The connections between the golden ratio, Fibonacci numbers and the mesmerising patterns found in the head of the sunflower are
fascinating. Our contribution pulls these elements together to highlight this magical sequence of numbers and their connection with growth, beauty and efficiency in nature. The golden angle (137.5°) is derived from the golden ratio and is responsible for the most compact arrangement of seeds in a sunflower head possible. The golden angle is calculated below:
360° / 1.618 = 222.5°
360° − 222.5° = 137.5°
We found it intriguing that manipulating the golden angle by as little as 0.1° has a significant effect on both the aesthetics and efficiency of the seed layout. Our visualisation explores the impact of changing this angle. The 13 different angles we chose to create the different seed layouts were derived from the Fibonacci numbers, the most famous sequence in maths (0, 1, 1, 2, 3, 5, 8...). We were struck by the connection between the golden ratio and this sequence: when the ratios of consecutive Fibonacci numbers are arranged in a sequence their values increasingly approach the golden mean.
In the same way that the golden angle is calculated using the golden ratio we derived a progressive set of angles using these Fibonacci ratios. To the nearest
decimal place, these were 0°, 180°, 120°, 144°, 135°, 138.5°, 137.1°, 137.6°, 137.5°. So for instance:
3/2 = 1.5
360° / 1.5 = 240°
360° − 240° = 120°
We put these angles into a mathematical formula to generate the seed dispersal within a fixed radius, with each composition made up of 400 dots. The final formation is made up of over 6,000 dots which demonstrates how beautifully dense the arrangement is when the golden angle is used. Our layout illustrates the distribution of seeds inside a sunflower head moving from rigid and inefficient formations to a beautifully elegant arrangement. We find it incredible that nature has evolved to find this perfect arrangement. We hope our contribution highlights how finely balanced the mathematics behind it are and how closely the Fibonacci sequence is tied to the golden ratio.