Fibonacci sequence and the golden ratio

In short: The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. It appears surprisingly often in nature and is closely related to the golden ratio.

In more detail: The sequence starts with 0, 1, 1, 2, 3, 5, 8, 13, 21 and continues infinitely according to the same formation rule.

In detail

• Formation rule: Each number in the sequence results from adding the two preceding numbers, creating a simple but powerful mathematical structure.
• Connection to the golden ratio: The ratio of consecutive Fibonacci numbers approaches the golden ratio of about 1.618 more and more closely as the values grow.
• Occurrence in nature: The arrangement of leaves, petals, pine cones, and sunflower seeds often follows Fibonacci patterns.
• Spiral shapes: The logarithmic spiral that can be constructed from Fibonacci squares resembles the spirals of snail shells and galaxies.
• Application in computer science: The Fibonacci sequence often serves as a classic example for recursion and dynamic programming when learning to code.
• Financial markets: Some traders use Fibonacci ratios for technical analysis of price movements, though the scientific validity of this method is disputed.
• Historical background: The sequence was named after the Italian mathematician Leonardo Fibonacci, although it was already known in Indian mathematics before him (exponentielles-wachstum, bildkomposition-goldener-schnitt).