construct op• Data kinds: none → pairs (an op determined by its arguments alone — it takes no image or data input)
• Call: import fullseye as fs; fs.ledger.ifs_fractal(preset='sierpinski', maps=None, n_points=60000, seed=0, burn_in=32) (to call the implementation directly, import mathops; mathops.ifs_fractal(preset='sierpinski', maps=None, n_points=60000, seed=0, burn_in=32); from the registry, opsmath.get("ifs_fractal"))
Chaos game on an iterated function system — the dimension is a closed form.
Picks a map at random (by weights, or by area if none are given), applies it, and plots the orbit. After a short burn-in the orbit lands on the attractor and stays there, so the picture is the attractor and not a path to it.
★Why this earns its place — two numbers that must agree and were computed two different ways:
• *Moran's equation.* For similarities with ratios r_i satisfying the open set condition, the similarity dimension d is the unique root of sum(r_i**d) == 1 — a closed form read off the maps, before anything is drawn. Sierpinski gives log 3 / log 2 = 1.5850, the Koch curve log 4 / log 3 = 1.2619, Cantor dust log 4 / log 3 as well.
• *Box counting.* This repository's existing fractal_dimension operator measures the dimension from the drawing. The two must agree, and they are not the same computation: one is algebra on the maps, the other is a regression on a rasterised image.
Hutchinson's theorem gives a third, structural check: the attractor is
invariant, so applying every map to the point set maps it back into
itself.
maps overrides preset: a sequence of (a, b, c, d, e, f) meaning x -> [[a, b], [c, d]] x + [e, f].
Returns pairs (n, 2).
Raises ValueError: unknown preset; a map that is not 6 numbers; a map that is not a contraction (spectral norm >= 1 — the orbit would escape); n_points below 1 or over the cap; negative burn_in.
HALCON: no operator.
Every mathops op validates its input before computing (nothing slips through silently):
• complex input raises ValueError — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• masked arrays with masked elements raise ValueError — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• NaN/Inf raises ValueError on every input (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• poc_theorems_as_pictures — py -3.11 examples/poc_theorems_as_pictures.py
pairs as input)neighbour_index_gaps · curve_locality
construct)circle_packing_apollonian · ford_circles · phyllotaxis_pattern · neighbour_index_gaps · ifs_similarity_dimension · space_filling_curve · curve_locality
*Provenance: mathops.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.