Fibonacci spirals.

Step 1 of 5

Spokes

In the Fibonacci sequence each term is the sum of the two before: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$⁠ These numbers turn up in a sunflower’s spirals, and one simple rule links them. In a sunflower head each new seed grows at the centre, turns a fixed angle from the seed before it, and is pushed outwards as newer seeds follow.

Turn between seeds is locked at $\frac13$⁠ of a turn, $120^\circ$⁠: every third seed points the same way again, and the seeds fall into $3$⁠ straight spokes with empty gaps between them. Predict what happens at $\frac38$⁠ of a turn.

Step 2 of 5

Fibonacci’s fractions

When the turn is $\frac{p}{q}$⁠ of a turn, the seeds fall into $q$⁠ spokes, with empty space between them: such a plant wastes room. To use all the space the turn must be far from every simple fraction. The fractions $\frac12, \frac13, \frac25, \frac38, \frac{5}{13}, \frac{8}{21}, \dots$⁠, whose numerators and denominators are Fibonacci numbers, close in on exactly such a number. Find the turn for the next fraction in the row, $\frac{5}{13}$⁠.

Step 3 of 5

The golden angle

These fractions close in on $1 - \frac{1}{\varphi}$⁠, where $\varphi = \frac{1 + \sqrt5}{2} \approx 1.618$$\varphi =$$\frac{1 + \sqrt5}{2} \approx$$1.618$⁠ is the golden ratio, which the ratio of consecutive Fibonacci numbers also closes in on: $\frac{55}{34} \approx 1.618$$\frac{55}{34} \approx$$1.618$⁠. That turn, 137.5°, is the golden angle. It is the number hardest to approximate by a simple fraction, so no seed ever lands exactly behind another, and the seeds fill the head evenly. Work out what fraction of a turn the golden angle is.

Step 4 of 5

Counting spirals

At the golden angle there are no spokes, but the eye finds spirals: each seed sits close to the seeds a Fibonacci number of steps before it, because after $34$⁠ or $55$⁠ turns of the golden angle a seed points almost the same way again. Choose On under Show the spirals and count.

Step 5 of 5

A pine cone

The same rule works on a pine cone: the scales grow along an axis, each one turned by the golden angle from the last. A pine cone has far fewer scales than a sunflower has seeds. Predict how many spirals you will see, then switch Plant.