construct op• Data kinds: none → table (an op determined by its arguments alone — it takes no image or data input)
• Call: import fullseye as fs; fs.ledger.circle_packing_apollonian(curvatures=(-1.0, 2.0, 2.0, 3.0), depth=4, min_curvature=0.0) (to call the implementation directly, import mathops; mathops.circle_packing_apollonian(curvatures=(-1.0, 2.0, 2.0, 3.0), depth=4, min_curvature=0.0); from the registry, opsmath.get("circle_packing_apollonian"))
Apollonian gasket from a Descartes quadruple — every circle a theorem.
Four mutually tangent circles satisfy the Descartes circle theorem
(k1 + k2 + k3 + k4)**2 == 2 * (k1**2 + k2**2 + k3**2 + k4**2)
where k = 1/r is the curvature (negative for the enclosing circle). The theorem is quadratic in k4, so a triple of mutually tangent circles has two solutions and the second is k4' = 2*(k1+k2+k3) - k4; recursing on that reflection fills the gasket. The centres follow the complex form k4*z4 = k1*z1 + k2*z2 + k3*z3 +- 2*sqrt(k1*k2*z1*z2 + ...), so no geometry is fitted — every circle is produced by an exact algebraic step.
★Why this earns its place: the drawing carries its own proof. Each circle can be checked against Descartes to machine precision, tangency is |z_i - z_j| == |r_i +- r_j| exactly, and an integral quadruple stays integral for ever — start from (-1, 2, 2, 3) and every curvature in the infinite packing is an integer (Lagarias-Mallows-Wilks). A drawing routine that is slightly wrong cannot keep integers integral.
curvatures : 4 floats
A Descartes quadruple. The default (-1, 2, 2, 3) is the smallest
integral gasket. Must satisfy the theorem to 1e-9 relative.
depth : int >= 0
Reflection levels. Level 0 is the four seed circles; each further level
adds 4 * 3**(level-1), so the total is 2 * 3**depth + 2.
min_curvature : float
Drop circles smaller than 1/min_curvature (0 = keep all).
Returns a table: x, y, radius, curvature, depth (the enclosing circle has negative curvature and positive radius).
Raises ValueError: not four curvatures; the quadruple does not satisfy Descartes; every curvature negative or zero (no packing); depth negative or so large the packing exceeds the cap; non-finite input.
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
table as input)dynsys_poincare_section · dwt_inverse · chebyshev_eval_nd
construct)ford_circles · phyllotaxis_pattern · 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.