geodesic_dome — 3D `polyhedron` op

• Data kinds: none → mesh (an op determined by its arguments alone — it takes no image or data input)

• Call: import fullseye as fs; fs.ledger.geodesic_dome(frequency=3, radius=1.0, hemisphere=False) (to call the implementation directly, import render3d; render3d.geodesic_dome(frequency=3, radius=1.0, hemisphere=False); from the registry, ops3d.get("geodesic_dome"))

Usage

Geodesic sphere from a subdivided icosahedron — Euler decides the shape.

Each icosahedron face is cut into `frequency**2` triangles and every vertex

is pushed out to the sphere. The counts are then forced, not chosen:

`F = 20*f**2, E = 30*f**2, V = 10*f**2 + 2`, and Euler's formula

`V - E + F == 2` holds exactly.

★Why this earns its place: Euler also forces the *irregularity*. If every

vertex had degree 6 then `2E = 6V, which contradicts V - E + F = 2`;

counting degrees gives `sum(6 - deg(v)) == 12` over all vertices, so a

triangulated sphere built from hexagons must contain exactly twelve

pentagons — no more, no fewer, at any frequency. That is why a football has

twelve black patches, and it is an integer this operator must reproduce

exactly. Nothing about the drawing can be "close enough".

Returns a `mesh (V, F): V is (n, 3) float, F is (m, 3)`

int. With `hemisphere=True the vertices below z = 0` and the faces

touching them are dropped (then Euler no longer gives 2, and the

twelve-pentagon count applies to the full sphere only — said here rather

than letting the gate quietly change meaning).

Raises `ValueError: frequency < 1; radius <= 0`; the mesh would

exceed the cap.

HALCON: no operator (`gen_sphere_object_model_3d` makes a UV sphere, whose

poles are degenerate — a different object).

References (sample data, literature)

• 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.

Runnable examples (verified samples that actually call this op)

• geodesic_dome — py -3.11 examples_3d/geodesic_dome.py

Ops the type connects to (they accept `mesh` as input)

mesh_to_voxel · mesh_to_points · to_points · fuse_to_voxel · ambient_occlusion · cast_shadow · supersample_mesh · render_beauty

Same category (`polyhedron`)

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*Provenance: render3d.py — 3D operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*

geodesic_dome — 3D polyhedron op