Fibonacci Numbers

The Fibonacci sequence is one of the most famous and fascinating number sequences in mathematics. It begins with 0 and 1, and each subsequent number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... This simple rule generates a sequence with extraordinary properties that appears in the most unexpected places in nature, art and science.

Origin of the sequence

The sequence is named after the Italian mathematician Leonardo of Pisa, known as Fibonacci, who presented it in his book Liber Abaci (1202) through a famous problem about rabbit reproduction. However, this sequence was already known in India centuries earlier by mathematicians such as Pingala (200 BC) and Virahanka (700 AD), who studied it in the context of Sanskrit poetic metre.

The golden ratio (φ)

One of the most remarkable properties of the sequence is its relationship with the golden ratio (phi, φ ≈ 1.6180339...). When dividing each Fibonacci number by its predecessor, the result converges towards φ. This irrational number appears in geometry, architecture, Renaissance art and is considered a symbol of harmony and beauty. The golden rectangle, whose side ratio is φ, was used by the Greeks in the design of the Parthenon and by artists such as Leonardo da Vinci.

Fibonacci in nature

The presence of Fibonacci numbers in nature is astonishing. Sunflower spirals typically have 34 and 55 spirals (both Fibonacci numbers). Pine cones display spirals in quantities that are consecutive Fibonacci numbers. Flower petals frequently follow this sequence: lilies have 3 petals, buttercups 5, daisies 34 or 55. Even the arrangement of leaves on stems (phyllotaxis) follows Fibonacci patterns to maximise exposure to sunlight.

Mathematical properties

The Fibonacci sequence has remarkable mathematical properties. Binet's formula allows any Fibonacci number to be calculated directly using the golden ratio, without needing to compute all the preceding ones. The sum of the first n Fibonacci numbers is F(n+2) − 1. Every third number is even, every fourth is divisible by 3, and every fifth is divisible by 5. Furthermore, the greatest common divisor of two Fibonacci numbers F(m) and F(n) is F(gcd(m,n)), an elegant property connecting the sequence with number theory.

Modern applications

In computer science, Fibonacci numbers appear in algorithm analysis, data structures such as Fibonacci heaps, and search techniques. In financial markets, Fibonacci retracements are widely used technical analysis tools among traders. In music, composers such as Bartók and Debussy have used Fibonacci proportions in their compositions.

The first 50 Fibonacci numbers

Click on any Fibonacci number to discover all its mathematical properties.