One number, hiding everywhere

Take a line and cut it in two, a longer piece and a shorter piece, so that the whole line is to the longer piece exactly as the longer piece is to the shorter. There is only one way to do it, and the ratio you get has a name that has echoed through geometry, art and biology for two and a half thousand years: the golden ratio, written \varphi (the Greek letter "phi").

Its value is a specific irrational number,

In this lesson we'll pin down where that value comes from, meet the one tidy equation that \varphi obeys, and watch it turn up in a self-copying rectangle, inside the humble regular pentagon, and at the far end of the Fibonacci sequence. That same five-fold structure is exactly what will later power Penrose tilings.

The defining cut, and the equation it forces

Call the longer piece 1 and the shorter piece x, so the whole line is 1 + x. The golden condition — whole is to longer as longer is to shorter — is

Cross-multiplying gives x(1 + x) = 1, i.e. x^2 + x - 1 = 0. It is neater, though, to name the ratio itself. If we set \varphi = 1/x (the long-to-short ratio), the very same condition becomes the cleanest quadratic in mathematics:

In words: squaring \varphi is the same as just adding one to it. That single identity is the real definition of the golden ratio — everything else in this lesson is a consequence of it. Rearranged the other way, dividing by \varphi gives an equally striking twin:

So \varphi is the one number whose reciprocal is itself minus one — its decimal tail 0.618\dots is literally \varphi with the 1 knocked off the front.

Worked example 1 — solve the equation, then check it

Where does (1+\sqrt5)/2 actually come from? Solve x^2 = x + 1, i.e. x^2 - x - 1 = 0, with the quadratic formula (a = 1,\ b = -1,\ c = -1):

The two roots are \tfrac{1+\sqrt5}{2} \approx 1.618 and \tfrac{1-\sqrt5}{2} \approx -0.618. A ratio of lengths must be positive, so \varphi = \tfrac{1+\sqrt5}{2}. (The negative root is its own curiosity — it equals -1/\varphi.)

Now verify the identity numerically, which is the fastest way to feel that it is true. With \varphi \approx 1.618:

They match — squaring \varphi and adding one to \varphi land on the same 2.618\dots Notice the decimals never change: \dots 618 in \varphi, \varphi^2 and 1/\varphi alike.

Worked example 2 — Fibonacci ratios home in on φ

Take the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots (each the sum of the previous two, first studied in Europe by Fibonacci) and divide each one by the term before it:

The ratios bounce up and down but the swings shrink, closing in on 1.6180339887\dots = \varphi. There's a clean reason. If the ratio settles to some value r, then because each term is the sum of the two before it, r must satisfy

That's \varphi's equation again — so the only positive limit the ratios can approach is \varphi itself. The Fibonacci numbers don't "contain" the golden ratio by coincidence; they are forced towards it by the same x^2 = x + 1.

Worked example 3 — φ inside the pentagon

Draw a regular pentagon and one of its diagonals (a line joining two non-adjacent corners). Measure the diagonal, measure a side, and divide:

exactly, for every regular pentagon. Draw all five diagonals and they trace a five-pointed star, the pentagram — and it is riddled with \varphi. Every diagonal is cut by the others into pieces whose lengths are in golden ratio, and the small inner pentagon the diagonals enclose is a shrunk copy of the outer one, smaller by a factor of \varphi^2. So a pentagram is a golden-ratio machine: five-fold symmetry and \varphi are two faces of the same thing. Hold onto that — it is precisely why Penrose tilings, built on five-fold shapes, are steeped in \varphi.

The golden rectangle that copies itself

A golden rectangle has its long and short sides in the ratio \varphi : 1. It has a magical property that follows straight from 1/\varphi = \varphi - 1: slice off the largest square you can, and what remains is another golden rectangle — the same shape, just smaller. That leftover is called the gnomon. You can peel a square off that one too, and off the next, forever — the rectangle contains a shrinking copy of itself all the way down. Trace a quarter-circle through each square and you get the famous golden spiral. Step through it.

The simplest continued fraction there is

The identity \varphi = 1 + \tfrac{1}{\varphi} can be fed into itself. Replace the \varphi on the bottom with 1 + \tfrac{1}{\varphi} again, and again, forever:

Every number in that tower is a 1. Written in the shorthand of continued fractions, \varphi = [1; 1, 1, 1, \dots] — all ones, the simplest continued fraction any number can have. Cutting it off early gives exactly the Fibonacci ratios \tfrac11, \tfrac21, \tfrac32, \tfrac53, \tfrac85, \dots from Worked example 2.

Here is the surprising payoff. Big numbers in a continued fraction mean a number is easy to approximate by a simple fraction (that's why \pi \approx \tfrac{22}{7} is so good — a 7 shows up early in \pi's expansion). Because \varphi's are all the smallest possible value, 1, it is the number that fractions approximate worst. That earns \varphi the title of the "most irrational" number — the hardest of all to pin down with a ratio.