The golden section search

Given a triplet $(a,b,c)$ that brackets the minimum, we choose a new point x that defines a new bracketing triplet $(a,x,b)$ or $(b,x,c)$ according to the rule:

$$\frac{x-b}{c-a} = 1 - 2\frac{b-a}{c-a}$$

This implies that $\vert b-a\vert=\vert x-c\vert$, and that at each iteration the interval is scaled of the same ratio $\lambda$.
Then we repeat the process with the new triplet. So the interval $(a,c)$ is divided in two parts, a smaller and a larger, and the ratio between the whole interval and the larger is the same between the larger and the smaller, or in other words:

$$\frac{1}{\lambda} = \frac{\lambda}{1-\lambda},$$

giving for $\lambda$ the positive solution

$$\lambda = \frac{\sqrt{5}-1}{2}.$$

This fraction is known as the golden-mean or golden-section, whose aesthetic properties come from ancient Pythagoreans.