The Fibonacci Sequence and the Golden Spiral

The Fibonacci sequence

Each number is the sum of the two before it:

$F(1) = 1, \quad F(2) = 1, \quad F(n) = F(n-1) + F(n-2) \text{ for } n \geq 3$

Step-by-step: Generate the first 15 terms

$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610$

How each is computed:

$F(3) = F(2) + F(1) = 1 + 1 = 2$

$F(4) = F(3) + F(2) = 2 + 1 = 3$

$F(5) = F(4) + F(3) = 3 + 2 = 5$

$F(6) = 5 + 3 = 8$

$F(7) = 8 + 5 = 13$

...continuing to $F(15) = 377 + 233 = 610$.

The golden ratio

The ratio of consecutive Fibonacci numbers converges to the golden ratio:

$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339..$

$F(14)/F(13) = 377/233 = 1.61803..$ — already very close to $\varphi$!

The golden spiral

Draw squares with side lengths $1, 1, 2, 3, 5, 8, 13, \ldots$ arranged in a spiral pattern. Draw a quarter-circle arc in each square. The resulting curve is the Fibonacci spiral, which approximates the golden spiral — a logarithmic spiral found throughout nature.

Fibonacci in nature

• Flower petals: Lilies have 3, buttercups 5, daisies 13 or 21, sunflowers 34 or 55.
• Pine cones and pineapples: Spirals count in Fibonacci numbers (8 and 13, or 5 and 8).
• Sunflower seed heads: Seeds arrange in opposing spirals of 34 and 55 (or 55 and 89).
• Nautilus shells: The growth pattern approximates a golden spiral.

Binet's formula (closed form)

$F(n) = \frac{\varphi^n - \psi^n}{\sqrt{5}}, \quad \text{where } \varphi = \frac{1+\sqrt{5}}{2}, \; \psi = \frac{1-\sqrt{5}}{2}$

This gives the exact $n$th Fibonacci number without computing all previous ones.

A surprising property

The GCD of two Fibonacci numbers equals a Fibonacci number: $\gcd(F(m), F(n)) = F(\gcd(m, n))$.

Try it in the visualization

Watch the sequence build term by term. Fibonacci squares tile a rectangle, and quarter-circle arcs form the golden spiral. The ratio $F(n)/F(n-1)$ converges to $\varphi$ — shown on a graph approaching the golden ratio line.