The Fibonacci Spiral: Golden Ratio, Golden Angle and Why Sunflowers Count in Fibonacci Numbers

A spiral built from squares, and a spiral built from an angle

There are two related but distinct constructions people call "the Fibonacci spiral," and this simulation shows both. The first: draw squares of side length equal to successive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, …) tiled edge-to-edge into an ever-growing rectangle, then draw a quarter-circle arc through each square. The result is a discrete, quantised spiral that approximates — but is not identical to — the true golden spiral, a logarithmic spiral that grows by exactly the golden ratio φ = (1+√5)/2 ≈ 1.618034 every quarter turn.

F(0)=0, F(1)=1, F(n) = F(n-1) + F(n-2)
F(n)/F(n-1) → φ = (1 + √5)/2 ≈ 1.618034...   as n → ∞

golden spiral (continuous): r(θ) = a · φ^(2θ/π)   — grows by φ every 90° turn

The Fibonacci ratio F(n)/F(n−1) converges to φ, but for small n the squares-based spiral's curvature visibly jumps at each new square, and it only asymptotically approaches the smooth logarithmic golden spiral. This distinction matters because the golden spiral itself — a self-similar curve that looks identical at every scale, satisfying r(θ) = a·φ^(2θ/π) — is what actually shows up in nature's approximate spirals (nautilus shells, spiral galaxies, low-pressure weather systems), and it is only loosely related to Fibonacci numbers specifically; any logarithmic spiral with growth factor φ per quarter-turn qualifies, Fibonacci or not.

The golden angle and phyllotaxis

Switch to phyllotaxis mode and the picture changes entirely: instead of arcs through squares, points are placed one at a time, each rotated from the last by the golden angle, ≈137.5° — the angle that divides a full circle in the golden ratio (360°/φ² ≈ 137.5077…°). Placing point n at radius √n and angle n × 137.5° produces the same interleaved spiral-arm pattern visible in a sunflower's seed head, a pine cone's scales, or a pineapple's eyes.

for (let n = 0; n < N; n++) {
  const theta = n * goldenAngle;      // ≈137.50776°, in radians
  const r = c * Math.sqrt(n);         // even area per point
  x = r * Math.cos(theta);
  y = r * Math.sin(theta);
}

Why 137.5° specifically — the "most irrational" angle

The golden angle is the choice that packs points as evenly as possible with no two points ever landing in exactly the same radial direction, no matter how many points you add. The reason is that φ is, in a precise continued-fraction sense, the hardest real number to approximate by a rational — its continued fraction is [1; 1, 1, 1, 1, …], all 1s forever, which makes its rational approximations converge more slowly (worse) than for any other irrational number. If a plant instead spaced new leaves or seeds by a rational fraction of a turn, like exactly 1/3 or 3/8 of a circle, radial gaps and clumps would appear as spokes; by using the number that resists rational approximation hardest, growth avoids ever lining up two elements along the same ray, which packs the maximum number of seeds into a given area with even coverage.

Counting the spiral arms: Fibonacci numbers again

Looking at a real (or simulated) golden-angle point cloud, the eye picks out two interleaved families of spiral arms — one curving clockwise, one counterclockwise — and counting them almost always gives two consecutive Fibonacci numbers: commonly 21 and 34, or 34 and 55 in a large sunflower head, matching decades of botanical surveys (a large literature going back to the 19th century tallies these counts across thousands of real sunflowers and pine cones). This is a direct consequence of how well specific ratios like F(n)/F(n+1) approximate the golden angle: the eye locks onto whichever spiral count corresponds to the best nearby rational approximation of φ, which is always a ratio of consecutive Fibonacci numbers because of how the golden ratio's continued fraction works.

Is this pattern universal in plants, or overstated?

It's real but not literally universal — botanists have documented Fibonacci-based (and occasionally non-Fibonacci, like Lucas-number-based) phyllotactic patterns across a majority of surveyed plant species, and the mechanism proposed by later researchers (notably work building on Douady and Couder's 1990s physical and computational models) shows that a simple local rule — each new primordium (growth bud) appearing at the least crowded available angular position relative to its immediate predecessors — spontaneously produces close to the golden angle without the plant "knowing" φ at all. It's an emergent optimum from local competition for space, not a hard-coded mathematical constant, which is also why occasional real plants deviate from it.

Frequently asked questions

Try it live

Everything above runs in your browser — open Fibonacci Spiral and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.