Fractals — self-similarity across scale, formalized against well-foundedness

A short note accompanying fractal_recursion.py. It formalizes what a fractal is, computes it exactly, and locates it precisely against this toolkit's recurring discipline: recursions must bottom out.

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The claim, made precise

A fractal is a set that is self-similar across scale: it is built from N reduced copies of itself, each scaled by a ratio r, so that magnifying it by 1/r reveals N copies of the whole. Its similarity dimension is

  D = log N / log(1/r)

which equals the Hausdorff dimension for the strictly self-similar (non-overlapping, open-set) case. Mandelbrot's definition is the classifier used here: a set is a fractal iff its dimension strictly exceeds its topological dimension. The catalogue computes exactly:

set	N	r	D	topological	verdict
segment (line)	2	1/2	1.0000	1	GROUNDED
filled square	4	1/2	2.0000	2	GROUNDED
Cantor dust	2	1/3	0.6309	0	FRACTAL
Koch curve	4	1/3	1.2619	1	FRACTAL
Sierpiński triangle	3	1/2	1.5850	1	FRACTAL
Sierpiński carpet	8	1/3	1.8928	1	FRACTAL
Menger sponge	20	1/3	2.7268	2	FRACTAL

A deterministic box-count over a generated Cantor point set recovers D = log2/log3 = 0.630930 to six figures — the closed form and the numerical estimate agree.

Where it sits in the toolkit: grounded as a set, ungrounded as a descent

This is the honest reason a fractal belongs here rather than being just a pretty picture. Everywhere else, the toolkit insists a recursion must bottom out: math is the well-founded root (dependency_graph), the human is the terminating authority, and fixed_point_governor refuses an "infrastructure of infrastructure of …" tower that never converges. A fractal is exactly the object whose point-wise descent never bottoms out — zoom into one location forever and the same structure keeps appearing, with no base case. That un-terminating detail is what the non-integer dimension measures.

And yet a fractal is not the ungrounded-regress case in disguise. There is a real, subtle split, and the tool draws it:

• As a point descent (keep zooming) it is ungrounded — no base case, infinite detail. That is the recursion fixed_point_governor would reject.
• As a set it is perfectly grounded. By Hutchinson's theorem, an iterated function system of contraction maps has a unique compact attractor A = ⋃ fᵢ(A) — the fixed point of the Hutchinson operator in the space of compact sets under the Hausdorff metric (Banach's theorem on a complete metric space). The fractal is that fixed point.

So a fractal is grounded as a fixed set and ungrounded as a descent — the same duality fixed_point_governor tests, seen through a magnifying glass. That is the precise, non-mystical place the fractal occupies in this family.

Honest scope

It models the idealized, exactly self-similar case. Real-world "fractals" — coastlines, lungs, turbulence, price series — are only statistically or approximately self-similar over a finite band of scales; the similarity-dimension formula does not apply outside that band, and pretending it does is the standard abuse of the word. The tool computes structure, not metaphysics: it does not claim the world is fractal, only classifies a declared scaling relation. Stdlib-only, deterministic, self-testing.