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🌀 Golden Ratio Spiral

Interactive golden ratio spiral simulator. Watch the golden rectangle subdivide into squares with the true golden spiral or the Fibonacci-sequence approximation, and see the ratio converge live on φ = 1.618.

Geometry2DEasy60 FPS

🌀 What It Demonstrates

The golden ratio φ = (1 + √5) / 2 ≈ 1.6180339887 is the unique positive number satisfying φ = 1 + 1/φ. Starting from a golden rectangle — one whose side lengths are in ratio φ : 1 — you can remove a square from one end and the leftover rectangle is itself a smaller golden rectangle, similar to the original. Repeating this forever produces a nested sequence of squares, and drawing a quarter-circle arc through each square traces an approximation of the true golden spiral, a logarithmic spiral with polar equation r = a·φ^2θ/π that grows by a factor of φ every quarter turn.

The golden ratio is closely tied to the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, …), where each number is the sum of the two before it: the ratio of consecutive Fibonacci numbers oscillates around φ and converges to it as the sequence grows. This simulator lets you compare the exact golden spiral against a Fibonacci-sequence approximation built from integer square sizes, which only approaches the true spiral in the limit.

How to Use

Drag the Squares slider (2–15) to change how many nested squares are drawn.
Toggle Fibonacci approximation to switch between the exact golden-ratio subdivision and integer Fibonacci-sequence square sizes.
Toggle Show squares and Show ratio labels for a cleaner or more annotated view, and Animate growth to watch the squares and spiral build up one at a time.
Watch the Statistics panel show the ratio of the latest two square sizes converging toward φ = 1.6180339887…

Did You Know?

The golden ratio's presence in nature and art is often exaggerated. Claims that the Parthenon, the Great Pyramid, or the "ideal" human body were deliberately designed around φ are mostly modern retrofits with no solid historical evidence, and many "golden spiral" overlays placed on seashells, hurricanes or galaxies in popular images are rough approximations chosen after the fact rather than measured, precise fits. What is genuine: φ appears exactly in the geometry of the regular pentagon and pentagram (the ratio of diagonal to side is φ), in certain phyllotaxis (leaf and seed arrangement) patterns approximated by Fibonacci-like spirals in sunflowers and pinecones, and, of course, in the pure algebra of the Fibonacci sequence itself.

About this simulation

This simulator draws the classic golden rectangle subdivision — a rectangle repeatedly split into a square and a smaller similar rectangle — together with a spiral drawn through the squares. You can compare the mathematically exact golden spiral, generated directly from powers of φ, against a Fibonacci-sequence approximation built from integer square sizes 1, 1, 2, 3, 5, 8, 13…, which only converges to the true spiral as the sequence grows.

🔬 What it shows

Each square's side length is either an exact power of φ (true golden spiral mode) or the next Fibonacci number (Fibonacci approximation mode). A quarter-circle arc is drawn inside each square, and consecutive arcs join into a continuous spiral. The stats panel reports the ratio of the two most recent square sizes, which is exactly φ in golden mode and only approaches φ in Fibonacci mode.

🎮 How to use

Set the Squares slider (2–15) for how many nested squares to draw. Toggle Fibonacci approximation to switch square-sizing modes. Toggle Show squares and Show ratio labels for annotation detail, and enable Animate growth to watch the construction build up step by step.

💡 Did you know?

Many popular claims about the golden ratio appearing deliberately in famous artworks, buildings or "ideal" body proportions are not well supported historically — these are frequently modern pattern-matching after the fact. The genuine, provable appearances of φ are mathematical: in the regular pentagon's diagonal-to-side ratio, in continued fractions, and in the limiting ratio of consecutive Fibonacci numbers.

Frequently asked questions

What exactly is the golden ratio?

The golden ratio φ is the positive solution to the equation φ = 1 + 1/φ, equivalently φ² = φ + 1. Solving this quadratic gives φ = (1 + √5) / 2, an irrational number approximately equal to 1.6180339887.

How is the golden spiral related to the golden rectangle?

Removing the largest possible square from a golden rectangle always leaves a smaller rectangle with the same aspect ratio φ:1. Repeating this indefinitely produces an infinite sequence of shrinking, rotated squares, and a smooth quarter-circle arc drawn through each one approximates a true logarithmic golden spiral.

What is the difference between the true golden spiral and the Fibonacci spiral shown here?

The true golden spiral scales by exactly φ every quarter turn, using continuous powers of φ for square sizes. The Fibonacci approximation instead uses the integer Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) for square sizes; the ratio of consecutive Fibonacci numbers only approaches φ as the sequence grows, so early squares in Fibonacci mode are visibly less golden than the true spiral.

Does the golden ratio really appear throughout nature and art as often as claimed?

Less often than commonly claimed. While Fibonacci-like spiral patterns genuinely appear in phyllotaxis (such as sunflower seed heads and pinecone scales) as a consequence of optimal packing growth angles, many popular claims about deliberate golden-ratio design in specific historical buildings, paintings or body proportions lack solid documentary evidence and are largely modern retrofits.

Why does the ratio of consecutive Fibonacci numbers converge to φ?

Dividing the Fibonacci recurrence F(n) = F(n-1) + F(n-2) by F(n-1) gives F(n)/F(n-1) = 1 + F(n-2)/F(n-1). As n grows, this ratio settles into a fixed point r satisfying r = 1 + 1/r, which is exactly the defining equation of the golden ratio.

⚙ Under the hood

Interactive golden ratio spiral simulator. Watch the golden rectangle subdivide into whirling squares with the true golden spiral or a Fibonacci-sequence approximation, and see the ratio of consecutive squares converge live on phi = 1.6180339887.

Canvas 2DGolden RatioFibonacci SequenceLogarithmic SpiralPhi

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install