For Fibonacci sequence, sum of two numbers divided by larger number equals larger number divided by smaller number {golden ratio}| {golden section}. Golden ratio equals 1.618034..., reciprocal equals 0.618033..., and square equals 2.618034....
construction
First, bisect square side. Then draw circle with center at bisection point and radius from bisection point to square far corner. Extend one square side until extension meets circle. Extended side has length 1.618...
geometry
If right triangle has one side equal 1 and hypotenuse equal 1.618..., angle is one radian. Golden rectangle has sides in golden ratio and has central point angle of 58 degrees. Pentagram and decagram have sides in golden ratio. Golden ratio is ratio of rectangle {golden rectangle} sides in logarithmic spirals.
music
In music, ratio 2^0.67 = 1.59 ~ 1.618... is similar to major-sixth/octave = 1.67, octave/major-fourth = 1.6, and minor-seventh/major-second = 1.59. Golden ratio and its inverse can make all music harmonics.
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Date Modified: 2022.0224