Exploring Noise Flow Fields
Noise Flow Field: If an Invisible Wind Blew Across the Canvas
Imagine a canvas that is completely blank. Now, we place thousands of tiny particles onto it and tell them: 'Move forward.'
However, we don't tell every single particle exactly where to go. That is because an invisible 'wind' is hidden across the canvas. The wind might blow to the left in one spot and to the right in another. A little further along, the wind might curve. In some places, it swirls into a vortex; in others, it flows forward like a river; and elsewhere, it gradually shifts direction.
The particles cannot see this 'wind map.' They simply follow the direction of the wind wherever they happen to be. As thousands of particles begin to move, they leave behind fine, delicate trails.
Gradually, a complex pattern emerges. It might resemble 🌬️ wind, 🌊 flowing water, ☁️ smoke, 🪶 feathers, or 🧵 silk—or even look like some unseen force flowing through space. This is a noise flow field.
Imagine a 'Wind Map'
The simplest way to understand a noise flow field isn't by looking at formulas first. Instead, imagine a map of the wind.
Suppose there is wind across the entire canvas. Here: → → → the wind blows to the right. Move to the side: ↗ ↗ ↗ the wind starts blowing up and to the right. Move further: ↑ ↑ ↑ the wind shifts to blow upwards. Then: ↖ ↖ ↖ it starts turning again. If we fill the entire canvas with arrows like these, we get a flow field—a map describing which direction the flow should move at every point on the canvas. In a flow field, particles simply need to 'go with the wind'.
Now, place a small particle into the field. It arrives at a spot where the wind direction is ↗, so it moves up and to the right. In the next step, it reaches a new position where the wind direction has changed to →, so it moves to the right. The step after that: ↘, so it begins moving down and to the right. Note: the particle itself doesn't know what the whole map looks like. It doesn't even need to ask, 'Where am I going?' It only needs to keep asking, 'I'm here right now—which way is the wind blowing?' and then: follow it.
If there's only one particle, nothing particularly special happens. It simply moves, turns, moves again, and turns again.
But if we introduce 100 particles, things start to get interesting.
With 1,000 particles, the trajectories begin to intertwine.
With 10,000 particles, a complete, flowing pattern emerges.
The most amazing part: we didn't draw the lines, yet they look so natural
Upon seeing the final pattern, you might wonder, 'Did mathematicians draw these curves directly?' The answer is no; we didn't even specify, 'Draw a curved line here.' We only defined where the particles should go and let them move on their own. As particles travel, they leave trails behind. The superposition of many particle trails creates the final pattern.
Therefore, a noise flow field doesn't involve drawing flow lines directly; instead, it allows the flow itself to generate the lines.
So, why does it look so natural?
If we were to randomly determine the direction at every single position, we might end up with something like: ↗ ← ↓ → ↖ ↑ ← ↘. With no discernible pattern, the particles' movement would be chaotic, resembling the static 'snow' on an old television. However, the 'noise' used in a noise flow field isn't just a series of completely independent random numbers; it is a type of noise characterized by continuity and smooth variation. One of the most common implementations is Perlin noise. Perlin noise generates values that change smoothly between adjacent positions, making it ideal for constructing continuous flow fields.
This means that if the direction here is 'right,' the adjacent spot is also roughly 'right.' The next spot might shift slightly to 'up-right,' and further along, the direction continues to turn gradually. It avoids abrupt changes—like going from 'right' at one spot to 'down' at the very next. Instead, the direction shifts incrementally—→ → ↗ ↗ ↑ ↖ ↖ ←—allowing the particles to turn smoothly. That is why noise flow fields often create such a natural, organic feel.
What exactly is “noise”?
At this point, you might ask: “But it looks so regular—why is it called noise?”
The “noise” here shouldn't be thought of simply as completely random chaos. Perlin noise is more akin to a smooth, continuous, yet random variation. For example: 0.41, 0.43, 0.45, 0.48, 0.52, 0.55... The numbers change, but they don't make sudden, drastic jumps like 0.42 → 0.97 → 0.03 → 0.81.
Therefore, it is neither completely regular nor completely chaotic; it sits right between order and randomness. This is precisely why it is so well-suited for generating natural-looking visuals. One of the design goals of Perlin noise is to produce numerical changes that are more continuous and natural than standard random numbers.
And so, we turned the noise into “wind.”
Noise itself consists merely of numerical values, but we can convert these values into angles. For instance: a noise value at a specific location → converted into an angle → yields a direction → becomes an arrow → becomes part of a flow field. When this calculation is applied across the entire canvas, the progression becomes: noise → direction → flow field → particles → trajectories → noise flow field.
A Recap of the Process
Let's put the entire process together:
Step 1: Generate a field of smoothly varying noise.
Step 2: Convert the noise into directional vectors.
