As a general rule, the divergence angle between successively arising leaves is fixed at the golden angle of 137.5°, i.e., an irrational number 6. In marked contrast to these vertical arrangements, nascent leaves in the bud, or leaf primordia at the shoot apical meristem are more regularly arranged, but their arrangement in no way conforms to a fraction, i.e., a rational number. Anatomically, vascular bundles are formed by connecting what are known as the leaf traces of respective leaves, which are readjusted when the phyllotaxis fraction transitions as the plant grows ( Fig. The transition on a stem is often very conspicuous, because the denominator of the phyllotaxis fraction represents the number of vertical ranks of leaves. This change in fraction is called phyllotaxis transition. In fact, the phyllotaxis fraction is not a determined trait of each species but, rather, may vary from one part of a plant to another. Some references present willow as 5/13 and others as 3/8 without citing sources 3, 4. A list of these sequences and representative plants is often presented as follows: 1/2 for elm, lime and linden 1/3 for beech and hazel 2/5 for oak, cherry, apple, holly and plum 3/8 for poplar, rose and pear 5/13 for almond etc. Although the rule lacks a rational basis, it empirically describes not only the most commonly observed sequence of the phyllotaxis fractions-1/3, 2/5, 3/8, 5/13 and so on-but other rare sequences as well 1, 2.
Accordingly, 2/5 is obtained from 1/2 and 1/3 by adding their numerators and denominators, respectively. The Fibonacci rule is to add the previous two numbers to obtain the next number. In a 3/8 phyllotaxis, for instance, every eighth leaf emerges above one below it after three turns of a spiral of successive leaves, so that eight straight ranks are visible along the stem ( Fig. In the early nineteenth century, Schimper and Braun reported that regularity is expressed by means of common fractions obeying a Fibonacci rule. Mathematically regular arrangements of plant leaves, flower petals and other homologous organs, a phenomenon known as phyllotaxis, have attracted the attention of biologists, physicists and mathematicians.
The model also effectively explains the observed diversity of rational and irrational numbers in phyllotaxis. This model accounts for not only the high precision of the golden angle but also the occurrences of other angles observed in nature. This angle is the optimal solution to minimize the energy cost of phyllotaxis transition.
Here, I propose a new adaptive mechanism explaining the presence of the golden angle. However, it remains unknown whether phyllotaxis has adaptive value, even though two centuries have passed since the phenomenon was discovered. Algebraic and numerical interpretations have been proposed to explain the golden angle observed in phyllotaxis. This mathematical regularity originates from leaf primordia at the shoot tip (shoot apical meristem), which successively arise at fixed intervals of a divergence angle, typically the golden angle of 137.5°. As plants grow, these fractions often transition according to simple rules related to Fibonacci sequences. Plant leaves are arranged around a stem axis in a regular pattern characterized by common fractions, a phenomenon known as phyllotaxis or phyllotaxy.
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