The Golden Spiral

Two of the most common ways of visualizing the Golden Ratio (Φ) are the Golden Spiral and the Golden Rectangle. Both of these geometric figures are drawn by constructing rectangles whose sides are 1 and Φ. These proportions can be formed over and over again by adding or removing squares from the original rectangle and repeating the process. Constructing an arc passing through opposite corners of the rectangles forms the Golden Spiral shown below. The Golden Spiral is a special type of logarithmic spiral that grows continuously by a factor of Φ each quarter turn. Different variations of logarithmic spirals can be found in countless examples throughout nature. The Milky Way Galaxy, hurricanes/cyclones, sea shells, and even some fingerprints. In rare examples such as the Nautilus sea shell, a growth pattern closely related to Φ yields a fairly accurate Golden Ratio. Like a perfect circle, the perfect Golden Ratio is nowhere to be found in nature. However, similar spirals can be seen all over if you look closely. The swooping flight of a hawk toward its prey traces out a similar spiral. The reason for this is because the hawk’s line of vision is sharpest when equal to the pitch of the spiral. Another similar example is the circling of some insects toward a light source. Many more cases can be made for the approximations of the ratio existing in everything from DNA to snowflakes. However, when one is looking for a pattern they will find it in everything. Therefore, many cases of the ratio in nature are recorded due to people wanting to find it and making the necessary “stretches” to argue its existence. Next post I will write about another well known spiral, the Archimedean Spiral.