Visualize the math! See the rules of the complex mathematical world – “Nature’s Mathematical Games”
I have another dream.
My first dream “Virtual Mirage Machine” is just a product of technology, it can help us visualize abstract mathematics, prompt us to build new insights, and allow us to ignore the boring number structure in mathematical problems.
Not least, it can make it easier for mathematicians to explore the world of the mind. But because mathematicians occasionally create new landscapes as they linger in mathematical gardens, the Mirage can also play a creative role.
In fact, the Virtual Mirage Machine or similar products will soon be released.
Classify complex math operations into simple schemes
I call the second dream “morphomatics” (morphomatics), this is not a technology, but a way of thinking. As far as creativity is concerned, morphological mathematics is of great importance. But I don’t know if that will happen, or even if it’s possible.
I hope the answer is yes, because we all need it.
The three examples in the previous chapter “droplet, fox and rabbit and flower petal” have very different structures from each other, but they all show the same philosophical view of how the universe works. Do not derive simple patterns directly from simple laws, as the laws of motion lead to the elliptical orbits of the planets. Instead, they race through a giant tree of leafy complexity that eventually collapses into a fairly simple pattern at just the right scale.
The simple narrative of “water dripping from the tap” is accompanied by a series of extremely complex and incredible changes.
While we have evidence from computer simulations, we still don’t know “why” these changes result from the laws of fluids. This is a simple result, but the cause isn’t simple.
In the mathematical computer game consisting of foxes, rabbits and grass, many complex and random rules are included. However, the important features of this artificial ecology can be represented by a dynamic system of four variables with up to 94% accuracy.
The number of petals is the result of complex interactions of all primordia, but by virtue of the golden angle, these interactions lead only to the various Fibonacci numbers. Fibonacci numbers are the clue to every Sherlock Holmes, not the culprit hiding behind the scenes. Mathematician Moriarty in this matter is not Fibonacci, but dynamics, mechanisms of nature, not “numbers of nature”.
In these three mathematical stories there is a common message: the patterns of nature are “emerging phenomena”, they explode from the ocean of complexity, just like Botticelli’s Venus (Sandro Botticelli, 1445-1510) suddenly appeared in the shell, without warning, and beyond the matrix.
They are not a direct result of the profound simplicity of the laws of nature, which do not apply at this level. They undoubtedly descend indirectly from the profound simplicity of nature, but the paths between cause and effect are so complex that no one can trace every step.
create a new kind of math
If we really want to catch the emergence of patterns, we must first have a new scientific method that can keep up with the traditional method that emphasizes laws and equations. Computer simulations are part of it, but we need more. It’s not satisfying just to have a computer tell us that a certain pattern exists, we want to know “why”.
This means that we need to develop a new kind of mathematics that can treat models as models and not just as accidental results of small-scale interactions.
I’m not trying to change the existing way of thinking about science, which has gotten us very, very far, and I’m asking for another system that complements it.
One of the most striking features of recent mathematics is that it has begun to focus on general principles and abstract structures, and attention has shifted from quantitative to qualitative issues. The great physicist Ernest Rutherford (Ernest Rutherford, 1871-1937) once said: “Qualitative is a poor quantitative description,” but this mentality is no longer justified.
Rutherford’s famous quote should be reversed: quantitative is a poor qualitative description. Because numbers are just one of many mathematical properties that help us understand and describe nature. If we try to squeeze all the degrees of freedom into a finite number system, we’ll have absolutely no way of understanding tree growth or sand dune formation.
The time was ripe for a new mathematics. Rutherford’s criticism of qualitative reasoning was mainly that it was sloppy; this new mathematics has considerable rigor, but also includes greater conceptual flexibility.
We need an effective mathematical theory of model study, which is why I call my dream “morphological mathematics”. Sadly, many branches of science are now going in the opposite direction.
For example, DNA is often thought to be the only answer to the shape and pattern of organisms, but current theories of biological development are not sufficient to explain why the organic and inorganic worlds share so many mathematical models. Maybe DNA encodes the rules of dynamics, not just the patterns that govern the completion of development. If this is the case, current theories clearly ignore many key steps in the development process.
Establish an appropriate natural mathematical system
The idea that mathematics is closely related to natural forms originated with Thompson and, in fact, dates back to the ancient Greeks and even the Babylonians. However, it is only in recent years that we have begun to develop what could be called proper mathematics.
Earlier mathematical systems themselves were too rigid, created to accommodate the constraints of pencils and paper.
For example, Thompson noted that there are many types of organisms whose shapes closely resemble the shape of fluids, but if you want to simulate organisms, the equations used in today’s fluid mechanics are too simple.
If we look at a single-celled organism under a microscope, the amazing thing is that its movements seem to have a precise purpose, as if it really knows where to go. In fact, it responds to its surroundings and internal states in a very specific way.
Biologists are gradually unraveling the mysteries of cell movement mechanisms, which are far more complex than traditional fluid mechanics. One of the most important features of cells is their so-called ‘cytoskeleton’, a kind of woven tubular network that looks like a bale of straw and functions as a rigid scaffolding inside the cell.
The cytoskeleton is an incredibly flexible and dynamic structure, under the influence of some chemicals it can completely disappear without a trace, wherever it needs support, it can grow there.
In fact, what cells rely on to move is to disassemble some skeletons and replace them with others.
The main component of the cytoskeleton is the microtubules, which I mentioned when talking about symmetry. As I mentioned in that chapter, this unusual molecule is shaped like a long tube and is made up of two units, alpha-tubulin and beta-tubulin, arranged in a black-and-white checkerboard pattern.
Microtubules can grow by adding new units and can also curl up from the tip like a banana peel. Its curl rate is much greater than its growth rate, but both tendencies can be stimulated with appropriate chemicals.
——This article is taken from “Nature math games November 2022,world culturePublished, please do not reprint without permission.
