Take a bite of the pagoda vegetable and the fractal secret behind it is...

As the weather warms up, I put away my thick cotton clothes and put on light spring clothes. However, without the cover of thick clothes, the fat covering my body that I accidentally added during the Chinese New Year has become particularly prominent...

In order to welcome the spring with a brand new look, I also started the "painless elimination of the whole body fat layer". However, while running on the treadmill, I thought of the pagoda vegetables I had eaten at noon to lose weight. Suddenly, I felt that I was caught in a huge mystery. Why did the shape of this vegetable become more and more strange the more I thought about it? ? ?

Image source: Tuchong Creative

(Copyrighted image from the gallery, no permission to reprint)

After careful thinking, the editor believes that thinking hard can also achieve the goal of "painless elimination of fat covering the whole body", and then replace daily exercise with daily thinking pagoda vegetables... Unexpectedly, the story behind the small pagoda vegetable is so fascinating...

Predictability

In the math and physics classes where we learn knowledge from childhood to adulthood, we gradually feel that the world is full of order: the curve direction of a function with a known expression and domain is predictable; the product of a chemical reaction is predictable if the reactants and reaction conditions are known; the initial position and motion law of a moving object are known, and the speed and position at any subsequent time are predictable...

Our impression of this “predictability” that science brings comes in part from Galileo Galilei and Isaac Newton’s studies of the swing of pendulums.

In 1581, Galileo observed the swing of a chandelier and realized that the swing had a predictable pattern. After a period of observation, Galileo found that although the swing amplitude was different, the time it took to swing the chandelier back and forth was the same.

To further explore this interesting phenomenon, Galileo conducted experiments using pendulums of different sizes but the same length to measure their swing periods, and he used his own pulse to time them.

It was eventually proven that the time a pendulum swings does not depend on the size of the object, nor on its position, but only on its length. Starting with Galileo's research, the swing of the pendulum became predictable.

After Galileo, Newton used differential equations to obtain the exact mathematical relationship between the length (l) of a pendulum and its period (T):

This has enabled us to make greater progress in "predictability", and the motion patterns of the pendulum's swing can be predicted not only qualitatively, but also precisely quantitatively.

We know that Newton discovered the laws behind many phenomena and invented mathematical methods such as calculus as powerful tools to help us understand the basic laws of the universe.

Among them, the three laws of motion of Newton, which we are most familiar with, describe the motion laws of macroscopic objects in a concise and beautiful way. They also let us realize that describing the laws behind motion phenomena with mathematical formulas, especially differential equations, can accurately describe how motion evolves over time, that is, predictability.

Predictability is undoubtedly fascinating, but if you think about it carefully, can all phenomena be described by this scientific idea based on "predictability"?

Chaos

Starting from the question of “whether it is predictable”, we can think of many examples that are very close to our daily lives: long-term weather forecasts, the development of animal populations, etc. These examples seem to hide a more fascinating “unpredictability”.

We cannot obtain accurate information about atmospheric motion through differential equations. How are these examples different from examples like "pendulum swinging"?

When we think of "uncertainty", we may feel unfamiliar and vague about its concept. From the perspective of scientific research, uncertainty is defined as "a certain random relationship between the system at different times before and after, and in statistical terms, it is mainly manifested as a causal relationship between the present and the future."

"Uncertainty" attracted researchers and gradually developed into an emerging discipline - Chaos.

Chaos has been a topic of scientific research since the late 1880s, when Henri Poincaré investigated the three-body problem in celestial mechanics.

It wasn't until 1963 that Lorenz, a meteorologist at MIT, showed that deterministic predictability was an illusion, giving rise to a still-thriving field: chaos theory.

Chaos theory holds that even the simplest equation (without any random factors) is known, and as long as there is a slight deviation during the operation, the result will be very different from the original idea.

The Butterfly Effect - Sensitive Dependence

At that time, there were two ways to predict the weather: first, using linear programs to predict the weather, the premise of which is that tomorrow's weather is a well-defined linear combination of today's weather characteristics; second, using fluid dynamics equations that simulate atmospheric flow to more accurately predict the weather.

When comparing the two calculation methods, Lorenz found that the weather data two months later obtained by the computer simulation was very different from the previous ones. However, Lorenz found that the "error" in this calculation actually came from the rounding of the initial values ​​in the simulation process.

Thus, Lorenz discovered a defining property of chaos - its sensitive dependence on initial values. The spheres in the figure below represent iterations of the Lorenz equation.

In 1972, at a conference, Lorenz gave a presentation titled "Predictability: Can the flapping of a butterfly's wings in Brazil cause a tornado in Texas?"

He used a butterfly as a metaphor for a tiny, seemingly insignificant perturbation that can change the course of the weather - what is known as the "butterfly effect".

After reading this, you may naturally have a question: Computer simulations usually introduce rounding errors at some point, and this error will be amplified by chaos, so can Lorenz's solution reflect the real chaotic trajectory?

The answer is yes, due to a property called shadowing: although for any given initial condition the numerical trajectory will differ from the exact trajectory, there is always an initial condition nearby whose exact trajectory is approximated by the numerical trajectory for a predetermined period of time.

