golden Ratio φ

Substitute \phi  for a:

(4) Golden Ratio φ

Simple algebraic manipulation:

(5) Golden Ratio φ

Solve for \phi:

(6) Golden Ratio φ Golden Ratio φ =1.618033988749895