What is the golden ratio?

The golden ratio φ = (1 + √5) / 2 ≈ 1.618 is an irrational number that appears throughout mathematics, art, and nature.

What is Binet's formula?

Binet's formula gives the nth Fibonacci number directly: F(n) = (φⁿ − ψⁿ) / √5, where ψ = (1 − √5) / 2.

How is the golden spiral constructed?

Each step places a quarter-circle arc in a square whose side equals the next Fibonacci number, producing the characteristic spiral shape.

Where do Fibonacci numbers appear in Pascal's triangle?

The sums of the shallow diagonals of Pascal's triangle equal successive Fibonacci numbers.

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Fibonacci & Golden Ratio Visualizer

Visualize the Fibonacci sequence, animated golden spiral, and ratio convergence toward φ.

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About this tool

Explore the Fibonacci sequence and its deep connection to the golden ratio φ ≈ 1.618. Watch an animated golden spiral drawn on canvas, examine how successive ratios converge to φ, and discover Fibonacci numbers hidden inside Pascal's triangle. Formulas including Binet's closed form are shown alongside every visualization.

How to use

1 Step 1: The tool displays the first 20 Fibonacci numbers and key formulas automatically.
2 Step 2: Click Animate Spiral to watch the golden spiral draw itself on the canvas.
3 Step 3: Review the ratio table to see F(n+1)/F(n) converging toward φ.
4 Step 4: Scroll down to explore Fibonacci numbers highlighted in Pascal's triangle.

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