Phyllotaxis, spatial optimization and irrationality

Tools: Rhinoceros, Grasshopper, Python

“Everything is what it is because it got that way.”
Sir D’Arcy Wentworth Thompson

Phyllotaxis

Phyllotaxis is the arrangement of leaves on a plant stem. The structural beauty of phyllotactic patterns has been known for centuries.

There are three main types of phyllotaxis:

• alternate where the leaves arise from each node and arrange in a spiral pattern
• opposite (paired phyllotaxy) where two leaves face each other on the same node
• whorled characterized by the insertion of three or more leaves on a single node

In the alternate case each new leaf on a plant stem is positioned to the previous one at a certain angle. This angle is called the divergence angle and it is constant. The helices paving the plant in the two opposite directions are called parastichies. The number of parastiches (dextral and sinistral) is used to characterize the different phyllotaxic patterns.

One of the best examples of the spiral phyllotaxis is a sunflower.

A remarkable observation, due to Alexander Braun in 1831, established the link between the number of parastiches and the Fibonacci sequence. Indeed, in the extreme majority of cases, the numbers of parastiches found in a plant, in one direction and the other, are two consecutive terms of the Fibonacci sequence.

Also, the most common divergence angle in nature is the golden angle.

The constant interest of researchers in the problem of phyllotaxis has led to a variety of models describing the pattern. To generate this pattern in grasshopper I used the H. Vogel’s formula for a planar model² :

r=c√n ; θ=137.508

import rhinoscriptsyntax as rs
import math

thetalist = []
radiuslist = []

for n in range(N):
    t = n * a
    r = c * math.sqrt(n)
    thetalist.append(t)
    radiuslist.append(r)

As you can see the divergence angle (a) is fixed to θ, the golden angle. This angle is a result of the division of the full circumference of the circle according to the Golden Ratio. So that the ratio of lengths of the big arc to the little arc is the same as the ratio of lengths of the full circumference of the circle to the big arc.

The Golden ratio manifests in nature in a variety of ways due to its efficiency in terms of optimization. Nature is always looking for forms that cost the least amount of energy and material. Every time you need to divide an area with a minimum effort and maximum efficiency, you will probably come up with the same pattern strategy and the same number. But why is it so ‘special’?

Let’s go back to our sunflowers. Imagine you are in the center of the circle and you need to evenly distribute the seeds over its surface. You can rotate and throw the seeds. The angle of rotation is fixed. What would it be? Let’s start with a fraction of a circle – 360/n. The problem with this is that every time you take a fraction, you will eventually return to the same position, so the seeds will be placed in lines with a dead space between them.

So what you will be looking for is something that could not be a fraction. Something as far from a fraction as possible so that it cannot be expressed or even approximated as a ratio of two numbers… something irrational. As irrational as possible. The most irrational number.

In this model, each seed is released from the center with absolute angle accuracy. The seeds move along the radius independently of each other. However, absolute measurement accuracy is unachievable in nature. Besides, the seeds are not mathematical points – they press against each other, and under this pressure, they move and deform. This leads us to the assumption that Fibonacci phyllotaxis does not arise due to absolutely accurate measurement of angles, but is a result of the dynamic actions that stabilize the arrangements.

In fact in a small percent of the total number of plants the phyllotaxis is described not by the Fibonacci sequence, but by other numerical sequences like Lucas numbers. However, the Lucas numbers, have the same property as the Fibonacci numbers: their ratios also tend to the Golden Ratio.

The distance between the seeds can be different in a different parts of the plant. The younger seeds are usually smaller than the old ones. The distance between the points can also be different depending on how you calculate the r parameter.

The planar formula can be easily converted into the cylindrical one by adding the height parameter:

import rhinoscriptsyntax as rs
import math


thetalist = []
Hlist = []
radiuslist = []

for n in range(N):
    t = n * a
    H = n * h
    r = c * math.sqrt(n)
    thetalist.append(t)
    Hlist.append(H)
    radiuslist.append(r)

And of cause by changing the r variable you can get another types of revolved forms.

Although the irrationality of the numbers is just a concept that exist only in our mind, it has allowed the plants to develop the best survival strategy. Keep this in mind the next time someone calls you “irrational” 😉

Resources:

1. D’Arcy Wentworth Thompson, On Growth and Form,  1917
2. H.Vogel, A better way to construct the sunflower head. Mathematical Bioscience 44, 179-189 (1979)
3. https://www.youtube.com/watch?v=DRjFV_DETKQ&ab_channel=Numberphile