As n increases, F(n+1)/F(n) approaches φ
F(n) = (φⁿ - ψⁿ) / √5
Where ψ = (1 - √5) / 2 ≈ -0.618. This closed-form expression allows direct calculation of any Fibonacci number without computing all preceding terms.
The Fibonacci sequence and golden ratio appear throughout the natural world, from the microscopic to the cosmic scale. These patterns emerge because they represent optimal packing and growth strategies.
The chambered nautilus grows in a logarithmic spiral, adding chambers in golden ratio proportions.
Seeds arrange in 34 and 55 spirals (consecutive Fibonacci numbers) for optimal packing.
Scales spiral in 8 and 13 directions, maximizing exposure to light and rain.
Spiral galaxies often display logarithmic spiral arms following golden proportions.
Phyllotaxis: leaves spiral around stems at 137.5° (golden angle) for optimal sunlight.
Male bees have 1 parent, females have 2—creating a Fibonacci family tree.
Many flowers have Fibonacci petal counts: lilies (3), roses (5), daisies (34, 55, 89).
DNA's double helix measures 34Å long by 21Å wide—a Fibonacci ratio.
A golden rectangle can be subdivided infinitely, each time producing a smaller golden rectangle and a square—demonstrating perfect self-similarity.