The golden ratio and the Fibonacci series in plants

We will try to explain in a simple and graphic way the importance of the golden ratio and the Fibonacci sequence, how they are calculated, and how they appear in nature.

Obtaining the golden ratio or divine proportion

The golden ratio is the proportion between the sides of a rectangle constructed from a square. The golden rectangle has fixed proportions that correspond to the number Phi = 1.618..., a real number with infinitely many decimal places without repeating.

1. We construct a square and draw a line from the midpoint of the base to one vertex.
2. We then rotate this line, drawing an arc. With that measurement we create a rectangle.
3. We now have a golden rectangle.

Calculating the golden ratio

Let's assume the side of the square measures one meter. If the base of the rectangle is "x", we obtain the following proportion:

The two sides of the outer rectangle are in proportion to the smaller rectangle. Using this proportion and solving the equation, we obtain the golden ratio.

If we continue making rectangles using this process, they maintain the ratio. If we make a large number of increasingly smaller rectangles, the measurements of the square and the rectangle tend to become equal; then we assign them the value of one unit, and a relationship appears between the measurements of the rectangles and the growth of the Fibonacci sequence, as we can see in the following figure: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

This rectangle is part of our daily lives, for example, in the proportions of computer screens, the dimensions of ID cards or credit cards, in works of art, in the proportions of the human body, etc. It has been shown to be more visually pleasing.

Archimedes' Spiral and its relationship to the golden ratio

If we draw arcs of circles with a radius equal to the side length of each square, we obtain the curve of the golden spiral. This spiral is found in nature, for example, in Romanesco broccoli, sunflower seeds, and pinecones.

We observe that the number of spirals formed on the right and left are consecutive numbers in the Fibonacci sequence. If there are 21 on the left (in red), then there are 34 on the right (in blue), and 34/21 = 1.618...

The Fibonacci sequence in plants

In 1202, Fibonacci discovered a sequence of numbers that appear in nature. The sequence is: "1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89..." As we can see, each number is the sum of the two preceding numbers; therefore, the next number in the sequence would be 144. If we divide each number by the previous one, we find that it tends toward the golden ratio, the result being a limit. This sequence appears in the reproduction of rabbits and cows, but we will focus on plant growth.

The Golden Angle

We observed in a previous article that the artichoke flower follows a golden ratio, such that no two leaves ever overlap, ensuring that all receive sunlight. This is because they grow at a golden angle, obtained by forming a circle with the length of the longer side plus the shorter side of the rectangle. Measuring the arc corresponding to the shorter side, we find that this arc corresponds to an angle of 137.5º.

The Fibonacci sequence also leads to the golden ratio. The ratio of each quotient in the 360-degree sequence corresponds to the golden ratio of 137.5 degrees, with greater accuracy as you progress further along the sequence. In other words, this angle, and therefore the Fibonacci numbers and the golden ratio, are related. We can observe how the branches of a tree, viewed from above, form approximately this angle between two consecutive branches.

It is also observed in:

• The distribution of leaves on the plant.
• The distribution of branches on the tree.
• The ratio between main and secondary branches.

In some flowers and fruits, it is particularly noticeable, such as:

• Fir cones and pineapples.
• Palm trees and artichoke flowers.
• Romanesco broccoli, sunflowers, and daisies.
• Cauliflower, cycads, and roses.
• The thorns on rose stems and many others.

All symmetries, in one way or another, grow according to the golden ratio, the Fibonacci angle, or the Fibonacci number.

The Golden Angle of Rose Flower Growth

In the case of the rose, we measure the angles of consecutive petals, from largest to smallest. We will never find two petals overlapping due to their growth at this 137.5-degree angle, as seen between points "a" and "b," between "b" and "c," and so on. This angle means that no matter how many petals grow, no two will ever align in the same direction. Not all plants grow in this order; there are always variations, but it is proven that the golden ratio is present wherever symmetry and beauty exist.

How does this relationship between the golden ratio and plants occur?

How does all this happen? We have all observed the harmony of certain plants, the symmetry of some fruits, and the beauty of flowers. This arrangement is not random; it follows an order called phyllotaxis, derived from the words "phyllo" (leaf) and "taxis" (order).

This order is maintained by meristems, composed of polyhedral stem cells that constantly reproduce through a process called mitosis, which consists of six phases. This reproduction is generally spiral, with the curvature corresponding to Archimedes' spiral. That is, the angle of rotation is the golden ratio, and a new cell appears with each turn.

Water entering the vacuoles, an important part of the cell where it is stored along with salts and sugars, influences their growth and multiplication.

Important: This is why water control is fundamental for the plant's healthy development. If the plant suffers water stress, this reproduction is disrupted.

These meristems can be permanent, like those of branches, or temporary, such as those that give rise to flowers. These cells are like the information code, constituting the DNA of plants. These chains follow spirals with golden turns, and inside them are the nitrogenous bases with pentagonal structures that also have the golden angle in their measurements.

Influence of the golden ratio on plant development

This decisively influences:

• The runoff of rainwater onto the plants, ensuring it reaches the roots more effectively.
• Ensuring all leaves receive sunlight more efficiently, preventing self-shading.
• Promoting pollination with this spectacular geometric distribution.

In other words, the golden ratio allows the plant to grow harmoniously. We simply need to provide it with constant and controlled water, as well as the appropriate temperature and nutrients to maintain its regeneration; then its development and beauty will approach perfection.