Why do Fibonacci numbers occur on so many different occasions in both mathematics and the real world?
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. This pattern occurs throughout nature, as well as in human culture and history. Here are some reasons why Fibonacci numbers seem to pop up everywhere:
- The golden ratio, which is an irrational number that appears frequently in architecture, art, and natural patterns, is equal to (1 + √5) / 2. It can also be expressed as F(n),where n refers to the current number in the Fibonacci sequence. Many structures such as trees, flowers, and shells use this ratio.
- The Fibonacci sequence has many connections with other mathematical concepts, including linear algebra, probability theory, and geometry. For example, it can be used to describe how points on a line move when they are influenced by a gravitational field (see below). In computer graphics, the Fibonacci sequence is commonly used to generate fractal shapes.
- The Fibonacci sequence arises naturally when looking at patterns in biological growth or animal behavior. For instance, the way certain animals build their nests, or how plants spread their seeds, involve the repetition of specific patterns of Fibonacci numbers.

Here's another interesting connection between Fibonacci sequences and the real world: If you start with 1 and 0 and continue adding the next pair of numbers, the resulting sequence will always end with 5. This is because 0+1=1 and 1+2=3, and therefore any additional term in the sequence must have the same pattern. For example, 3+4=7, 7+5=12, 12+13=25, etc., leading us back to the initial sequence of 1 and 0. This property is known as the "Fibonacci sequence identity" or the "$f$ theorem."