🐚 Fibonacci Spiral — Golden Ratio Visualizer
Watch the golden angle pack points into a spiral, a sunflower head, and a classic square-and-arc tiling. Discover why the same ratio φ≈1.618034 shows up in shells, pinecones and flower heads.
Watch the golden angle pack points into a spiral, a sunflower head, and a classic square-and-arc tiling. Discover why the same ratio φ≈1.618034 shows up in shells, pinecones and flower heads.
This tool visualizes the Fibonacci spiral, the pattern formed when points are placed at a constant golden angle of 137.5077° apart and pushed outward as the square root of their index. The same golden-angle packing that draws the spiral also fills a disc like a sunflower head, and the classic Fibonacci squares construction shows where quarter-circle arcs approximate the same curve using whole-number side lengths 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
In Spiral and Sunflower mode, point i sits at radius r=c·√i and angle θ=i·137.5077°, the golden angle defined by 360°/φ² where φ=(1+√5)/2≈1.618034. Squares mode instead tiles the plane with squares whose sides are consecutive Fibonacci numbers and connects them with quarter-circle arcs, the classic golden-rectangle construction.
Switch modes with the Spiral / Sunflower / Squares buttons. Sliders control the number of points, the golden angle in degrees, and the dot size; the color-mode menu recolors the seeds and the overlay checkbox draws the connecting spiral or Fibonacci arms. Save PNG exports the current canvas and Reset restores the defaults.
Sunflower heads typically show 34 spiral arms turning one way and 55 the other — both Fibonacci numbers — because packing seeds at the golden angle is the only way to avoid gaps as the flower head keeps growing.
The Fibonacci spiral is the pattern traced when points are placed one after another, each rotated by the golden angle (about 137.5077°) from the previous point and moved slightly further from the center as r=c·√i. It is closely related to, but not identical to, the geometric spiral built from quarter-circle arcs inscribed in squares whose sides follow the Fibonacci sequence.
The golden angle is what you get when you split a full 360° turn in the golden ratio φ: it equals 360°/φ², which works out to approximately 137.5077°. Because this angle is an irrational fraction of a full turn, repeatedly rotating by it never returns exactly to a previous direction, so points spread out as evenly as possible with no two ever falling on the same radial line.
The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…, where each number is the sum of the two before it. As you go further along the sequence, the ratio of consecutive terms gets closer and closer to φ=(1+√5)/2≈1.618034, which is exactly the ratio used to define the golden angle and the golden rectangle.
A growing plant adds new seeds, scales or leaves one at a time near a central growth point, each rotated from the last by a roughly constant angle. Evolution favors the golden angle because it packs new growth as densely as possible without gaps or excessive overlap, which is why sunflower heads, pinecones, pineapples and many succulents all display Fibonacci-numbered spiral counts.
Spiral mode plots points outward without limit using r=c·√i. Sunflower mode uses the same golden-angle math but scales the radius to fill a fixed disc with elliptical seeds, mimicking an actual flower head. Squares mode instead builds the classic geometric construction: squares with Fibonacci-number side lengths tiled in a spiral, each joined by a quarter-circle arc.