The Math | patterns-in-nature

The Math Behind the Golden Spiral

The math that represents the Golden Spiral, which is also known as the the Golden Ratio, is called the Fibonacci sequence. This sequence is what the graph of the Golden Spiral is based off of.

What is the Fibonacci Sequence?

The Fibonacci Sequence is a sequence of numbers that is based off of the rule that any number in the sequence is equal to the sum of the two numbers preceding it. The Fibonacci Sequence is also the basis of the graph of the Golden Spiral, and it is a very prominent pattern in nature.

Some numbers from the sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765

How is the Golden Spiral related to the Fibonacci Sequence?

The Golden Spiral is a graph that represents the numbers of the Fibonacci Sequence. The spiral does this by following the rule that after every quarter of a turn the radius of the spiral is increased by a factor of phi, or increased to the next number in the Fibonacci Sequence. For example, in the picture below the radius at one point of the spiral is 21 and after a quarter of a turn that radius has increased to 34.

The Equaion behind the Golden Spiral?

The equation that represents the Golden Spiral is an equation that finds the radius of the spiral after a certain number of quarter-turns. This equation also represents the Fibonacci Sequence where the nth number of the sequence equals a certain value.

The equation is called Binet's formula and looks like this: