Professor Alain Goriely, delivering a lecture for Gresham College, illuminated the deep mathematical principles underpinning the shapes of plants, demonstrating that structures universally found in nature—such as flowers, pinecones, and pineapples—encode Fibonacci numbers, the golden ratio, and golden angles. As part of a series on the geometry of nature, Goriely showcased how mathematical objects appear in everyday plants. The lecture focused on the classification of phylum's (leaves, petals, etc.) on a plant stem, known as phyllotaxis.
While some plants exhibit simple geometric symmetry—like trifold symmetry, pentagons, hexagons, 10-sided, or even 21-sided polygons—the spiral pattern is the one that truly dominates and forms the king of mathematical structures in nature, exemplified by the Romanesco broccoli. Goriely encouraged the audience to look closely at Romanesco broccoli, which displays a fractal structure where spirals are composed of smaller buds that are themselves spirals, with the same structure repeating at different length scales.
The signature of this spiral geometry lies in two crucial observations. First, when counting the spirals (parastichy numbers) on structures like the sunflower head (capitulum), pinecones, or pineapples, they consistently appear as consecutive Fibonacci numbers. The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on) systematically appears, and studies show that 95% to 97% of plants counted are either directly Fibonacci or closely related to the sequence.
Second, the angle of rotation, or divergence angle, between successive leaves or primordia in these spirals is highly conserved at approximately 137.5 degrees. This angle was so perplexing that Charles Darwin wrote a letter in 1861, lamenting that the occurrence of these specific angles "is enough to drive the quietest man mad".
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Goriely explained that the mathematical link between the Fibonacci sequence and the 137.5-degree angle is the golden ratio, Phi (Φ), approximately 1.618. The ratio of consecutive Fibonacci numbers converges to the golden ratio. Furthermore, the golden ratio defines a special geometric property for rectangles, and when applied to a circle, it defines the golden angle, which is precisely 137.5 degrees. This golden angle is critical because it ensures optimal packing; due to the golden ratio being an irrational number, using this angle creates patterns that are completely packed with no empty space, maximizing the use of available area. The golden ratio is, in a mathematical sense, the number "furthest away from any rational point," making it the most efficient slope for dense packing.
The mechanism driving this perfect order is physical and chemical, occurring at the plant’s tip in the shoot apical meristem. Goriely explained that the plant hormone auxin regulates growth. New primordia appear at the spot where the auxin concentration is maximal, which is defined as the point "furthest away from the previous one," or the largest available space. Computational models based purely on reaction-diffusion equations and mechanics show that this physical search for the largest available space naturally results in the primordia organizing themselves at the special golden angle, with Fibonacci numbers emerging as a consequence of this process.
Beyond botany, Goriely detailed an unusual application of phyllotaxis: embedding the Fibonacci code into the script of the Hollywood film Sherlock Holmes: A Game of Shadows. The villain, Professor Moriarty, used the Fibonacci sequence—which naturally appears when inclining lines on Pascal's triangle—to encode secret messages for his henchmen, using a book on horticulture as the cipher.
Goriely's research also extends to carnivorous flora, such as the pitcher plant (Nepenthes). These modified leaves act as passive traps. Through mathematical modeling of the pitcher's rim, called the peristome, and the mechanics of friction, scientists can predict the stability and tumbling points of insects. Goriely suggested that the plant's morphology and inclination are tuned for maximum efficiency in capturing prey, proving that the study of plants integrates number theory, differential geometry, and evolutionary dynamics.