Step 3: These vectors form an invisible flow field.
Step 4: Place a large number of particles onto the canvas overlaid with the flow field.
Step 5: Each particle continuously queries the direction of the flow field at its current location.
Step 6: The particles move along the direction indicated by the flow field.
Step 7: The particles leave trails behind them.
Finally, a flowing noise-based flow field pattern emerges.
Why does it sometimes resemble water and other times smoke?
This is because flow fields can vary in scale and behavior. If the flow field changes very slowly, the direction remains relatively consistent over large areas, causing particles to form broad, smooth flows. If the flow field changes with finer detail, particle movement becomes more complex, potentially creating intricate curls and interweaving patterns. Thus, simply adjusting a few parameters allows the same system to produce vastly different visual effects.
One crucial parameter is the noise scale—or the fineness of the turbulence—which can be roughly understood as 'how much the wind varies.' If the scale is large, the wind direction changes slowly, and particles may form 🌊 expansive streamlines. If the scale is small, the wind direction shifts more finely, and particles may create 🌀 more complex, dense textures.
There is also an interesting question: can the wind itself move? Certainly. If we extend the noise beyond just horizontal and vertical dimensions to include a time-varying dimension, the wind direction at a specific point can differ slightly from one moment to the next. Consequently, the entire flow field begins to shift slowly; as particles continue to move, the 'wind' influencing them changes as well. The result is no longer a static image, but a world that constantly breathes, flows, and evolves.
What you see is 'wind,' yet there is no wind
If we displayed only the particles themselves, you would see nothing but a multitude of tiny dots. However, the picture changes completely when each particle leaves a trail behind it. A particle (●) moves (——), continuing its path. Eventually, a complete streamline emerges. The intertwining streamlines of countless particles create the noisy flow field we see. Thus, the ultimate aesthetic appeal stems from the combination of particle motion and the accumulation of trails.
There is no actual wind, water, smoke, or even physical particles in the image; everything is the result of mathematical calculations.
Yet, when a vast number of particles move according to a smoothly varying directional field, our eyes naturally interpret the pattern as flow. In this way, mathematics transforms into motion, motion into trails, trails into visuals, and visuals evoke associations with the natural world.
How do noise flow fields differ from attractors?
If you have already explored mathematical attractors, a noise flow field offers a very different experience.
For instance, the De Jong attractor focuses on a single point being iteratively calculated based on a recurrence formula.
The Ikeda attractor centers on the feedback between rotation, scaling, and position.
The Lorenz attractor focuses on three-dimensional dynamical systems and chaos.
In contrast, a noise flow field does not focus on where a point is ultimately attracted; instead, it emphasizes how a multitude of particles move along an 'invisible flow field,' making it more akin to a mathematical wind map.
Noise flow fields do not necessarily require 'attraction.' While attractors typically emphasize the formation of a stable structure after long-term system evolution, a noise flow field may have no such 'destination.' Particles might simply move forward and reappear upon hitting a boundary, or new particles might be continuously generated. The entire flow field itself can even change over time. Therefore, it is better understood as a flow-field-driven generative art system rather than an 'attractor' in the strict sense.
An Invisible Map
If we were to distill the concept of a noise flow field into the simplest possible story, imagine it like this: there is an invisible map on a canvas. Every point on this map features an arrow. These arrows aren't entirely random; they curve gradually and connect with one another to form a continuous flow. Then, we introduce thousands of particles. The particles don't know what the map looks like; they only know two things: 'Where am I right now?' and 'Which way is the wind blowing here?' So, they begin to move, leaving trails in their wake. Ultimately, the paths traced by these particles reveal that invisible map.
What you see is not merely a static image, but the traces left by movement. Without particle movement, there would be no trails. Without a flow field, the particles wouldn't know where to go. Without noise, there would be no such smooth, natural shifts in direction. When these elements combine—noise → direction → flow field → particles → trails → noise flow field—an invisible 'mathematical wind field' finally becomes visible to us.
Continue Exploring the World of Attractors
Like attractors, noise flow fields demonstrate a fascinating concept: simple rules can create complex visual worlds.
If you enjoy creating abstract textures through parameters, you might want to explore the De Jong attractor and the Clifford attractor.
If you are interested in rotation, feedback, and chaos, try exploring the Ikeda attractor.
If you prefer strange motion in 3D space, you can experiment with the Aizawa attractor and the Lorenz attractor.
Noise flow fields offer a different kind of experience: instead of waiting for a single point to reach its destination, you observe thousands of points flowing together along an invisible map.
You cannot see the wind, the flow field, or even the mathematical rules governing them; you can only see the paths they trace. Ultimately, these paths combine to form a flowing mathematical painting.
You can also explore more amazing mathematical patterns or read more about noise on Wikipedia.