The Golden Ratio

Problem: Find point $P$ on segment $AB$ such that AP:PBAB:AP.

Solution:

If we let $r$ be the length of $AP$ and $s$ be the length of $PB$, then the ratio above can be written as

$\displaystyle\frac{r}{s} =\frac{r+s}{r}$

Simplifying the equation we have $r^2 = rs + s^2$ which is equivalent to $r^2-rs - s^2$, a quadratic equation in $r$. So, we solve for r using the quadratic formula with $a = 1$, $b = s$ and $c = -s^2$:

$r = \displaystyle\frac{-b \pm \sqrt{b^2-4ac}}{2a} = \frac{-s \pm \sqrt{s^2-4(1)(-s^2)}}{2(10)}$.

Simplifying this gives us $r = \frac{s + s\sqrt{5}}{2}$. Dividing both sides by s, we have

$\displaystyle\frac{r}{s} = \frac{1 + \sqrt{5}}{2} \approx 1.61803398874989.$

The ratio that given above is called the Golden Ratio.  The Golden Ratio seems to be an ordinary irrational number, but it was extensively used in ancient constructions.

Many ancient mathematicians, especially the Greeks, deemed anything that has a property of a Golden ratio somehow divine. In fact, they have integrated in in their architecture. Many proportions of the Parthenon, for instance, are said to exhibit the Golden Ratio.

The video below shows some facts (and fiction) about the Golden Ratio.

8 thoughts on “The Golden Ratio”

1. I find your timing for this post somewhat humorous. Today, three of my colleagues and I created a lesson which shows the golden ratio in relation to the measure of the forearm (elbow to wrist) and cubit (elbow to tip of middle finger). I enjoy reading your posts and tutorials.

2. Guillermo, you can see some of the things I do with my students relating to the Golden Mean by clicking (the lower left corner) on my A Map to Calculus at http://www.mathman.biz/html/map.html . There are lessons about the ratios of the Fibonacci numbers, which form a n infinite sequence which has a limit , the golden mean, about counting the row on a sunflower head and stalk, about the golden triangle, the golden angle and the sides and angles of a regular pentagon, all related.

Of course, one great website is by Ron Knott, with even more ideas about the Fibonacci numbers and the Golden mean.

Keep up the fine work, Guillermo. By the way, more people this month have come to my website from The Philippines than even the USA.