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 :)
Thanks in advance for any comments and suggestions!