The golden ratio

2020-11-24T15:39:01+02:00

The golden ratio is obtained when a line segment is divided into two parts, [[$ a $]] and [[$ b $]], so that the ratio of the shorter part to the longer part is the same as the ratio of the longer part to the whole segment.

[[$ \displaystyle\frac {a} {b} = \displaystyle\frac {b} {a + b} $]]

In general, the inverse of the ratio is used as the numerical value of the golden ratio [[$ \displaystyle\frac {b} {a} = 1,618034… $]] .

If a rectangle is divided into two parts by a line that is parallel to the side of the rectangle, a similar rectangle with the original rectangle is usually not formed. However, a rectangle is said to be a golden rectangle if it can be divided into a square and a smaller rectangle so that the smaller rectangle is similar to the original rectangle.

The rectangles [[$ ABCD $]] and [[$ BCFE $]] are similar if the lengths of their sides are in the golden ratio.

[[$ \displaystyle\frac {AB} {BC} ≈ 1,618034… $]].

Example 1

The end of the Parthenon in Athens is a golden rectangle. The height of the end is [[$ 19 $]] m. How wide is the end?

Solution:

The ratio of the length of the longer side of the golden rectangle to the shorter side is [[$ 1,618034… $]]. Let's signify the width of the temple with [[$ x $]] and solve the equation.

[[$ \quad \begin{align} \frac {x} {19 \text{ m}} &= 1,618034... \space ||·19 \text{ m} \\ \\ x &= 19 \text{ m} · 1,618034... \\ \\ x &\approx 31 \text{ m} \end{align} $]]

Answer: The width of the temple is about 31 m.

4. The golden ratio

The golden ratio

Example 1

Exercises