Explore the Fibonacci sequence, golden ratio, and mathematical beauty
Watch how the ratio of consecutive Fibonacci numbers converges to φ (phi), the golden ratio.
Calculate Fibonacci numbers directly using Binet's formula: F(n) = (φⁿ - ψⁿ) / √5
Visualize the Fibonacci spiral constructed from Fibonacci-sized squares.
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones.
Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...
Rule: F(n) = F(n-1) + F(n-2), with F(0) = 0, F(1) = 1
The golden ratio, φ (phi), is approximately 1.618033988... It's the limit of the ratios of consecutive Fibonacci numbers.
Mathematical definition: φ = (1 + √5) / 2
Unique property: φ² = φ + 1 and 1/φ = φ - 1
Binet's formula calculates any Fibonacci number directly without computing all previous numbers:
F(n) = (φⁿ - ψⁿ) / √5
where φ = (1 + √5)/2 ≈ 1.618 and ψ = (1 - √5)/2 ≈ -0.618
Despite using irrational numbers, it always produces integers!
Fibonacci numbers and spirals appear throughout nature:
Cassini's identity states: F(n-1) × F(n+1) - F(n)² = (-1)ⁿ
This means the difference alternates between +1 and -1.
Example: For n=5: F(4)×F(6) - F(5)² = 3×8 - 25 = -1 ✓