Each seed is planted one fixed turn after the last. Drag to orbit · scroll or pinch to zoom.
Any simple fraction p/q of a turn sends every q-th seed to the same bearing, so the seeds collapse onto q straight arms and waste the space between them.
Every seed is measured against its nearest neighbours. The gaps in the index that keep coming up are the spiral families — tap one to trace it.
| p/q of a turn | θ | arms |
|---|
Each row is one Fibonacci number over the next but one. They close in on 1/φ² = 0.381966… — and the limit is the one turn that is not a fraction at all.
| ratio | value | θ = 360°/ratio² |
|---|
Where’s the maths? This is number patterns and ratio — the same Secondary 1 topic as “find the next term” — wearing a very unusual costume. A sunflower builds its head with one rule repeated: put down a seed, turn through a fixed angle θ, push the older seeds outwards, put down the next seed. Seed number k therefore sits at bearing kθ and at a distance c√k from the centre, so the whole head is a sequence written in polar coordinates rather than in a list. Everything then depends on the single number θ. If θ is a simple fraction of a turn, the seeds line up into a few straight arms and most of the head is wasted; if θ is as far from every simple fraction as a number can be, the seeds never line up and the head packs evenly. That “worst-approximated” number is the golden ratio, which is why the spirals you can count on a real sunflower are Fibonacci numbers — 13 and 21, or 21 and 34, or 34 and 55. The spirals are not planted; they are what your eye makes of a lattice built by one turn repeated a few hundred times.
Beyond the syllabus: the golden ratio's irrationality argument is enrichment.
The rule on screen is: seed k at angle kθ, radius c√k. The square root is not decoration. Area grows like radius squared, so if the k-th seed is to have the same room as the first, its distance from the centre must grow like √k. That one choice makes the density even; the angle θ then decides how the seeds are shared out around the ring. In radians the golden angle is 2π(1 − 1/φ) ≈ 2.3999 rad — the same turn measured the Secondary 3/4 way.
Why fractions fail. Suppose θ = p/q of a turn, say 2⁄5 = 144°. After q = 5 seeds you have turned through p whole turns and you are pointing exactly where you started, so seeds 1, 6, 11, 16, … all sit on one bearing. Only 5 bearings are ever used and the head becomes 5 straight arms with empty wedges between them. Tap the trap buttons: the number of arms is always the denominator.
The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … where each term is the sum of the two before it: F(n+1) = F(n) + F(n−1). Divide one term by the one before and the answers zig-zag in on a limit: 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.6154, 34/21 = 1.6190, … converging on the golden ratio φ = (1 + √5)/2 = 1.618034…. Turn that into an angle and you get 360°/φ² = 137.5078°, the golden angle.
Where the spirals come from. Nothing draws them. Seed k and seed k + m land close together whenever mθ is nearly a whole number of turns — and for the golden angle the values of m that do that best are exactly the Fibonacci numbers. Your eye joins each seed to the neighbour m steps away and reads a curve. Two families are usually clearest, one clockwise and one anticlockwise, and their counts are consecutive Fibonacci numbers; the app finds them by measuring, not by assuming. Since F(n) + F(n+1) = F(n+2), the two counts always add up to the next family out.
Best of all the bad approximations. Every number can be squeezed between fractions. φ is the number that resists hardest: its continued fraction is 1 + 1/(1 + 1/(1 + …)), all ones, which converges more slowly than any other. Slow convergence means φ is never close to a simple fraction, so 137.5078° is never close to p/q of a turn, so the seeds never fall into arms. The plant is not doing number theory; plants that packed badly simply grew fewer seeds.
Drag to orbit · scroll or pinch to zoom · drag the Coarse and Fine sliders to change the turn angle.