construct op• 数据种类:无 → pairs(仅由参数决定的算子 —— 不接受图像或数据输入)
• 调用: import fullseye as fs; fs.ledger.ifs_fractal(preset='sierpinski', maps=None, n_points=60000, seed=0, burn_in=32)(要直接调用实现,import mathops; mathops.ifs_fractal(preset='sierpinski', maps=None, n_points=60000, seed=0, burn_in=32);从台账取用则 opsmath.get("ifs_fractal"))
迭代函数系统的混沌游戏 —— 维数有闭形式。
> 以下的详细说明为原文 —— 摘要与标题已翻译。
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 的每个算子都先校验输入再计算(不让任何东西无声通过):
• **complex 输入一律 ValueError** —— 强制转成 float64 会无声丢掉虚部(numpy 只发一个 ComplexWarning,然后返回一个「看着合理却是错的」实数)。请显式写出 .real/.imag/abs(),或改用支持复数的 complexops。
• **含被掩元素的 masked array 一律 ValueError** —— 拒绝「剥掉掩码直接使用下面原值」的隐式转换。请显式选择填充还是丢弃。
• **所有输入中的 NaN/Inf 一律 ValueError**(明确给出个数后拒绝 —— 它会污染整个结果)。
• 形状严格:不对 1-D 与 2-D 做隐式提升或广播(向量槽位收到矩阵、矩阵槽位收到向量都是 ValueError;请显式 reshape)。
• 尺寸上限:接受矩阵的算子与 stat_histogram 的 bins,超过 mathops.MAX_ELEMENTS(2^26 ≈ 6700 万个元素)即 ValueError。
• 示例数据目录(下载 URL / 许可证) —— 2-D 用 skimage.data(BSD/公有领域)加合成图,3-D 给出真实数据源(Stanford/PDS 等)的下载 URL。
• 算子来历与参考文献 —— 该算子族所依据的研究/方法出处。
• 算法的正典(作者・年份)与用途见上面的族使用指南。
• poc_theorems_as_pictures — py -3.11 examples/poc_theorems_as_pictures.py
pairs 作为输入)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 算子登记表。本条目由 tools/opdocs.py md 自动生成(请勿手工编辑)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.