# Are there "continuously generated" space-filling curives (or other fractals) that allow "differentiable approximation"?

I think all fractals I am aware of are based directly or indirectly on the iterated application of some function or substitution rule.

Typically, e.g. space-filling curves are presented as substitution of a pattern into itself (or rewriting rules) with discrete approximation iterations, and all visualization tools that seem to allow "smooth" zoom-in and refinement are apparently based on implementation and presentation tricks.

But fractal-like shapes in nature do not emerge in discrete iterations, everything evolves in a smooth and organic way. So I wondered, aren't there mathematically well-defined fractal-like processes that have this property?

That is, functions that depend on some real-valued parameter $$t: 0 \to \infty$$ which continuously either

• refines the fractal curve with increasing $$t$$, adding increasingly more detail (e.g. homeomorphically stretch the curve into an increasingly "rough" shape)
• act in some holistic way on "all scales" at the same time (e.g. modeled as some "force field" acting on the curve)

and yielding the full, non-differentiable fractal curve in the unreachable limit.

In any case I'm interested in suggestions where this is a natural representation, and not some artificial "piecewise" construction to get this effect.

If this is also something I could actually implement (in code), visualize and play with, that would be great!

Or, I'm also happy to learn a bit about why this is impossible, if this happens to be the case :)

• This "But fractal-like shapes in nature do not emerge in discrete iterations, everything evolves in a smooth and organic way" seems to be a huge assumption. How do you know this?
– Jeff
Commented May 24, 2023 at 17:22
• How about the Lorenz attractor? More generally, many phenomena that we see in the context of discrete dynamics have parallels in the context of continuous dynamics - i.e., differential equations with time as the independent variable. Commented May 24, 2023 at 19:35
• @Jeff good point, it is a huge assumption. But aren't most processes taking place in nature modelled using differential equations, at least when trying to do precise calculations? And most importantly, the underlying physics (aside from "wave function collapse events" and the rest of the quantum box that I do not want to open here) is stated in continuous terms. So I think this is a reasonable intution to have. Commented May 24, 2023 at 19:55
• @MarkMcClure ah I actually did not think of the Lorentz attractor as a fractal, I just know it as a classic example for chaotic dynamics. But just recently I stumbled on the fact that the Mandelbrot set and the logistic function are closely related. I guess I need to have another look at these things from a different perspective. Thanks for the hint! Commented May 24, 2023 at 19:59
• @apirogov Well, there is a limiting object that's a compact set and has slices that are Cantor sets. Its box-counting dimension has been estimated to be just over 2. It's pretty fractal, I'd say. :) Commented May 24, 2023 at 20:13