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-1742bad0ada4c7f514e86bd4e74531db_l3 Golden Ratio φ

Figure-1-PHI Golden Ratio φ This relationship establishes quicklatex.com-b2ca1fc843398c7dffa6a4c19378d223_l3 Golden Ratio φ

(2)   quicklatex.com-168fe8208998d6206fdf63c17cda289c_l3 Golden Ratio φ

Let b=1 then:

(3)   quicklatex.com-c42b1aef8e8664f67e30f8c1bb1d15b6_l3 Golden Ratio φ

Substitute \phi  for a:

(4)   quicklatex.com-903235a992475592144cc9a76b32812d_l3 Golden Ratio φ

Simple algebraic manipulation:

(5)   quicklatex.com-1ef993d31a255315ff2bb3e067b141ef_l3 Golden Ratio φ

Solve for \phi:

(6)   quicklatex.com-aeda3fb6293b07198c1d9bb4f9c567d9_l3 Golden Ratio φ

quicklatex.com-b2ca1fc843398c7dffa6a4c19378d223_l3 Golden Ratio φ =1.618033988749895