Chaotic Attractor

Through the study of chaotic systems, Lorenz formally proposed the Lorenz equation in 1963. Its typical trajectory often converges to a non-integer bounded structure, as shown in the figure above, which is called a chaotic attractor.

The introduction of chaotic attractors can help us understand when the trajectory of a chaotic system will become "chaotic" due to its sensitive dependence on initial values.

First, the trajectory on the attractor will exhibit chaotic behavior different from that of the linear system. In addition, any point within the attractor's domain will also produce a chaotic trajectory that converges to the attractor.

Due to the existence of chaotic attractors, unlike the periodic trajectory of a simple pendulum, there is no periodic trajectory in a chaotic system, or it can be said that the periodic trajectory is divergent.

This is also the essential characteristic of chaos: non-periodicity means sensitive dependence, and sensitive dependence is the root cause of non-periodicity.

Is the above concept a bit confusing? It doesn’t matter, isn’t the pagoda dish coming soon?

When talking about chaos, we can’t help but think of another concept: fractal. Compared with the abstract concepts mentioned above, fractal is more concrete and there are many examples in our daily life.

From the above three pictures we can see that fractals seem to refer to the similarity of graphics from small scale to large scale. So, what is the exact definition of fractals?

Fractal structures or fractal processes can be roughly defined as having a characteristic form that remains constant across scales, that is, having self-similar properties.

A structure is fractal if its small-scale form is similar to its large-scale form.

If we think about it carefully, the strange feeling that pagoda vegetables give us seems to come from fractals. Its shape is different from the geometric shapes we usually come into contact with.

From Chaos to Fractals

In the above, we introduced chaos and fractals respectively. What is the relationship between the two?

Chaotic attractors are usually fractal. We can consider the trajectory of points in the phase space near a chaotic attractor: under the influence of the chaotic attractor, the points in the nearby phase space will show a nonlinear trend, that is, they are stretched and contracted by the chaotic attractor in different directions.

Under the combined effect of stretching and contraction, points in phase space form "filaments", and because the trajectory is bounded, these "filaments" will naturally fold.

When this effect of a chaotic attractor is repeated indefinitely, the result is a fractal.

Similar to how we can obtain relevant physical information through images, the geometric structure of chaotic attractors can be quantitatively related to their dynamic properties.

Concepts such as chaos and fractals sound very abstract. Compared to the meteorological system studied by Lorenz, is there a more vivid and simple example that can reflect the ideas of chaos theory?

Chaos in biology

It turns out that chaos theory is highly relevant to the field of biology, and the scientist who used this idea to conduct biological research also surprised the editor - Alan Mathison Turing.

Turing gave profound thought to the process of embryonic development and believed that this complex process could be described by simple mathematical formulas.

At the beginning, the cells inside the embryo are exactly the same and will self-organize according to simple rules. The process of self-organization is repeated continuously until a certain stage when it suddenly presents a complex pattern, gradually forming various different cells and eventually developing into different organs - this process is called morphogenesis.

Turing tried to use mathematics to explain how living things gradually evolved from a natural, uniform state into uneven, repeating patterns, that is, the process from self-organization to pattern emergence.

On the other hand, the famous Belousov oscillation experiment is also an example of self-organization leading to the spontaneous formation of patterns.

He discovered that mixing two solutions formed a colored liquid, which then turned clear, then colored again... and the process continued over and over again.

The spontaneous random ripples of Belousov's solution show that the system can change spontaneously and irregularly without being disturbed by external conditions. This is also an example of the process from self-organization to pattern formation.

Fractal of Pagoda Vegetable

Having learned so much about chaos and fractals in mathematics and biology, we still need to keep our original aspiration in mind - why does the pagoda vegetable grow fractals?

First, we need to understand how plant organs develop - throughout development, plant meristems regularly produce organs in a spiral, opposite, or whorled manner.

Think about ordinary cauliflower. Its special structure comes from the fact that the primary flower primordia produced by each meristem do not eventually develop into the flowering stage, but repeatedly produce more identical primary flower primordia, which is similar to an "avalanche" effect in the development process.

The self-similarity of the structure of the pagoda vegetable is because although the meristem cannot ultimately form flowers, during the development process, the primary flower primordium briefly undergoes a "soul-penetration" process, that is, it briefly maintains the "memory" of the flower.

This short-lived process affects the growth of the meristem, producing additional mutations that induce the formation of cone-shaped structures, ultimately forming cone structures with self-similar characteristics, that is, fractals.

I never thought that there are so many complex knowledge points hidden behind the ordinary pagoda dish. It turns out that the most advanced knowledge only needs the simplest way to show it~

Now, both chaos and fractals are gradually integrated with physics, mathematics, biology, chemistry and other fields, and have produced many novel and interesting results. What other interesting related phenomena do you know?

References

[1] Chen Lu. Research on control and synchronization of a hyperchaotic system with self-organizing structure[D]. Northeast Normal University, 2019.

[2] Wang Xiang. Research on distributed chaos theory and its applications[D]. Dalian University of Technology, 2021.

[3] Physics Today 66, 5, 27 (2013).

[4] The Secret Life of Chaos, BBC.

[5] Sean Bailly, L'art fractal du chou romanesco, Pour la Science, Septembre, 9, (10-11), (2021).

Editor: Norma

Source: Institute of Physics, Chinese Academy of Sciences