Introduction
The golden ratio is a ratio such that, given two quantities a and b, (a+b)/a=a/b We can solve this equation to find an explicit quantity for the ratio. LHS=a/a+b/a=1+b/a 1+b/a=a/b We set the ratio equal to a certain quantity given by r. r≡a/b Then we can solve for the ratio numerically. 1+1/r=r r+1=r^2 We can see from the above result that the golden ratio can also be described as a ratio such that in order to get the square of the ratio, you add one to the ratio. r^2-r-1=0 We can then apply the quadratic formula to solve for the roots of the equation. r=(1±√(1^2-4(1)(-1) ))/2=(1±√5)/2 The positive root is then the golden ratio. (1+√5)/2=1.618…≡ϕ The golden ratio, ϕ, is sometimes also called the golden mean or the golden section. The golden ratio can be frequently observed in man-made objects, though they are generally “imperfectly golden” – that is, the ratio is approximately the golden ratio, but not exactly. Some everyday examples include: credit cards, w/h=1.604, and laptop screens, w/h=1.602 (Tannenbaum 392). Visualizations of the golden ratio can be seen below (Weisstein):
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