As the weather warmed up, Xiaobian also put away the thick cotton clothes and changed into light spring clothes. however... Without the cover of thick clothes, the whole body fat cover that was accidentally replenished during the New Year became particularly prominent...
In order to greet the spring with a new look, Xiaobian also began the "painless elimination of the whole body fat cover", but running on the treadmill, thinking of the pagoda dish eaten at noon in order to lose weight, suddenly felt that he was caught in a huge mystery, how the shape of this vegetable became more and more strange???
After some careful thinking, Xiaobian believes that hard thinking can also achieve the purpose of "painless elimination of fat covering throughout the body", and then replace the daily serving of exercise with the daily serving of thinking pagoda dish... Unexpectedly, the story behind the little pagoda dish is so fascinating...
Predictability
In our mathematical classrooms, which we grew up learning, we gradually felt that the world is full of order: the functions of known expressions and defining domains, the curve direction of which is predictable; the reactants and reaction conditions of known chemical reactions, the products are predictable; the initial position and laws of motion of known moving objects, the speed and position of the next arbitrary moment are predictable...
Part of our impression of this "predictability" that science brings stems from Galileo Galilei and Issac Newton's study of the pendulum swing of the clock.
In 1581, when Galileo observed the swing of the chandelier, he realized that there was a predictable law of the oscillation. After a period of observation, Galileo found that although the swing amplitude was different, the time to swing the chandelier back and forth was the same.
Image credit: pixabay
To further explore this interesting phenomenon, Galileo experimented with swing cycles with pendulums of different sizes and lengths, timing them with his own pulse.
It was finally confirmed that the swing time of the pendulum did not depend on the size of the instrument, nor on its position, but only on its length. At the beginning of Galileo's study, the swing of the pendulum became predictable.
After Galileo, Newton used differential equations to obtain the exact mathematical relationship between the length (l) of the pendulum and the oscillation period (T):
This allows us to make greater progress in "predictability", where the laws of motion of the pendulum swing can be predicted not only qualitatively, but also 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 fundamental laws of the universe.
Among them, the three laws of Newton, which we are most familiar with, describe the laws of motion of macroscopic objects in a concise and beautiful way. It also makes us realize that the laws behind the phenomena of motion are described by mathematical formulas, especially differential equations, which can accurately describe how motion evolves over time, that is, predictability.
Predictability is undoubtedly fascinating, but think about it, can all phenomena be described in terms of this scientific thinking based on "predictability"?
Chaos
From the perspective of "predictable or not", we can think of many examples that are very close to daily life: long-term weather forecasts, the development of animal populations, and so on. There seems to be a more fascinating "unpredictability" hidden in these examples.
We can't get accurate information about the motion of the atmosphere through differential equations, how are these examples different from examples like "pendulum swing"?
From the perspective of scientific research, uncertainty is defined as "a certain random relationship between different moments before and after the system, and in a statistical sense, it is mainly manifested as a causal relationship between the present and the future".
"Uncertainty" attracted researchers, and gradually developed an emerging discipline , Chaos.
In the late 1880s, from henri Poincaré's study of the three-body problem in celestial mechanics, chaos began to appear in the field of scientific research.
Edward Lorenz | Image source [3]
It wasn't until 1963 that lorenz, a meteorologist at the Massachusetts Institute of Technology, thought predictability of certainty was an illusion, and from it emerged a field that was still thriving— chaos theory.
Chaos theory holds that even the simplest equation (without any random factors) is known, and that if there is a slight deviation in the course of operation, the result will be very different from the original idea.
The Butterfly Effect – Sensitivity Dependence
At the time, there were two ways to predict weather: first, predicting the weather using a linear program, provided that tomorrow's weather was a well-defined linear combination of today's weather characteristics; and second, predicting the weather more accurately by simulating fluid dynamics equations that simulated atmospheric flow.
When comparing the two calculations at one time, Lorenz found that the computer simulations showed that the weather data two months later was very different from the past. However, Lorenz found that the "error" in this calculation actually stemmed from the rounding of the initial value during the simulation.
From this, Lorenz discovered a defining property of chaos—a sensitive dependence on the initial value. Here the sphere in the figure below represents an iteration of the Lorentz equation.
Lorenz attracts sub-| Image source [3]
In 1972, at a conference, Lorenz recounted a lecture titled "Predictability: Will Brazilian Butterflies Flapping Their Wings Cause Tornadoes in Texas?" " report.
He uses a butterfly as a metaphor for a tiny, seemingly inconsequential perturbation that can change the course of the weather — what we know as the "butterfly effect."
Reading this, you may naturally have a question: computer simulations usually introduce rounding errors at some point, and this error is amplified by chaos, so can Lorenz's solution reflect the real chaotic trajectory?
Of course, this is because of a property known as "shadowing": although the numerical trajectory is different from the exact trajectory for any given initial condition, there is always an initial condition nearby, and its 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 Lorentz equation in 1963, whose typical trajectory tends to converge to a non-integer bounded structure, known as the Chaotic Attractor, as shown in the figure above.
