Fibonacci, quasicrystals and the beauty of flowers

Abstract

T he appearance of Fibonacci sequences and the golden ratio in plant structures is one of the great outstanding puzzles of biology. Here I suggest that quasicrystals, which naturally pack in the golden ratio, may be ubiquitous in biological systems and introduce the golden ratio into plant phyllotaxy. The appearance of golden ratio-based structures as beautiful indicates that the golden ratio may play a role in the development of consciousness and lead to the aesthetic natural selection of flowering plants. Fibonacci and the Golden RatioIn mathematics and the arts, two quantities are in the golden ratio (τ or 1.6180339…) if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is thought to be aesthetically pleasing and thus appears often in art and architecture. The Fibonacci numbers are simply the arithmetic consequence of multiplying each number by τ, and rounding to the nearest integer or, in other words, the next number in the Fibonacci sequence is the sum of the previous two numbers in the sequence thus 1, 1, 2, 3, 5, 8, 13, 21, 34, … The Fibonacci sequence was first discovered by Leonardo da Pisa (also known as Fibonacci) in 1202 and has puzzled scientists for centuries due to its frequent occurrence in nature, including in spiral molluscan shells and plants. The ubiquity of the Fibonacci sequence in nature seems to indicate that it arises through a process that is fundamental to life.

Beyond the 21-cm Notch-to-nipple Myth: Golden Proportions in Breast Aesthetics

“…The concept of the Fibonacci series and its occurrence in nature has been discussed repeatedly over time. 2,18 It is believed that the spiral pattern optimizes stereotactic growth and organization of modules (modularization) to, for example, capture light in photosynthesis and growth providing an optimal organization of subunits in a biologic pattern. 2 Phyllotaxis, the structure of pinecones, the packing patterns of florets of sunflowers and cauliflower, [18][19][20][21] and coronary artery branching 5 have all been shown to correlate with the Fibonacci sequence.…”

Generalized Fibonacci Numbers, Cosmological Analogies, and an Invariant

“…With the help of these analogies, it is easy to derive a Lagrangian and a Hamiltonian associated with the ODEs satisfied by the functions F (e,o) . Then, the conservation of the Hamiltonian (or energy) yields a quantity I, given by Equation (39) that is constant across the Fibonacci sequence; i.e., it does not depend on the index n. This invariant may turn out to be useful in view of the many applications of the Fibonacci numbers in mathematics and in the natural sciences (e.g., [12][13][14][15][16][17][18]).…”

Generalized Fibonacci numbers, cosmological analogies, and an invariant

“…Then, the conservation of the Hamiltonian (or energy) yields a quantity I, given by Eq. (4.8) that is constant across the Fibonacci sequence, i.e., does not depend on the index n. This invariant may turn out to be useful in view of the many applications of the Fibonacci numbers in mathematics and in the natural sciences (e.g., [12][13][14][15][16][17][18]).…”