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)   quicklatex.com-4e7412ab7d60febff4bd50314ebd5b8e_l3 Golden Ratio φ

Figure-1-PHI Golden Ratio φ This relationship establishes quicklatex.com-770335440e88fda366377dc488cc5391_l3 Golden Ratio φ

(2)   quicklatex.com-cb59009bf6d7ada2ca2babd766fd4ddd_l3 Golden Ratio φ

Let b=1 then:

(3)   quicklatex.com-f9ad52dd08b8716c64958d4ee44b7e22_l3 Golden Ratio φ

Substitute \phi  for a:

(4)   quicklatex.com-c40d07252ff0f9dd822a362a78d932f1_l3 Golden Ratio φ

Simple algebraic manipulation:

(5)   quicklatex.com-0a0beafbfce567b126209f8108d87a54_l3 Golden Ratio φ

Solve for \phi:

(6)   quicklatex.com-4c42694c6820fa02535db837166403c2_l3 Golden Ratio φ

quicklatex.com-770335440e88fda366377dc488cc5391_l3 Golden Ratio φ =1.618033988749895