Fibonaccitiling

Fibonaccitiling refers to a method of tiling a rectangle using squares with side lengths that correspond to Fibonacci numbers. A Fibonacci number is a number in the sequence where each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.

When constructing a Fibonaccitile, one typically starts with a 1x1 square. Adjacent to this, another 1x1 square is placed. A 2x2 square is then added to form a 2x3 rectangle. Subsequently, a 3x3 square is placed alongside the 2x3 rectangle, creating a 5x5 square. This process continues, adding squares with side lengths corresponding to the next Fibonacci number in sequence to the growing rectangular or square shape.

The resulting arrangement of squares often forms a larger rectangle that has a side length ratio approaching the golden ratio, phi (approximately 1.618). This is because the ratio of successive Fibonacci numbers approaches phi. Visualizations of Fibonaccitiles are often used to illustrate the relationship between the Fibonacci sequence and the golden ratio, and they can be found in various artistic and geometric contexts. The tiling is not limited to perfectly filling a square; it can also be used to approximate a golden rectangle.