Fibonacci Explorer

Explore the Fibonacci sequence, golden ratio, Binet’s formula, and visual spiral patterns

Overview

The Fibonacci Explorer generates the famous sequence where each number is the sum of the previous two (0, 1, 1, 2, 3, 5, 8, 13…). Discover how consecutive Fibonacci ratios converge to the golden ratio (φ ≈ 1.618), visualize the beautiful spiral pattern, and use Binet’s formula to calculate any Fibonacci number directly.

Tips

  1. Watch the golden ratio emerge: Generate at least 15-20 numbers and calculate the ratio between consecutive terms to see how it converges to approximately 1.618.

  2. Use Binet’s formula for large numbers: Instead of calculating all previous terms, jump directly to any Fibonacci number using the closed-form formula F(n) = (φⁿ - ψⁿ) / √5.

  3. Visualize the spiral: Create the Fibonacci spiral by drawing squares with Fibonacci side lengths and connecting opposite corners with quarter-circle arcs to see the natural growth pattern.

  4. Spot divisibility patterns: Notice that every 3rd Fibonacci number is even, every 4th is divisible by 3, and every 5th is divisible by 5.

  5. Verify mathematical identities: Test Cassini’s identity F(n-1) × F(n+1) - F(n)² = (-1)ⁿ with different values to see this elegant relationship in action.