The Sunflower: Fibonacci’s Golden Ratio in Full Bloom

Photographer: Patti WeeksSummary Author: Patti Weeks 

Why does the center of a sunflower develop in such a fascinating way? The answer can be traced back to the 13th century Italian mathematician Leonardo Fibonacci. In his 1202 book Liber Abaci, he proposed this hypothetical mathematical problem: How many pairs of rabbits would be produced in a year, starting with one breeding pair, and adding their subsequent breeding offspring pairs each month. The solution reveals the sequence 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Each new number is attained by adding the previous two numbers. This is the Fibonacci Sequence — and can continue ad infinitum beyond the rabbit explosion!

Furthermore, if you divide each number in this sequence by the previous number, you approach what’s labeled in mathematics by the Greek letter Φ (pronounced 'fi'): ~1.618… And the higher in the sequence you go, the closer you get to the “real phi,” which actually can never be reached, because it cannot be written as a fraction —it’s an irrational number! It's been assigned the description of the “golden ratio.” (Click here to read about other methods of arriving at the golden ratio.) 

The crisscrossing spirals we see in the center of the sunflower create a pattern of densely packed seeds with no gap between them, creating the most systematic arrangement to maximize the sunflower’s chances of reproductive success. These overlapping “golden spirals,” (technically called parastichies) happen to be adjacent Fibonacci numbers — typically 34 clockwise and 55 counterclockwise; the larger number divided by the smaller results in the golden ratio! And the “golden angle” describes the spatial relationship of each seed to its neighboring seed. (Scroll down in this link to see an animation of a sunflower’s seed formation.) As a non-mathematician, I'm simply satisfied to know that the Fibonacci sequence in nature reflects a robust and efficient method of growth, ultimately perpetuating the particular species — and creates a mystifying, yet stunning beauty.

I snapped this photo in a sunflower field along a highway in eastern North Carolina on August 4, 2024. A Fibonacci number can almost always be seen in the number of petals in a flower — I count 34 in my picture! The golden ratio can also be observed in other natural phenomena, such as pinecones and pineapples, and approximated in others, including hurricanes and galaxies. It was described as the Divine Proportion by Leonardo Da Vinci as early as the 1500s in the comparison of proportions of human body parts. There's such an admiration for this mathematical phenomenon that Fibonacci Day is celebrated annually on 11/23 (November 23) — the first four numbers of the Fibonacci sequence!

 

Related Links:Sunflower Seed-head, Real and SimulatedThe Golden Ratio: Is it Myth or Math5 Fun Facts to Celebrate Fibonacci Day