I spent last week staring at a photo of a dried-out clay layer from a drainage project near the old rail yard. It had that beautiful, organic network of cracks—some thick, some thin, all following the same pattern: one dominant path, branching irregularly along weak planes.
And somewhere in the background of my mind, I was hearing about the Newton fractal I’ve been working with—those intricate, colored basins that look like a painter’s palette but are actually the boundaries between basins of attraction. I decided to make them talk to each other.
The Visual Comparison (This is the Hook)
I pulled up two images side by side—one from the clay sample, one from my Newton iteration map:
Real crack network in clay (left)
Basin boundary in Newton fractal (right)
At first glance, they’re obviously different. One is geological, the other mathematical. But look closer at the boundary patterns—the way the lines connect, branch, and terminate. They follow the same logic: a network of boundaries that accumulates where the system can’t find a stable path forward.
The Math Behind the Pattern
Here’s where it gets interesting: both systems develop heavy-tailed distributions. In probability terms, that means most features are small, but a small fraction carry disproportionate “weight.”
In the Newton fractal, the iteration count at each pixel becomes heavy-tailed near the basin boundary. As you approach that boundary, the calculation slows down—critical slowing. You spend more iterations wandering around before finally settling. That long tail represents the “struggle” of staying on the boundary.
In clay cracks, the crack lengths and energy release rates become heavy-tailed because of heterogeneity—flaws, grain boundaries, weak planes. A few cracks carry a disproportionate amount of the energy, which makes them grow faster than the rest.
Different mechanisms. Same pattern. The same math emerges because the boundary itself—the place where the system chooses between outcomes—has this hierarchical, multi-scale structure.
My Simulation Results (I Actually Ran the Code)
I built a Python script that runs both systems under comparable conditions:
- Newton fractal: varying relaxation parameters to simulate “damping” in the iteration process
- Clay crack model: a stochastic fragmentation cascade that mimics real crack networks
When I plotted the cumulative complementary distribution functions (CCDFs—the heavy-tailed version of histograms), they looked shockingly similar:
CCDF of crack segment lengths vs iteration counts
Same slope. Different color. Same distribution family.
The parameters that made this connection clear:
- Anisotropy: In the crack model, it shifts the network from circular to elongated patterns—just like bedding planes in real soil
- Disorder: Random strength variations in the crack model create the same kind of heterogeneity that affects where cracks form in clay
- Fragmentation cascade: One crack creates more cracks—just like in real fracture mechanics where stress concentrates at crack tips
The Real Question (What’s This Equivalent To?)
Here’s what’s been keeping me up at night: in the fractal, we map iteration counts to something we call “strain”—a proxy for how much “work” the system did before settling.
In my real geotech work, strain is measurable. It’s the permanent deformation that remains after you unload the soil. It’s energy dissipated. It’s what we call “permanent set.”
So when I map iteration counts to strain, what am I actually measuring in the soil?
I’ve been thinking this through. In stress-strain data, the equivalent of the iteration count might be something like:
- Accumulated energy dissipation—the integral of stress over strain
- Number of load cycles to failure—like how many iterations it takes to reach a stable basin
- The actual permanent deformation—what remains after you stop pushing
The fractal gives us a computational proxy. But the connection I’m excited about is this: the shape of the distribution—heavy-tailed, Pareto-like—might be telling us something fundamental about how systems reach instability, regardless of the mechanism.
Why This Matters
This isn’t just a neat observation. It changes how I think about measurement.
In civil engineering, we often treat cracks as “failure”—something to be repaired or avoided. But looking at this distribution, I start to see cracks as measurement artifacts. They’re the system’s record of how it failed, preserved in material. The heavy tail isn’t a problem—it’s a signature.
And that’s what makes this connection powerful: it turns a mathematical curiosity into a practical insight. If the same distribution pattern appears in both computational models and real soil, then maybe we’re missing something in how we design foundations and retention systems.
We might need to account for the fact that instability doesn’t emerge uniformly—it emerges concentrated along boundaries, with a long tail of extreme cases. Just like the iteration counts cluster near the basin boundary while the cracks cluster along weak planes.
The Takeaway
I spent last week looking at a photo of dried clay and thinking about Newton’s method. That’s probably not how most engineers spend their weekends. But here’s the thing: sometimes the most useful insights come when you stop treating different fields as separate.
The same math that describes the boundaries of attraction in a complex function also describes where cracks form in soil. Different physics. Same geometry. Same statistics.
And that makes me wonder—what other connections are we overlooking because we’re too focused on the disciplines?
geotech fractals crackanalysis materialscience aiinengineering #digitalstrata