Take a line segment and cut it into two pieces, one longer than the other. There's exactly one way to cut it so that the ratio of the whole segment to the longer piece equals the ratio of the longer piece to the shorter piece. That specific ratio is the golden ratio, and it shows up so often in math, nature, and art that it earned its own Greek letter: phi ($\varphi$).

Euclid, writing around 300 BCE, called this the division of a line into extreme and mean ratio — a clunky name for a simple idea. Say the whole segment has length $a+b$, where $a$ is the longer piece and $b$ is the shorter piece. The condition is:

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

In words: the whole is to the long part as the long part is to the short part. Notice this is a self-referential setup — the ratio of the pieces has to match the ratio of the whole to a piece. That self-similarity is the whole point, and it's why phi keeps reappearing throughout this book, from spirals to pentagons.

To find the actual number, let's give the ratio itself a name. Let $\varphi =a/b$ (this is the number we're solving for). Divide both the numerator and denominator of the left side of the equation by $b$:

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

$$\frac{\varphi +1}{\varphi }=\varphi$$

Multiply both sides by $\varphi$:

$$\varphi +1={\varphi }^2$$

Rearranged, this is a quadratic equation (an equation where the highest power of the unknown is 2):

$${\varphi }^2-\varphi -1=0$$

Apply the quadratic formula, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, with $a=1$, $b=-1$, $c=-1$:

$$\varphi =\frac{1\pm \sqrt{5}}{2}$$

Since $\varphi$ was defined as a ratio of lengths, it has to be positive, so we take the plus sign:

$$\varphi =\frac{1+\sqrt{5}}{2}\approx 1.6180339887\dots$$

The other root, $\frac{1-\sqrt{5}}{2}\approx -0.618$, isn't thrown away — it turns out to be $-1/\varphi$, and it will matter later when we look at phi's algebraic quirks.