FIBONACCI SEQUENCE FOUND IN NATURE SERIES

“We do wish that one day this will be proven,” he said. He explained how physicists know that the body-centered cubic lattice is the least energy configuration of atoms for many metals, which is also currently unproven. And numerical solutions would never be accepted as proof.”Ĭao added that there are many similar “embarrassments” in physics. Even worse, it is difficult to make oneself believe that the least energy solution he finds is really the global minimum. “There is no general method to find the least energy configuration for a given confining geometry, and the numerical solution costs enormous time of both the computers and the scientists. “The patterns on a sphere are now referred to as the Thomson problem, which has been generalized as the Generalized Riesz Problem,” Cao said.

Cao explained why this proof is so difficult. I conjecture that Fibonacci spirals are the least energy configuration on conical supports, but I cannot prove it.”įor plants, Cao’s conjecture is supported even in very old literature, although scientists have yet to discover a mathematical proof. However, Fibonacci spirals do appear on conical receptacles in nature. “I only know that the Fibonacci spiral patterns are not the least energy pattern for a sphere (try to imagine a football) or flat plane (we can make Fibonacci spirals on street pavement, but it is not self-assembled). “The least energy configuration for particles is dependent on the geometry of the space in which the particles are confined-just look at the thorn bundles of various cacti,” Cao explained. Still, the scientists noted that conical surfaces do not have to be perfect to produce Fibonacci spirals, which may explain the common occurrence in nature. On the other hand, spheres produced triangular patterns, while conical shapes with irregularities (such as the shape of a strawberry) produced an ‘X-pattern’. The scientists experimented with different shaped structures, and found that only conical shapes produced Fibonacci spirals with definite chirality. “We are seeking help on this question from simulations.” “Modeled as the least energy configuration on a confining conical support, one element more or less may reverse the chirality,” said Cao. While the chirality is considered random in plants, Li, Ji and Cao suggested that a geometrical factor may tip the balance. The scientists couldn’t determine why one form rather than the other would emerge. The spherules revealed Fibonacci spiral patterns, sometimes growing in the sinister form and sometimes in the dexter form.

The thermal stress caused the shell to become unstable, and when the conical-shaped structures were cooled, spherules grew on the most stressed sites. In this example, the scientists heated a mixture of SiO and Ag 2O to prepare the silver-cored, silicon-dioxide-shelled microstructures onto substrates at 1270 K, a temperature above the melting point of silver but below that of silicon dioxide, in order to achieve the proper elasticity difference in the subsequent cooling process. On an elastically mismatched structure, which consists of a stiff layer on a compliant surface, a variety of buckling occurs. In their experiment, the scientists used a technique from stress engineering, which is often used for the mass fabrication of micro- and nanostructures.

FIBONACCI SEQUENCE FOUND IN NATURE SERIES

Because their microstructures were very small, the next series (21x34) would have required more than 700 “spherules,” creating so much stress that the structure would break. For example, Li, Ji, and Cao produced a series of spirals of 3x5, 5x8, 8x13, and 13x21. The numbers of spirals on a surface are two consecutive numbers in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, etc.). The patterns consist of spirals that curve around a surface in both the “sinister” form (clockwise) and the “dexter” form (counterclockwise). This is the best support for this energy principle of phyllotaxis (or “leaf arrangement,” often credited to D'Arcy Thompson) before a rigorous mathematical proof is available.”įibonacci spiral patterns appear in many plants, such as pinecones, pineapples, and sunflowers. Our experimental results provide a vivid demonstration of this energy principle. “We conjecture that the Fibonacci spirals are the configuration of least elastic energy. “Patterns that evolve naturally are generally an optimized configuration for an assembly of elements under an interaction,” Cao explained to. Scientists conjecture that Fibonacci spirals are the least energy configuration on conical shapes. Fibonacci spiral patterns grow on conical-shaped microstructures, shown above in the sinister form.

Fibonacci sequence found in nature