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.phyllotaxis_pattern(n_points=400, angle_deg=None, scale=1.0, power=0.5) (to call the implementation directly, import mathops; mathops.phyllotaxis_pattern(n_points=400, angle_deg=None, scale=1.0, power=0.5); from the registry, opsmath.get("phyllotaxis_pattern"))
Vogel's spiral — the angle that packs best, and the spirals it makes.
Point `k sits at r = scale * k**power, theta = k * angle_deg`. With
the golden angle `180*(3 - sqrt 5) = 137.50776...` degrees (the default)
this is the arrangement of sunflower florets, pine-cone scales and the leaves
of most plants.
★**Why this earns its place — two independent theorems, each with a control
group**:
• *The golden angle packs best.* Sweep the divergence angle and the minimum
nearest-neighbour distance is maximised at 137.50776 deg; a fraction
of a degree either side is measurably worse. Nothing in the formula says
this — it has to be measured.
• *The visible spirals are consecutive Fibonacci numbers.* Take each point's
nearest neighbours and look at the difference of their indices: the
differences concentrate on 1, 2, 3, 5, 8, 13, 21, 34... At a non-golden
angle they do not.
Returns `pairs (n, 2) of (x, y)`.
Raises `ValueError: n_points < 1` or over the cap; non-finite angle
or scale; `power outside (0, 1]`.
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 · neighbour_index_gaps · ifs_fractal · 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.