Fibonacci spiral
Quarter circles drawn in squares of side 1, 1, 2, 3, 5, 8 …: the popular approximation of the golden spiral.
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Formulas and key values
- Construction rule
Draw a quarter circle of radius Fₖ in the k-th square, turning 90° further each time.
- Arc lengthIn the graph ≈ 84.82
- Ratio of successive radii
Fₖ is the k-th Fibonacci number (1, 1, 2, 3, 5, …) and n the number of squares.
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Start the video01 · Definition
What is the Fibonacci spiral?
Lay squares of side 1, 1, 2, 3, 5, 8, … edge to edge, each new one along the long side of the rectangle formed so far, and draw a quarter circle across each square. The arcs join smoothly into a spiral, and the squares always tile a rectangle of sides Fₙ × Fₙ₊₁.
It is often shown as the golden spiral, but it is only an approximation. The ratio of successive radii Fₖ₊₁/Fₖ tends to the golden ratio φ, so the growth per quarter turn approaches φ; yet the curvature jumps at every joint, where a true logarithmic spiral changes smoothly.
02 · Properties
Key properties
- The k-th quarter circle has radius Fₖ, the k-th Fibonacci number.
- n squares tile a rectangle of sides Fₙ × Fₙ₊₁, so F₁² + … + Fₙ² = FₙFₙ₊₁.
- Total length of n arcs: (π/2)(Fₙ₊₂ − 1).
- Fₖ₊₁/Fₖ → φ ≈ 1.618, so it approaches the growth of the golden spiral.
- Unlike the golden spiral, its curvature jumps from 1/Fₖ to 1/Fₖ₊₁ at each joint.
03 · Questions
Fibonacci spiral: common questions
- How is the Fibonacci spiral constructed?
- Construction rule: F_k+1 = F_k + F_k-1, Draw a quarter circle of radius Fₖ in the k-th square, turning 90° further each time.. Fₖ is the k-th Fibonacci number (1, 1, 2, 3, 5, …) and n the number of squares.
- What is the arc length of the Fibonacci spiral?
- Arc length: L = (π/2)(F_n+2 - 1). Fₖ is the k-th Fibonacci number (1, 1, 2, 3, 5, …) and n the number of squares.
04 · Keep exploring