construct opなし → pairs(引数だけで決まる op —— 画像やデータの入力を取らない)import fullseye as fs; fs.ledger.ifs_fractal(preset='sierpinski', maps=None, n_points=60000, seed=0, burn_in=32) (実装を直接呼ぶなら import mathops; mathops.ifs_fractal(preset='sierpinski', maps=None, n_points=60000, seed=0, burn_in=32)、台帳から引くなら 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:
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.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.
mathops の全 op は入力を検証してから計算する(黙って通さない):
ValueError — float64 への強制変換は虚部を黙って捨てる(numpy は ComplexWarning だけ出して「もっともらしく間違った」実数を返す)。.real/.imag/abs() を明示するか、複素対応の complexops を使う。ValueError — マスクを剥がして下の生値を使う暗黙変換を拒否。埋める/落とすを明示する。ValueError(件数を明示して拒否 — 結果全体に伝播するため)。ValueError。reshape を明示する)。stat_histogram の bins は mathops.MAX_ELEMENTS(2^26 ≈ 6700 万要素)超で ValueError。py -3.11 examples/poc_theorems_as_pictures.pypairs を入力に取れる)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. この per-op ノートは tools/opdocs.py md が自動生成(手編集しない)。
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.