construct op• 資料種類:無 → pairs(僅由參數決定的運算子 —— 不接受影像或資料輸入)
• 呼叫: import fullseye as fs; fs.ledger.ifs_fractal(preset='sierpinski', maps=None, n_points=60000, seed=0, burn_in=32)(要直接呼叫實作,import mathops; mathops.ifs_fractal(preset='sierpinski', maps=None, n_points=60000, seed=0, burn_in=32);從台帳取用則 opsmath.get("ifs_fractal"))
迭代函數系統的混沌遊戲 —— 維數有閉形式。
> 以下的詳細說明為原文 —— 摘要與標題已翻譯。
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.
mathops 的每個運算子都先檢驗輸入再計算(不讓任何東西無聲通過):
• **complex 輸入一律 ValueError** —— 強制轉成 float64 會無聲丟掉虛部(numpy 只發一個 ComplexWarning,然後回傳一個「看似合理卻是錯的」實數)。請明確寫出 .real/.imag/abs(),或改用支援複數的 complexops。
• **含被遮罩元素的 masked array 一律 ValueError** —— 拒絕「剝掉遮罩直接使用下面原值」的隱式轉換。請明確選擇填補還是丟棄。
• **所有輸入中的 NaN/Inf 一律 ValueError**(明確給出個數後拒絕 —— 它會汙染整個結果)。
• 形狀嚴格:不對 1-D 與 2-D 做隱式提升或廣播(向量槽位收到矩陣、矩陣槽位收到向量都是 ValueError;請明確 reshape)。
• 尺寸上限:接受矩陣的運算子與 stat_histogram 的 bins,超過 mathops.MAX_ELEMENTS(2^26 ≈ 6700 萬個元素)即 ValueError。
• 範例資料目錄(下載 URL / 授權) —— 2-D 用 skimage.data(BSD/公有領域)加合成圖,3-D 給出真實資料源(Stanford/PDS 等)的下載 URL。
• 運算子來歷與參考文獻 —— 該運算子族所依據的研究/方法出處。
• 演算法的正典(作者・年份)與用途見上面的族使用指南。
• poc_theorems_as_pictures — py -3.11 examples/poc_theorems_as_pictures.py
pairs 作為輸入)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 運算子登記表。本條目由 tools/opdocs.py md 自動產生(請勿手動編輯)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.