Fibonacci Sequence
A seemingly simple sequence: 1, 1, 2, 3, 5, 8... where each term is the sum of the two preceding ones, widely appearing in nature and mathematics, closely related to the Golden Ratio.
Fibonacci Sequence
A Magical Sequence
1, 1, 2, 3, 5, 8, 13... Why does this seemingly simple sequence appear repeatedly in the breeding of rabbits, the spirals of sunflowers, the shells of nautiluses, and even the fluctuations of the stock market? This is the famous Fibonacci Sequence.
Its definition is simple: starting from the third term, each term is the sum of the two preceding ones. This sequence, proposed by the medieval mathematician Fibonacci, is a simple mathematical expression of a fundamental pattern in nature.
Marvelous Mathematical Connections
- To the Golden Ratio: The Fibonacci sequence has an inseparable relationship with the Golden Ratio (φ ≈ 1.618). As the sequence progresses, the ratio of a term to its predecessor gets infinitely closer to the golden ratio.
- Ubiquity: This sequence appears in various branches of mathematics, such as number theory and combinatorics.
Nature's Fingerprints
The Fibonacci sequence is ubiquitous in biology, as if it were a favored "design pattern" of nature:
- Number of Petals: Many flowers have a number of petals that is a Fibonacci number. For example, lilies have 3, buttercups have 5, and delphiniums have 8.
- Seed Arrangement: The seeds on a sunflower head are arranged in spirals curving left and right. The number of spirals in each direction is usually a pair of adjacent Fibonacci numbers, like 34 and 55, or 55 and 89. This is an optimal packing arrangement.
- Branching of Trees: Some trees' growth patterns also follow this rule, where the trunk produces branches, which in turn produce sub-branches, often in numbers related to the Fibonacci sequence.
Recommended Reading
- (Book) The Fabulous Fibonacci Numbers by Alfred S. Posamentier and Ingmar Lehmann.
- (Book) The Book of Numbers by John H. Conway and Richard K. Guy.