The introduction of chaotic attractors makes it easy to understand when the trajectory of chaotic systems will be "chaotic" due to their sensitive dependence on initial values.
First, trajectories located on the attractor exhibit chaotic behavior different from those of linear systems, in addition to which any point within the attractor's attraction domain also produces a chaotic trajectory converging toward the attractor.
Because of the existence of chaotic attractors, unlike the pendulum trajectory, there is no periodic trajectory, or it can be said that the periodic trajectory is divergent.
This is also the essential feature of chaos: aperiodic means sensitive dependence, and sensitive dependence is the root cause of aperiodicity.
Fractal
Isn't it a bit of a cloud coming out of the above concept? It doesn't matter, pagoda dish is not coming!
When it comes to chaos, it is always inseparable from another concept- fractals. Fractals are more figurative than the abstract concepts mentioned earlier, and there are many examples in everyday life.
Pagoda dishes are | Image credit: pixabay
Fractals of branches | Image credit: pixabay
Mandelbro set | Image credit: pixabay
From the above three pictures, we can see that fractals seem to refer to the similarity of small-scale to large-scale graphics, so what is the accurate definition of fractals?
Fractal structures or fractal processes can be roughly defined as having a characteristic form that remains constant on a scale, i.e. having a self-similar property.
If the small-scale form of a structure is similar to the large-scale form, then it is fractal.
If you think about it, the strange feeling that pagoda dishes give us seems to come from fractals, which are different from the geometries we usually come into contact with.
From chaos to fractals
In the previous article, we introduced chaos and fractals separately, what does the relationship between the two look like?
Chaotic attractors are usually fractal. We can consider the trajectory of the midpoint in phase space near the chaotic attractor: under the influence of the chaotic attractor, the points in the nearby phase space exhibit a nonlinear tendency, that is, they are stretched and contracted by the chaotic attractor in different directions.
Under the combined action of stretching and contraction, the points in the phase space will form "filaments", and since the trajectory is bounded, these "filaments" will naturally fold.
When this effect of the chaotic attractor is repeated indefinitely, the result is fractals.
Similar to the physical information we can obtain from images, the geometry of the chaotic attractor can be quantitatively related to its dynamic properties.
The concepts of chaos and fractals sound abstract, but are there more vivid and simple examples of the meteorological systems Lorenz studied that reflect the ideas of chaos theory?
Chaos in Biology
Chaos theory turned out to be highly relevant to the field of biology, and scientists who conducted biological research with this kind of thinking also surprised the editor - Alan Mathison Turing.
Turing thought deeply about the process of embryonic development, and he thought that this complex process could be described by simple mathematical formulas.
In the beginning, the cells inside the embryo are exactly the same, self-organizing according to simple rules, and the process of self-organization is repeated until a certain stage suddenly appears in complex patterns, gradually forming different cells, and eventually developing into different organs - this process is called morphogenesis.
Turing attempted to use mathematics to explain how living organisms evolved from natural, uniform states to uneven repetitive patterns, i.e., 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 found that the two solutions were mixed to form a colored liquid, and the liquid became clear and then became colored... It's a cycle of this process.
Belousov oscillation reaction | Image source
The pattern of random ripples spontaneously generated by Belousov's solution indicates that the system can change spontaneously and irregularly without interference from external factors. This is also an example of self-organization to pattern formation.
Fractals of pagoda dishes
After learning so much about chaos, fractals in the field of mathematics and biology, we still have to not forget the original intention - so why does the pagoda dish grow fractals?
First, we need to understand how plant organs develop —throughout development, plant meristems regularly produce organs in spirals, paragenesis, or rotational patterns.
Think of ordinary cauliflower, whose special structure stems from the fact that the primary flower primordium produced by each meristem does not eventually develop to the flowering stage, but instead reproduces more of the same primordial primordium, similar to an "avalanche" effect in a developmental process.
Cauliflower | Image credit: pixabay
The self-similarity of the structure of pagoda vegetables is because although the meristem cannot eventually form flowers, in the process of development, the primary flower primordial base briefly appears in a process of "soul piercing", that is, temporarily maintaining the "memory" of the flower.
This short-lived process affects the growth of meristems, producing additional mutations that induce the formation of conical structures that eventually form conical structures with self-similar characteristics, also known as fractals.
The fractal structure of pagoda cuisine | Image credit: pixabay
I can't imagine that there are so many complex knowledge points hidden behind the ordinary pagoda dishes, and it is true that the most high-end knowledge only needs the most simple way to show it
Now, whether it is chaos or fractals, they are gradually integrated with physics, mathematics, biology, chemistry and other fields, and have cross-developed a lot of new and interesting results, what other interesting phenomena do you know about it?
bibliography
CHEN Lu. A study on the control and synchronization of a superchaotic system with self-organizing structure[D].Northeast Normal University, 2019.
WANG Xiang. Distributed chaos theory and its application[D].Dalian University of Technology,2021.
[3] Physics T oday 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).
The images of the emojis that are not indicated in the article are all from the network
Edit: Norma