# The Golden Spiral

## The construction of the golden spiral known as the Golden Spiral or Fibonacci Spiral

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Geometric Constructions

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29 June 2023

### The construction of the golden spiral

The golden spiral is a fascinating geometric concept derived from the Fibonacci sequence, a series of numbers where each number is the sum of the two preceding ones (0, 1, 1, 2, 3, 5, 8, 13, 21, etc.). It’s closely related to the golden ratio, denoted by the Greek letter phi (φ), which is approximately equal to 1.618.

To construct a golden spiral, you start with a golden rectangle. A golden rectangle is a rectangle whose side lengths are in the golden ratio (the ratio of the longer side to the shorter side is φ).

Here’s how you can construct a golden spiral:

2. Divide the square into a smaller square and a rectangle. The rectangle’s longer side has the same length as the original square, and its shorter side has the length of the smaller square.
3. Remove the original square, leaving only the smaller square and the rectangle.
4. Repeat steps 2 and 3 with the rectangle (the larger side becomes the length of the shorter side of the next rectangle, and the shorter side remains the same).
5. Continue this process indefinitely.

As you repeat this process, you’ll end up with a sequence of rectangles whose proportions approximate the golden ratio. If you draw quarter circles in each square, starting from the corner opposite the smaller side of the rectangle, and if you connect the points where the circles intersect, you’ll create a spiral that approximates the golden spiral.

To start the construction draw an arc from one corner of the rectangle until it intersects the adjacent side.

You then lead a segment perpendicular to the side that was intersected, from the intersection point to the opposite side.

Repeat the process to form another square…

.. and so on.

By drawing arcs with sequences of squares, one can construct the logarithmic spiral known as the Golden Spiral

The construction created can be related to the Fibonacci numbers using the construction shown on the side.

The golden spiral is fascinating because it appears in various natural phenomena and is often considered aesthetically pleasing. It’s found in the shapes of galaxies, hurricanes, seashells, and even in the proportions of the human body. Its mathematical properties make it a subject of interest in fields like art, architecture, and design.

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