Golden Ratio: Introduction - Rhea

Introduction

The golden ratio is a ratio such that, given two quantities a and b,

\frac{a+b}{a} = \frac{a}{b}

We can solve this equation to find an explicit quantity for the ratio.

LHS = \frac{a}{b} + \frac{b}{a} = 1 + \frac{b}{a}

1 + \frac{b}{a} = \frac{a}{b}

We set the ratio equal to a certain quantity given by r.

r = \frac{a}{b}

Then we can solve for the ratio numerically.

1 + \frac{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 = \frac{1 \pm \sqrt{1^2 - 4(1)(-1)}}{2} = \frac{1 \pm \sqrt{5}}{2}

The positive root is then the golden ratio.

\frac{1 \pm \sqrt{5}}{2} = \phi

The golden ratio, $\phi$ , 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, $\frac{w}{h}=1.604$ , and laptop screens, $\frac{w}{h}=1.602$ (Tannenbaum 392).

Visualizations of the golden ratio can be seen below (Weisstein):

MathIsFun also has an interactive display that can construct a rectangle in the golden ratio given a certain fixed width or length.

Golden Ratio: Introduction - Rhea

Introduction

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