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Encyclopedia>Geometry>Planar Figures>The Golden Ratio>What Is the Golden Ratio?

Let’s explore a special number: the golden ratio. The golden ratio is a particular ratio that appears in many contexts, from nature, to art, and many more.

The golden ratio is perceived as more beautiful than other ratios. We do not know why exactly this is, but people respond positively to this specific ratio.

In art and architecture, you will find multiple examples of the use of the golden ratio. For example, the painting “Mona Lisa” by Leonardo da Vinci contains the golden ratio everywhere.

It is also said that the ratio between the distance from your navel down to your toes, and the distance from your navel up to the top of your head, is similar to the golden ratio. I have measured myself, and I found that the distance from my navel to my toes is a little short for that to be completely true, but with a little leeway I can at least say that it’s similar.

Rule

The Golden Ratio

Mathematically, the golden ratio is given by

ϕ = \frac{1 + \sqrt{5}}{2} = 1.618 \dots,

and you can use this to find where you’d need to split a line segment for it to be divided according to the golden ratio:

\frac{a + b}{a} = \frac{a}{b} = ϕ

Example 1

If you have a line segment $A B$ that you want to divide to match the golden ratio, select a point $C$ so that

\frac{A B}{A C} = \frac{A C}{C B}

Line of golden ratio

Example 2

Emma has painted a picture of a woman standing up. From her navel to her feet she measures $5.61 cm$ , while she measures $3.47 cm$ from her navel to the top of her head. Does the proportions of the woman follow the golden ratio?

To answer this question, we need to find the ratio between the two lengths. You find the ratio between the lower part of the body and the upper part of the body by dividing the distance from the navel to the feet by the distance from the navel to the top of the head.

\begin{array}{l} Ratio & = \frac{5.61 cm}{3.47 cm} \\ \approx 1.617 \end{array}

\begin{array}{l} Ratio & = \frac{Distance from navel to feet}{Distance from navel to top of head} \\ = \frac{5.61 cm}{3.47 cm} \\ \approx 1.617 \end{array}

This ratio is very close to the golden ratio, which is approximately

1.618

. That means it’s reasonable to conclude that the proportions on the woman that Emma has painted follow the golden ratio.

Example 3

You want to split a line segment $A B = 10 cm$ into two using the golden ratio. What will be the length of the two parts?

You know that $a + b = 10$ . If you solve this expression for $a$ , you get

a = 10 - b

If you insert this into the formula for the golden ratio, you get an equation with one variable. Then, you can solve the equation to find that variable:

\begin{array}{l} \frac{a}{b} & = ϕ \\ \frac{(10 - b)}{b} & \approx 1.618 \\ 10 - b & \approx 1.618 b \\ 10 & \approx 2.618 b \\ \frac{10}{2.618} & \approx b \\ 3.82 & \approx b \end{array}

That makes line segment $b \approx 3.82 cm$ . Line segment $a$ is then

a \approx 10 - 3.82 cm = 6.18 cm .

The line $A B$ is now divided according to the golden ratio.

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What Is the Golden Ratio?

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