Explore the De Jong Attractor
Four numbers create countless mathematical patterns
If you were asked to create an abstract painting, you might need brushes, paints, and colors—and it would likely take a long time to finish.
But if I told you that this painting requires only four numbers, you might not believe it.
The De Jong attractor is one such fascinating mathematical figure. It begins with a single point that moves continuously according to a fixed set of mathematical rules; the entire process is governed primarily by four parameters: a, b, c, and d.
By simply tweaking these four numbers, the original pattern can transform into something completely different: at times resembling smoke, clouds, or ink, and at others, looking like finely interwoven silk.
This is the most captivating aspect of the De Jong attractor: four numbers can unlock a world of mathematical patterns filled with infinite possibilities.
Forget the formulas for now
When you first encounter a De Jong attractor, there is no need to know the underlying mathematical formulas.
Just imagine a point on a canvas, currently at a specific location. The mathematical rules observe its position and then instruct it: 'Go here next.'
The point moves. Upon reaching the new location, the rules calculate again: 'Go here next.'
So, it moves again. Then it calculates and moves once more—repeating the process endlessly.
Step 1: Point → Calculate → Move
Step 2: Point → Calculate → Move
Step 3: Point → Calculate → Move
...
At first, you might not see anything—just a few scattered points.
But as the calculations accumulate and the trajectory builds up, a complex pattern gradually emerges.
It’s not randomly generated
At this point, you might be wondering: “Isn't this just randomly generated noise?”
Actually, no. Every step of the De Jong attractor is determined by specific rules. If the initial position, parameters, and calculation rules are identical, the final result will be identical too.
So, it doesn't involve scattering points at random; instead, it moves continuously according to rules. This is the key difference between it and ordinary random noise.
But why does it look so “random”?
That’s exactly what makes it interesting.
A single point does just one very simple thing: calculates its next position based on its current one.
However, this process repeats thousands upon thousands of times, with the result of one step serving as the input for the next. Thus, the first step influences the second, the second influences the third, and so on... After hundreds of thousands of calculations, simple rules can give rise to incredibly complex overall structures.
So, complex patterns don't necessarily require complex rules. Sometimes, all it takes is repeating a simple rule enough times.
Four numbers—four “dials”
One of the most fascinating aspects of the De Jong attractor is its simplicity regarding parameters. There are four main ones: a, b, c, and d.
You can think of them as four dials. Turn one, and the way the point moves changes. As the movement changes, the resulting trajectory shifts. And just like that: the pattern changes.
What happens if you change just one number?
This is an experiment well worth trying yourself.
First, select a set of parameters and take note of the current pattern. Then, modify only 'a'—leaving the other three numbers untouched—and regenerate the image. You might find that the original pattern has changed significantly.
Next, revert 'a' to its original value and modify only 'b'; you will get yet another structure.
At this point, you will realize: You modified just one number, yet you changed the entire mathematical world.
Why do these four numbers hold such power?
Because they don't directly dictate instructions like “Draw a line here” or “Draw a circle there.” Instead, they alter how the point moves next. Since the position of the next step determines the position of the step after that, the influence of the parameters propagates continuously.
You can think of it simply as: Parameter change → Change in movement → Change in next position → Change in every subsequent step → Change in final trajectory → Change in the entire pattern.
This is the core generative process of the De Jong attractor.
It looks like a cloud, yet no one is drawing clouds
De Jong attractors often produce soft visual effects: masses of points clustering to form areas of varying density—some regions are densely packed, while others are sparse.
Consequently, we see shapes resembling ☁️ clouds or mist, 🖋️ ink splatters, 🌀 swirls, 🎗️ ribbons, 🌫️ smoke, and so on.
Yet, none of these forms are explicitly specified by the mathematical formula; the formula never says, “Draw a cloud now.” It simply calculates continuously: “Where do I go next?” The cloud-like pattern is merely the natural result of layering countless simple movements.
This is the biggest difference between De Jong attractor and ordinary pictures
A normal image, no matter how complex it looks, is essentially pixels that already exist.
The de Jong attractor is different. The pattern you see is generated bit by bit during the calculation process. It is more like a painting being drawn mathematically.
So when you click Regenerate, you don't just change a picture from the gallery, but you run the math rules again.
This is why it is particularly suitable for exploring by yourself
You can know in advance: What the rules are. But you don't necessarily know: What pattern you will get next time.
It's a lot like opening a blind box, where you select a set of parameters, hit generate, and then: See what the math tells you.
Sometimes it's unremarkable, sometimes it's very beautiful. Sometimes you even get a structure that you didn't expect at all.
From Patterns Back to Mathematics
We now know that a De Jong attractor is a point in constant motion. So, what is the mathematics behind it?
It typically employs the following two recurrence relations:
xn+1 = sin(ayn) - cos(bxn)
yn+1 = sin(cxn) - cos(dyn)
It might look intimidating at first glance, but what it expresses is actually quite simple.
The first formula: Calculate the next x based on the current x and y.
The second formula: Calculate the next y based on the current x and y.
This yields new x and y values.
Then, feed the new x and y back into the formulas to continue the calculation—repeating the process.
And so, a single point transforms into an image.
The entire process can be summarized as: A point → calculate the next step → move → calculate again → move again → repeat many, many times → leave behind a multitude of paths → form a complex pattern.
That is the De Jong attractor. It doesn't explicitly state, 'This is what I am going to draw.' It only dictates, 'How to move to the next step.' The final pattern emerges naturally from the long-term accumulation of these movements.
Why is it called an 'Attractor'?
You can think of it this way: After moving for a long time, the point becomes confined to the vicinity of a specific structure.
The point never stops; it keeps moving. Yet, it doesn't wander off infinitely into the distance; in the long run, its trajectory concentrates within a specific region and structure. When these numerous paths overlap, we see the attractor.
Therefore, what we see is not a static, pre-designed image, but the 'footprints' left behind by the long-term motion of a dynamic system.
How do different attractors differ in appearance?
There are many types of attractors, and they are not required to look the same.
For example, the Lorenz attractor resembles a mathematical butterfly moving through 3D space.
The Aizawa attractor looks more like a complex 3D spatial structure.
The Clifford attractor often produces intricate, abstract textures.
Meanwhile, the De Jong attractor excels at creating soft, fine, and richly layered mathematical textures.
Thus, compared to other attractors, the De Jong attractor offers a different way to explore: using four numbers to discover new textures.
Continue Exploring the World of Mathematics
The De Jong attractor is a prime example within the realm of attractors for exploring how parameter changes affect the outcome.
If you enjoy the experience of “changing a few numbers to see what mathematics creates,” you might want to explore the Clifford attractor.
If you prefer observing motion in three-dimensional space, try the Aizawa attractor.
If you are interested in chaos, weather prediction, and the butterfly effect, you can explore the Lorenz attractor.
They all belong to the world of attractors, yet each possesses its own unique character: the Clifford attractor (parameters create abstract art), the De Jong attractor (parameters create mathematical textures), the Aizawa attractor (rules create 3D structures), and the Lorenz attractor (rules create chaos and a mathematical butterfly).
What makes the De Jong attractor particularly memorable are those four seemingly ordinary numbers: a, b, c, and d. They may appear to be mere parameters, but a slight adjustment to them can give rise to an entirely different mathematical world.
Continue exploring more amazing mathematical patterns.