Fullerene's hexagon count = Catalan-like solid C(n)'s total face count.
f-vector
Polyhedron's (vertices, edges, faces).
Face-size vector
Polyhedron's face-size histogram, as {size:count}.
Symmetry
Point-group symmetry, in Schoenflies notation. The same across Primal/Dual and Canonical/Optimized, but can genuinely differ between Fullerene and C(n) — pole contraction doesn't always preserve the symmetry group.
ρ (rho)
Polyhedron's largest ÷ smallest face area.
ι (iota)
Polyhedron's smallest ÷ largest face-to-center distance.
ε (epsilon)
Polyhedron's longest ÷ shortest edge length.
Primal / Dual
Dual shows the polar reciprocal of whatever Primal is currently showing (Fullerene or Catalan-like C(n)).
Canonical / Optimized
Canonical is the mathematically solved midsphere-tangent form. Optimized relaxes the currently displayed solid toward a different ideal per view: equalized edge lengths for the Fullerene and its dual (the dual allowed to go slightly concave), equalized face areas for C(n), and equalized edges plus regularized angles for C(n)'s dual.
Admissible?
Whether the fullerene has zero isolated hexagons and zero adjacent-pole-face edges.