A remarkable sequence of numbers that appears throughout mathematics, geometry, nature and the structure of the golden ratio.
The Fibonacci sequence begins with 0 and 1. Every subsequent number is the sum of the two preceding numbers.
Take the ratio of consecutive Fibonacci numbers:
As the Fibonacci numbers become larger, this ratio converges remarkably quickly towards the golden ratio.
Even though the Fibonacci sequence is generated using only addition, the ratio between successive terms converges towards a number that has its own fundamental mathematical identity.
Fibonacci numbers can be used to construct squares whose side lengths follow the sequence. Placing these squares together creates a Fibonacci rectangle.
Drawing a quarter-circle inside each square produces a continuous Fibonacci spiral.
The important point is that the curve is not an approximation made from lots of tiny straight lines. Each section of the spiral is a genuine quarter-circle with a radius equal to the corresponding Fibonacci number.
Because each arc finishes exactly where the next one begins, the result is one continuous curve.
Suppose the ratio between consecutive Fibonacci numbers tends towards a number φ.
From the Fibonacci rule:
Dividing everything by Fn gives:
As the sequence becomes very large, both ratios approach the same limiting value φ. Therefore:
Multiplying through by φ:
Solving the resulting quadratic equation gives:
Numerically:
What makes the Fibonacci sequence particularly beautiful is the way several apparently different ideas are connected:
A simple recurrence relation leads to a fundamental mathematical constant, which in turn appears naturally in geometry.