rotate:0.03angle°:137.537seeds:500.00preset:
1/φ continued fraction: [00; 2; 1; 1; 1; 1]
🌻 Phyllotaxis — Why Sunflowers Are Mathematical
The one-line takeaway: sunflowers grow in Fibonacci spirals because each
new seed is placed exactly one golden angle (137.507°) around from the last,
and that angle is the single most "irrational" way to never overlap.
HOW SEEDS ARE PLACED (Vogel's model, 1979):
For seed n = 0, 1, 2, 3, …
θ = n × φ (φ is the divergence angle — the one knob)
r = c × √n (√n keeps seed AREA-density constant)
x = r · cos(θ)
y = r · sin(θ)
That is the entire algorithm. There is no "loop count the spirals" logic.
The double-spiral structure you see is a side-effect of the angle, not
something programmed in.
WHY THE GOLDEN ANGLE SPECIFICALLY?
The golden ratio φ ≈ 1.6180339... is the "most irrational" number.
Its continued-fraction expansion is [1; 1, 1, 1, …] — all ones, forever.
That means rational approximations converge to it slower than to any other
number. In other words: no simple fraction p/q is ever close to φ.
If you used a rational angle (1/2 turn, 1/3 turn), every seed in the
same angular "slot" would stack radially, leaving gaps. Wasteful, ugly.
The golden angle (1/φ × 360°) is so irrational that seeds never line up
radially, giving perfect, gap-free packing. Nature didn't "choose"
Fibonacci — Fibonacci is just what you count when you look at the closest-
approach spirals of a perfectly-packed golden-angle lattice.
THE FIBONACCI CONNECTION
Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…) are the
denominators of the best rational approximations to 1/φ. When you count
clockwise and counter-clockwise spiral arms in a sunflower, you always get
consecutive Fibonacci numbers because those are the "almost-fits" of the
golden angle — angles at which seeds nearly line up, forming the visual ridges.
CONTROLS
• Divergence angle — the hero slider. At 137.507° the head crystallizes.
Drag away and watch it shatter.
• "Snap to Golden" button — eases the slider home and crystallizes.
• Seeds — grow from a single seed to 1 000.
• Color — toggle gradient (by generation) vs. monochrome.
• Spirals — overlay that traces and counts the two spiral families
(parastichies), so the Fibonacci pair is shown, not just claimed.
• Presets — 1/2 turn (180°), 1/3 turn (120°), Golden (137.507°),
√2 turn (222.49°), Silver (151.14°).
• Auto / Manual — auto mode slowly sweeps the angle back and forth
while rotating, showing the knife-edge between packing and chaos.
• Rotate speed — how fast the whole pattern spins.