The Golden Ratio and Phylotaxis - Report Example

Introduction

Phyllotaxis refers to a fundamental aspect of the plant form, depicted by regular arrangement of lateral organs. The arrangements include scales on a cone axis, florets in a composite flower head, and leaves on a stem. Jean and Erickson demonstrated interest in the field of phyllotaxis on different levels. In their interests, biologists and mathematicians proposed models ranging from pure geometric descriptions to complicated physiological hypotheses tested through computer simulations. The two common models were by Jean and Erikson. Both models are associated with problems in packing. The first operates in a plane while the other reduces phyllotaxis to the issue of packing circles on surface of a cylinder. Vogel was originally concerned with the structure of a sunflower head while reviewing phylotaxis as the packing problems while operating on a plane. Conversely, Erickson reviewed in detail the problem of packing circles on the surface of a cylinder. In both models, fascinating mathematical associations dominate Phylotaxis. One relationship is that the traceable numbers of spirals through a phyllotactic pattern are primarily Fibonacci sequence integers. Fibonacci sequence is a number starting with two one and iteratively constructing the next number as the sum of the last two numbers. The most provable fact about the sequence is that the ratio of consecutive Fibonacci numbers come together to a golden ratio. For instance, a pineapple has 8 rows of scales sloping to the left and thirteen rows sloping to the right. The ratios of consecutive Fibonacci numbers Fk-1/Fk converge towards the golden mean, τ. The Fibonacci angle 360 τ -2 that is approximately equivalent to 137.50. This paper investigates the golden ratio and phylotaxis.

Fibonacci Numbers and nature

Evidently, Phylotaxis or the arrangement of seeds in some flowers and the arrangement of leaves on a stem obeys the Fibonacci sequence mathematical relationship. According to Okabe , the regular patterns on flowers and plant leaves are characterized by common fractions that transition based on simple rules. The mathematical regularity arises from leave primordia at the tip of the shoot or the shoot apical meristem that arises at fixed intervals of the divergent angle known as the golden angle. The general rule is that the divergent angle between leaves arising successfully is constant at the golden angle of 137.50. The golden angle lessens the energy cost of the phylotaxis transition. Okabe confirmed that the regularity of the phylotaxis phenomena is expressed using common fractions that obey Fibonacci rule ½, 1/3, 3/5,…. The rule involves adding the previous two numbers to obtain the next number both the numerator and the denominator. The meaning of the fraction, say ½ is that every third leave emerges above the one below it or after three turns of a spiral of successive leaves such that three straight leaves are visible along the stem. A list of representative plants that follow the phylotaxis fractions are ½ for lime and elm; 1/3 for hazel and beech; 2/5 for cherry and plum; 3/8 for rose and pear among others. Besides the flowers, phylotaxis fraction is variant from one part of the plant to another through what is known as phylotaxis transition. On the stem, phylotaxis transition characterises the number of vertical levels of leaves. Structurally, vascular bundles are formed through the connection of leaf traces of respective leaves that adjust with the phylotaxis fraction transitions of a growing plant. Fibonacci numbers are also evident in idealized rabbit, bee, and cow populations. Unlike using the phylotaxis fractions, idealized bees, rabbits, and cow populations follow the Fibonacci numbers. For instance, in the case of idealized rabbits shown below, several observations are visible:

The comparisons that can be made from the diagram above include among others:

• Rabbits of the same generation were born the same month
• The numbering is such that the same generation of new rabbits is numbered in the order of the parent’s number
• Rabbits numbered with Fibonacci number are children of the first parents on the tree

Packing Problem

When dealing with Phylotaxis in plants and animals, the most prominent question is on what is the best way to pack objects. Ordinarily, the shape of the object would determine the best way to pack the objects. For a square, packing would be in a square array, while in a round object, packing would be in a hexagonal arrangement. Evidently, circular seeds would be arranged best in a hexagonal arrangement, but that is not the case when it comes to the arrangement of leaves round a stem, or the packing of flower heads with seeds. From phylotaxis, nature seems to use identical pattern to position seeds on a seed head as used in positioning petals round a flower edge. Furthermore, nature seems to ensure that all the aspects of phylotaxis maintain their efficiency with the ongoing plant growth. The spiral growth patterns of seeds heads and stems is made possible by one fixed angle that yields the optimal design no matter the size of the plant. With the angle fixed for a given leaf, the leave will cause least obstruction to the leaves below it and will be least obstructed by future leaves above it. Uniform packing of seeds will always remain regardless of the seed head with the use of a single fixed angle of rotation between new cells. In such arrangement of seeds, petals or leaves, the Golden Ratio appears in plants as 137.50. The implication is that there are Phi leaves per turn or there are 0.618 = Phi turns per leaf. The arrangement ensures that each leave obtains optimal exposure to light, while forming the least shadow on the others. In terms of rain, the golden ratio ensures that each leaf has the best possible area exposed to falling rain such that rain is directed back along the leaf and down the stem into the roots.

Golden ratio and Fibonacci number

The golden ratio, golden mean, or golden section, occurs when taking the ratios of distances in simple geometric figures. The Fibonacci number forms the best whole number approximations to the golden ratio. The Fibonacci number, Fn is represented as F0=1, F1=2, F3=3, F4=5, where Fn = Fn-1+ Fn-2 and the golden ratio is

Ҩ = (1+ √ 5)/2 = Fn+1/Fn.

Notice that the consecutive number divides the first number and this calculation is repeated numerous iterations. Every iteration brings us nearer to an estimate of the golden ratio or Ø (1.618...). The example demonstrates that golden relationships are co-dependent of the Fibonacci numbers, and none can exist independent of the other.

Phylotaxis, Fibonacci Sequence, and the golden ratio

The regular spiralling patterns displayed from the arrangement of seeds on flowers and leaves on a stem display familiar set of numbers like 13 and 21 when counted. Consequently, these plants display numbers from the Fibonacci sequence. The reason why plants display Fibonacci numbers confirms that the Fibonacci sequence is closely linked to the golden ratio and is both irrational and the worst possible good rational approximations amongst all irrational numbers. The property makes the golden ratio the most irrational number. It makes the arrangement of successive leaves separated by the golden mean optimally distributed and realizable through inanimate physical systems. Additionally, the occurrence of the golden ratio is natural along the length of a pentagon’s diagonal, and in other systems that display pentagonal symmetry.

Conclusion

In summary, buds placed along a spiral and separated by an angle of 137.50 or the golden angle are packed most efficiently. Research suggests that the most new primordia always occurs in the least crowded spot along the growing plant or the Meristem. As the plant grows continuously, successive primordium forms at one point along the meristem. The primordium then moves drastically outwards at a rate that is proportional to the growth of the stem. The second primordium is also placed in the least crowded spot and as far as possible from the first primordium. As primordia increases, the divergence angle converges to a constant value of 137.50. The result is the creation of the Fibonacci spirals. Though the Fibonacci rule explained might not apply to all situations like the law of nature, it has a fascinating relevant frequency that is worth understanding in order to gain explanations to occurrences in life.