Oloid Geometry
The oloid is a geometric solid formed as the convex hull of two linked circles. This research generalizes the classical construction, deriving exact closed-form results for an entire one-parameter family of oloids.
Every theorem is backed by Python code verifying results to machine precision (12+ decimal places, some to 80 digits), and key results carry formal proofs in Lean 4.
The Generalized Oloid
Geometry of the Convex Hull of Two Linked Circles at Variable Dihedral Angle
Takes the classical oloid (where the two circle planes are perpendicular) and generalizes it to an arbitrary dihedral angle, deriving exact closed-form results for the entire one-parameter family. Key findings include angle-independent structural invariants, a conservation law for ruling lengths, a clean volume scaling law, and closed-form surface area via complete elliptic integrals.
Contains 14 original theorems and propositions, extending foundational work by Dirnbock and Stachel (1997).
Research Topics
- Computational and Differential Geometry — convex hulls, ruled surfaces, developable surfaces
- Algebraic Geometry — elliptic curves, genus classification, moduli spaces
- Elliptic Integrals — complete elliptic integrals, Legendre parameters
- Conformal Geometry — cross-Gram matrices, conformal moduli, Mobius invariants
- High-Precision Numerical Verification — 80-digit computations, convergence analysis
Verification Approach
Results are verified through two complementary methods:
- Python numerical verification — Every theorem is accompanied by runnable code using NumPy, SciPy, SymPy, and mpmath, verifying results to machine precision and beyond (up to 80 digits).
- Lean 4 formal proofs — Key results carry machine-checked proofs in Lean 4, covering the ruling formula, genus classification, and conservation laws.
The paper is bilingual (English and Polish) and transparently acknowledges AI-assisted derivation where applicable.
License
Released under Creative Commons Attribution 4.0 International (CC BY 4.0).