Fibonacci Spirals in a Sunflower

A sunflower adds seeds one at a time, each turned by the same angle from the last. At the golden angle, about 137.5°, the seeds pack with no gaps, and the spirals you see come in Fibonacci numbers, because ratios of consecutive Fibonacci numbers close in on the golden ratio.

How it works

The Fibonacci recurrence and the golden ratio

In one sentence

Vogel's model places seed $k$ at polar coordinates $(c\sqrt k,\, k\alpha)$ . With $\alpha = 2\pi/\varphi^2$ , the visible spiral families (parastichies) number consecutive Fibonacci numbers, because the convergents of the continued fraction of $\varphi$ are $F_{n+1}/F_n$ .

General formula

{\htmlData{sym=0}{\theta_k}} = k{\htmlData{sym=2}{\alpha}}, \quad {\htmlData{sym=1}{\rho_k}} = {\htmlData{sym=3}{c}}\sqrt{k}, \qquad {\htmlData{sym=2}{\alpha}} = \frac{360^\circ}{{\htmlData{sym=4}{\varphi}}^2} \approx 137.5^\circ, \qquad \frac{F_{n+1}}{{\htmlData{sym=5}{F_n}}} \to {\htmlData{sym=4}{\varphi}}

Turning by an irrational fraction of a circle never repeats exactly, so each new seed falls into the biggest gap left.

What the symbols mean

$\theta_k$
Angle of seed k
How far around the center the kth seed sits.
$\rho_k$
Distance of seed k
How far from the center the kth seed sits.
$\alpha$
Turning angle
The fixed angle between consecutive seeds.
$c$
Scale
Sets the seed spacing.
$\varphi$
Golden ratio
(1 + √5)/2 ≈ 1.618, the limit of Fibonacci ratios.
$F_n$
Fibonacci number
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

Each new seed appears at the center and is pushed outward, turned by a fixed angle $\alpha$ from the one before. In Vogel's model seed $k$ sits at angle $k\alpha$ and distance $c\sqrt k$ , so every seed gets the same area.
If the angle is a simple fraction of a turn, the seeds line up in straight rays with gaps between them. The golden angle is the 'most irrational' choice, and the seeds fill the disk evenly.
The best fractional approximations of $\varphi$ are ratios of consecutive Fibonacci numbers ( $\frac85, \frac{13}{8}, \frac{21}{13}, \dots$ ), so the visible spiral families come in Fibonacci counts.
That's the Fibonacci recurrence at work: $F_{n+1}/F_n$ converges to $\varphi$ , landing alternately above and below it.

Learn the math

Recurrence RelationsDiscrete · Lesson 21, Sequences and Recurrences Solving Linear RecurrencesDiscrete · Lesson 22, Sequences and Recurrences